Mathematical quantization [1 ed.] 1584880015, 9781584880011, 9781420036237

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Mathematical Quantization

© 2001 by Chapman & Hall/CRC

Studies in Advanced Mathematics Series Editor STEVEN G. KRANTZ Washington University in St. Louis

Editorial Board R. Michael Beals Rutgers Uni versity

Dennis de Turck University of Pennsylv ania

Ronald DeVore University of South Carolina

Lawrence C. Evans University of California at Berk eley

Gerald B. Folland University of Washington

William Helton University of California at San Die go

Norberto Salinas University of Kansas

Michael E. Taylor University of North Carolina

Titles Included in the Series Steven R. Bell, The Cauchy Transform, Potential Theory, and Conformal Mapping John J. Benedetto, Harmonic Analysis and Applications John J. Benedetto and Michael W. Frazier, Wavelets: Mathematics and Applications Albert Boggess, CR Manifolds and the Tangential Cauch y–Riemann Comple x Goong Chen and Jianxin Zhou, Vibration and Damping in Distrib uted Systems Vol. 1: Analysis, Estimation, Attenuation, and Design Vol. 2: WKB and Wave Methods, Visualization, and Experimentation Carl C. Cowen and Barbara D. MacCluer, Composition Operators on Spaces of Analytic Functions John P. D’Angelo, Several Comple x Variables and the Geometry of Real Hypersurf aces Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Functions Gerald B. Folland, A Course in Abstract Harmonic Analysis José García-Cuerva, Eugenio Hernández, Fernando Soria, and José-Luis Torrea, Fourier Analysis and P artial Differential Equations Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Inde x Theorem, 2nd Edition Alfred Gray, Modern Differential Geometry of Curv es and Surfaces with Mathematica, 2nd Edition Eugenio Hernández and Guido Weiss, A First Course on Wavelets Steven G. Krantz, Partial Differential Equations and Comple x Analysis Steven G. Krantz, Real Analysis and F oundations Kenneth L. Kuttler, Modern Analysis Michael Pedersen, Functional Analysis in Applied Mathematics and Engineering Clark Robinson, Dynamical Systems: Stability , Symbolic Dynamics, and Chaos, 2nd Edition John Ryan, Clifford Algebras in Analysis and Related Topics Xavier Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators John Scherk, Algebra: A Computational Introduction Robert Strichartz, A Guide to Distrib ution Theory and Fourier Transforms André Unterberger and Harald Upmeier, Pseudodifferential Analysis on Symmetric Cones James S. Walker, Fast Fourier Transforms, 2nd Edition James S. Walker, Primer on Wavelets and Their Scientif ic Applications Gilbert G. Walter and Xiaoping Shen, Wavelets and Other Orthogonal Systems, Second Edition Kehe Zhu, An Introduction to Operator Algebras Dean G. Duffy,Green’s Functions with Applications Nik Weaver, Mathematical Quantization

© 2001 by Chapman & Hall/CRC

Mathematical Quantization

NIK WEAVER

CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C.

© 2001 by Chapman & Hall/CRC

Library of Congress Cataloging-in-Publication Data Weaver, Nik. Mathematical quantization / Nik Weaver. p. cm.— (Studies in advanced mathematics) Includes bibliographical references and index. ISBN 1-58488-001-5 (alk. paper) 1. Geometric quantization. I. Title. II. Series. QC174.17 .G46 W43 2001 530.14′3—dc21

2001028679

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2001 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-001-5 Library of Congress Card Number 2001028679 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

© 2001 by Chapman & Hall/CRC

v

Prefa e It has been roughly one hundred years sin e physi ists began to realize that lassi al me hani s is fundamentally wrong in the atomi realm and hen e annot be a orre t des ription of nature. The subsequent transition to quantum me hani s was rather rapid; despite some false starts, and the truly alien quality of the new theory, by the late 1920s its basi framework was omplete. Sin e that time there has been a parallel development in mathemati s whi h has played out mu h more gradually.1 It began around 1930. Not long after John von Neumann published the de nitive mathemati al treatment of quantum me hani s [52℄, he and Garrett Birkho pointed out that the logi al stru ture of quantum systems was dierent from that of lassi al systems [7℄. Their des ription of the former is now known as quantum logi . This was the rst example of a quantum version of a

lassi al mathemati al subje t. Over the next several years von Neumann, together with Fran is Murray, initiated the study of what are now alled von Neumann algebras [49, 50, 53, 51℄, and quantum measure theory was born. But although Murray and von Neumann knew that their algebras were a non ommutative generalization of lassi al L1 spa es, it was only several de ades later that this point of view was openly embra ed in the popular expression \non ommutative measure theory." As other types of quantum stru tures arose | quantum topologi al spa es, quantum groups, quantum Bana h spa es, et . | it be ame in reasingly lear that they were all instan es of a single basi phenomenon. Also, examples su h as quantum omputation and quantum logi showed that the entral property they all shared was not non ommutativity, whi h only appears expli itly in algebrai stru tures, but 1 See

[22℄ for a similar histori al take.

© 2001 by Chapman & Hall/CRC

vi rather their ommon relation to Hilbert spa e. We have now rea hed a point where it is possible to give a simple, uni ed approa h to the general on ept of quantization in mathemati s. That is the aim of this book. The fundamental idea of mathemati al quantization is that sets are repla ed by Hilbert spa es. Thus, we regard latti e operations (join, meet, ortho omplement) on subspa es of a Hilbert spa e as orresponding to set-theoreti operations (union, interse tion, omplement) on subsets of a set. This already allows one to determine quantum analogs of some simple stru tures. But the real breakthrough is the fa t that the quantum version of a omplex-valued fun tion on a set is an operator on a Hilbert spa e. The reason for this is not obvious, but it has dramati

onsequen es: sin e topologies and measure lasses on a set an be de ned in terms of s alar-valued fun tions, we are then able to transfer these onstru tions into the quantum realm. With more work the analogy an be pushed even further. At ea h step one must formulate the given lassi al notion in just the right way to obtain a viable quantum version. This sometimes requires signi ant

reativity. However, as it is done in ase after ase, general quantization prin iples emerge. In this book I dis uss the following orresponden es.

lassi al theory

quantum analog

omputation propositional logi entire fun tions topologi al spa es measure spa es Bana h spa es Hilbert bundles metri spa es Riemannian geometry topologi al groups

quantum omputation quantum logi Bargmann-Segal multipliers C*-algebras von Neumann algebras operator spa es Hilbert modules Lips hitz algebras non ommutative geometry quantum groups

I use the quantum plane and tori, whi h are arguably the most fundamental non ommutative examples, to illustrate several of these topi s. My identi ation of the set/Hilbert spa e analogy as a basi prin iple may surprise some C*-algebraists who view non ommutativity as the de ning property of their subje t. I hope they will appre iate the e onomy of my approa h and the strength of its unifying power. Having said that, I should point out that no part of this approa h is original. In fa t, the basi analogy between sets and Hilbert spa es was brought out

© 2001 by Chapman & Hall/CRC

quite learly in Birkho and von Neumann's original paper [7℄. However, that paper gave rise to a line of resear h whi h developed in the dire tion of general latti e-theoreti issues whi h have little relevan e to the main quantization program, and perhaps for this reason it has not re eived the attention it deserves. The next major observation, that real-valued fun tions orrespond to self-adjoint operators, was made by George Ma key [45℄, but in this ase the axiomati formalism in whi h he ast his idea seems to have obs ured its signi an e. On e the fun tion/operator orresponden e is granted, I think my interpretation of C*-algebras and their relatives is fairly standard. My goal has been to write a sort of broad introdu tory survey, in luding some deep results but keeping the whole a

ount as nonte hni al as possible. I did not dis uss several prominent related topi s (nonselfadjoint operator algebras, subfa tors, K-theory) be ause I ould not t them dire tly into my themati framework, and I omitted quantum probability be ause there was not enough spa e. Likewise, with the ex eption of Chapter 7 on quantum eld theory, I also avoided presenting mu h in the way of appli ations. But I want to emphasize that there are many other appli ations to physi s; indeed, many of the ideas in this book were originally developed in onne tion with mathemati al physi s. Some referen es on this aspe t are [2℄, [6℄, [10℄, [17℄, [24℄, [33℄, and [69℄. The most signi ant appli ations within mathemati s but outside analysis proper are probably in knot theory [38℄ and index theory for foliations [12℄. The essential prerequisite is a good rst-year graduate ourse in analysis along the lines of [28℄. Readers need to be familiar with measure theory and fun tional analysis on at least the level of the Hahn-Bana h, Stone-Weierstrass, and Riesz representation theorems. Graduate level topi s outside fun tional analysis are tou hed on here and there, but I generally try to give enough ba kground so that an unfamiliar reader

an follow the development at some level. There are also o

asional instan es where a omplete proof of some fa t would have required a prohibitively long ex ursion. In these ases the reader will nd referen es to full treatments in the notes given at the end of the hapter. This work was partially supported by NSF grant DMS-0070634.

© 2001 by Chapman & Hall/CRC

Contents Prefa e

1 Quantum Me hani s 1.1 Classi al physi s . . . 1.2 States and events . . . 1.3 Observables . . . . . . 1.4 Dynami s . . . . . . . 1.5 Composite systems . . 1.6 Quantum omputation 1.7 Notes . . . . . . . . .

v . . . . . . .

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1 1 2 6 9 13 16 17

2 Hilbert Spa es 2.1 De nitions and examples . 2.2 Subspa es . . . . . . . . . 2.3 Orthonormal bases . . . . 2.4 Duals and dire t sums . . 2.5 Tensor produ ts . . . . . . 2.6 Quantum logi . . . . . . 2.7 Notes . . . . . . . . . . .

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19 19 23 28 31 35 40 43

3 Operators 3.1 Unitaries and proje tions . . . . 3.2 Continuous fun tional al ulus . 3.3 Borel fun tional al ulus . . . . . 3.4 Spe tral measures . . . . . . . . 3.5 The bounded spe tral theorem . 3.6 Unbounded operators . . . . . . 3.7 The unbounded spe tral theorem 3.8 Stone's theorem . . . . . . . . . . 3.9 Notes . . . . . . . . . . . . . . .

. . . . . . . . .

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. . . . . . . . .

45 45 50 54 57 61 63 66 68 72

4 The Quantum Plane 4.1 Position and momentum . . . . . . . . . . . . . . . . . . 4.2 The tra ial representation . . . . . . . . . . . . . . . . .

73 73 77

. . . . . . .

ix

© 2001 by Chapman & Hall/CRC

x 4.3

Bargmann-Segal spa e . . . . . . . . . . . . . . . . . . .

79

4.4

Quantum omplex analysis

. . . . . . . . . . . . . . . .

85

4.5

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5 C*-algebras

C (X )

91

5.1

The algebras

. . . . . . . . . . . . . . . . . . . . .

91

5.2

Topologies from fun tions . . . . . . . . . . . . . . . . .

95

5.3

Abelian C*-algebras

. . . . . . . . . . . . . . . . . . . .

99

5.4

The quantum plane . . . . . . . . . . . . . . . . . . . . .

101

5.5

Quantum tori . . . . . . . . . . . . . . . . . . . . . . . .

109

5.6

The GNS onstru tion . . . . . . . . . . . . . . . . . . .

116

5.7

Notes

123

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Von Neumann Algebras 6.1 6.2

l1 (X ) The algebras L1 (X ) The algebras

125

. . . . . . . . . . . . . . . . . . . .

125

. . . . . . . . . . . . . . . . . . . .

128

6.3

Tra e lass operators . . . . . . . . . . . . . . . . . . . .

131

B (H )

6.4

The algebras

. . . . . . . . . . . . . . . . . . . . .

135

6.5

Von Neumann algebras . . . . . . . . . . . . . . . . . . .

138

6.6

The quantum plane and tori . . . . . . . . . . . . . . . .

143

6.7

Notes

146

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 Quantum Field Theory

147

7.1

Fo k spa e . . . . . . . . . . . . . . . . . . . . . . . . . .

147

7.2

CCR algebras . . . . . . . . . . . . . . . . . . . . . . . .

150

7.3

Relativisti parti les . . . . . . . . . . . . . . . . . . . .

155

7.4

Flat spa etime

. . . . . . . . . . . . . . . . . . . . . . .

159

7.5

Curved spa etime . . . . . . . . . . . . . . . . . . . . . .

161

7.6

Notes

164

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Operator Spa es

V (K )

167

8.1

The spa es

. . . . . . . . . . . . . . . . . . . . . .

167

8.2

Matrix norms and onvexity . . . . . . . . . . . . . . . .

169

8.3

Duality

176

. . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4

Matrix-valued fun tions

. . . . . . . . . . . . . . . . . .

180

8.5

Operator systems . . . . . . . . . . . . . . . . . . . . . .

184

8.6

Notes

190

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Hilbert modules

191

9.1

Continuous Hilbert bundles . . . . . . . . . . . . . . . .

191

9.2

Hilbert

194

. . . . . . . . . . . . . . . . . . . .

9.3

L1 -modules

Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . .

197

9.4

Hilbert W*-modules

. . . . . . . . . . . . . . . . . . . .

202

9.5

Crossed produ ts . . . . . . . . . . . . . . . . . . . . . .

208

© 2001 by Chapman & Hall/CRC

xi 9.6

Hilbert

-bimodules

. . . . . . . . . . . . . . . . . . . .

211

9.7

Notes

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

217

10 Lips hitz algebras

219

X

10.1 The algebras Lip0 ( ) . . . . . . . . . . . . . . . . . . . 10.2 Measurable metri s . . . . . . . . . . . . . . . . . . . . . 10.3 The derivation theorem 10.4 Examples

219 226

. . . . . . . . . . . . . . . . . .

231

. . . . . . . . . . . . . . . . . . . . . . . . . .

236

10.5 Quantum Markov semigroups . . . . . . . . . . . . . . .

242

10.6 Notes

248

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 Quantum Groups 11.1 Finite dimensional C*-algebras . . . . . . . . . . . . . .

249 249

11.2 Finite quantum groups . . . . . . . . . . . . . . . . . . .

251

11.3 Compa t quantum groups . . . . . . . . . . . . . . . . .

256

11.4 Haar measure . . . . . . . . . . . . . . . . . . . . . . . .

260

11.5 Notes

263

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Referen es

© 2001 by Chapman & Hall/CRC

265

Chapter 1

Quantum Me hani s 1.1

Classi al physi s

No ba kground in physi s is really needed to understand the on ept of mathemati al quantization, whi h may be taken as nothing more than a formal analogy between sets and Hilbert spa es. However, knowledge of some physi s adds a layer of meaning to the mathemati s whi h an be quite illuminating. It an also help when one is unsure of the best way to translate some set theoreti onstru tion into the Hilbert spa e setting. The basi ideas of quantum me hani s are really quite simple, and in the setting of nite state systems the mathemati al apparatus is elementary. It involves nite dimensional Hilbert spa es, whi h are on retely just the ve tor spa es Cn equipped with the eu lidean inner produ t

h

v; w

i=

X n



ai b i ;

i=1

where v = (a1 ; : : : ; an ) and w = (b1 ; : : : ; bn ). The key properties of the inner produ t are (1) hv; v i  0 for all v 2 Cn , and hv; v i = 0 only if n v = 0; (2) hav1 + bv2 ; w i = ahv1 ; w i + bhv2 ; w i for a; b 2 C and v; w 2 C ; n and (3) hv; wi = hw; v i for all v; w 2 C . In nite dimensional Hilbert spa es are a bit deeper, but we an ignore them for now be ause the most fundamental prin iples of quantum me hani s are illustrated perfe tly well in nite dimensions. Our goal in this hapter is just to give an informal introdu tion to the physi al

on epts that underlie the math whi h follows. We do this here at a very basi mathemati al level, and in the following two hapters we will develop the essential mathemati al tools used in the in nite dimensional

ase. Thus, all of the math in this hapter will appear again in a more general form later. When this is done the reader should easily be able to infer its physi al interpretation by analogy with the nite dimensional

ase. 1 © 2001 by Chapman & Hall/CRC

2

Chapter 1: Quantum Me hani s

The following on epts appear in lassi al physi s:  phase spa e  states  events  observables  transformations These are best explained in the ontext of a spe i example. Consider a one-dimensional parti le, that is, a parti le onstrained to move in one dimension. In this example the state of the parti le is hara terized by the pair of real numbers (q; p) where q is the position of the parti le and p is its momentum. The parti le's position alone is not enough, be ause a omplete des ription of its state has to in lude how fast and in whi h dire tion it is moving. The phase spa e of a physi al system is the set of all possible states of the system. In our example the phase spa e an be identi ed with R2, as q and p an take on any real values independently of ea h other. An event is a subset of phase spa e, and this orresponds to a single bit of information about the system. For instan e, in the above example the right half-plane f(q; p) : q > 0g des ribes the event that the parti le lies to the right of the origin. Events are always true-or-false propositions. Any state either belongs to (\satis es") a given event or it does not. An observable is a real-valued fun tion on phase spa e. This kind of stru ture has the ee t of isolating some parti ular quality of a given state. For example, in the ase of a one-dimensional parti le, the oordinate fun tions (q; p) 7! q and (q; p) 7! p tell us, for a given state of the system, the parti le's position and momentum in that state. If the parti le has mass m and is free (meaning that there are no for es present, and hen e no potential energy) then the fun tion (q; p) 7! p2 =2m des ribes its energy. Finally, transformations are permutations of phase spa e. In general, a transformation des ribes the result of some a tion whi h an be taken on the system. This a tion ould be simply letting the system evolve undisturbed for a ertain length of time, or it ould involve intera tion with some external in uen e. In any ase, it must be reversible if the

orresponding map on phase spa e is to be a bije tion. 1.2

States and events

Now we des ribe the orresponding on epts in quantum me hani s. The hara teristi property of quantum systems is the possibility of superposition: distin t states an be superimposed. This is be ause the phase spa e of a quantum system is modelled by a ve tor spa e, in fa t a Hilbert spa e. States are des ribed by unit ve tors.

© 2001 by Chapman & Hall/CRC

3 In this hapter we will deal with nite state systems, systems whose

orresponding Hilbert spa es are nite dimensional and hen e of the form Cn . The term \ nite state" refers to the fa t that every state is a linear ombination of a nite set of states (i.e., a basis). Note that we take the s alar eld to be omplex; ex ept in a handful of ases, we will

ontinue to do so throughout the book, usually without omment. For example, the polarization of a photon is represented by a twodimensional omplex Hilbert spa e C2 . The ve tor (1; 0) represents a horizontally polarized photon and the ve tor (0; 1) represents a verti ally polarized photon. Events in nite state systems are modelled by subspa es of Cn . In the photon example, the subspa es = f(a; a) : a 2 Cg = f(a; 0) : a 2 Cg = f(0; a) : a 2 Cg

E1 E2 E3

indi ate the events that the photon is diagonally, horizontally, or verti ally polarized, respe tively. Before we explain how events are physi ally interpreted in relation to states, we need to introdu e the operations that an be performed on them and establish the basi properties of these operations. DEFINITION 1.2.1 the

ortho omplement of E E

the

Let E , E1 , and E2 be subspa es of

join of

? = fv 2 Cn : hv; wi = 0 for all w 2 E g;

_

meet

E2

= E1 + E2 = fv + w : v 2 E1

^

We say that E1 and E2 are

2

Let E be a subspa e of

2

E

? and v2 2 E .

Cn

PROOF k

v

2 2 g; E

E2

= E1 \ E2 :

orthogonal if hv; wi = 0 for all v 2 E1

and

E2 .

LEMMA 1.2.2 v1

and w

of E1 and E2 is the subspa e E1

w

. Then

is the subspa e

E1 and E2 is the subspa e

E1 and the

Cn

w

and let v

2 Cn

. Then v

= v1 + v2

for some

Consider the fun tion : ! R de ned by ( ) = k. This fun tion is ontinuous (by the triangle inequality), and it

© 2001 by Chapman & Hall/CRC

f

E

f w

4

Chapter 1:

Quantum Me hani s

satis es f (0) = kv k and f (w) > kv k whenever kwk > 2kv k. Therefore it attains a global minimum at some (perhaps not unique) ve tor w0 in the ompa t set fw 2 E : kwk  2kv kg. We laim that v0 = v w0 belongs to E ? . To see this, hoose a nonzero ve tor w 2 E . Then kv0 awk  kv0 k for all a 2 C, by minimality of 2 f at w0 . In parti ular, taking a = hv0 ; w i=kw k and performing a short

omputation, we get

2 2

h v0 ; w i jh v0 ; w ij 2 2

= v0 w  kv0 k : kv0 k

2 2

k k

k k

w

w

Thus hv0 ; wi = 0, as desired. We on lude that v0 2 E ? , and so v1 = w0 and v2 = v0 verify the on lusion of the lemma. PROPOSITION 1.2.3 Let E be a subspa e of Every v v1

2

2C

n

E and v2

Cn

.

Then E

_

E

?

=

Cn

an be expressed uniquely in the form v

2

E

?

. We have

k k2 = k 1k2 + k 2 k2 v

v

v

= f0g. = v1 + v2 with

and E

^

E

?

.

PROOF The rst assertion follows immediately from the lemma. Also, if v 2 E ^ E ? then hv; v i = 0, so v = 0, and thus E ^ E ? = f0g. To prove uniqueness, let v 2 Cn and suppose v = v1 + v2 = v10 + v20 with v1 ; v10 2 E and v2 ; v20 2 E ? . Then v1 v10 = v20 v2 . But v1 v10 2 E and v20 v2 2 E ? , and we know that E ^ E ? = f0g, so both sides must be zero. Thus v1 = v10 and v2 = v20 . This proves that the de omposition v = v1 + v2 is unique. Finally, sin e hv1 ; v2 i = 0 it follows that h i=h 1+ 2 so that k k2 = k 1 k2 + k 2 k2 . v; v

v

v

v ; v1

v

+ v2 i = hv1 ; v1 i + hv2 ; v2 i;

v

We an now explain the naive probabilisti interpretation of states and events. Let v 2 Cn be a unit ve tor and E  Cn a subspa e. If v 2 E then we regard the state as de nitely satisfying this event, but in general it satis es the event only with probability kv1 k2 , where ? 2 2 2 v = v1 + v2 with v1 2 E , v2 2 E . Sin e kv1 k + kv2 k = kv k = 1, the probability of satisfying any event and the probability of satisfying its ortho omplement add up to one. For instan e, a photon in the state (a; b) (with jaj2 + jbj2 = 1) will pass through a horizontally polarized lter with probability jaj2 , and through a verti ally polarized lter with probability jbj2 . In this naive interpretation, after one measures whether a state satis es an event, the state a tually be omes either v1 =kv1 k or v2 =kv2 k, with probability kv1 k2 or kv2 k2 . One says that the state has \ ollapsed" onto

© 2001 by Chapman & Hall/CRC

5 or E ? as a result of the measurement. The problem with this idea is that it is diÆ ult to identify whi h kinds of physi al intera tions should be lassi ed as measurements and to justify why they should have su h an ee t. We will dis uss this issue further in Se tion 1.5. The sense of this interpretation is seen in the lassroom demonstration that a horizontal lter followed immediately by a verti al lter passes no light, while a sequen e of horizontal, diagonal, then verti al lters does allow some light through. If a photon in the state (a; b) passes through a horizontally polarized lter (whi h it does with probability jaj2 ), then afterwards it is in the state (1; 0) and annot pass through a verti ally polarized lter. But it still has a 50% han e of passing through p apdiagonally polarized lter, after whi h it would be in the state (1= 2; 1= 2) and have a 50% han e of passing through a verti ally polarized lter. E

  -           Figure 1.1

      -            

Passage of light through polarized lters

Ortho omplementation, join, and meet of subspa es orrespond to

omplemention, union, and interse tion of lassi al events (i.e., subsets of a set), and they an be interpreted physi ally in mu h the same way: the states in E ? are pre isely those states whi h have zero probability of satisfying the event E , and the states in E1 ^ E2 are pre isely those whi h satisfy both E1 and E2 with full probability. A physi al interpretation of E1 _ E2 is not immediate but an be obtained by redu ing to the previous two operations via the following proposition. PROPOSITION 1.2.4

Let

E , E1 ,

and

E2

be subspa es of

?? = E ; (b) (E1 _ E2 )? = E1? ^ E2? ; and ( ) (E1 ^ E2 )? = E1? _ E2? . (a)

E

© 2001 by Chapman & Hall/CRC

Cn . Then

6

Chapter 1:

Quantum Me hani s

PROOF

(a) Let v 2 E . Then hv; wi = 0 for all w 2 E ? , so v 2 E ?? . This shows that E  E ?? . Conversely, any v 2 E ?? an be written ? v = v1 + v2 with v1 2 E and v2 2 E ; sin e hv2 ; v2 i = hv; v2 i = 0 we have v2 = 0, and hen e v = v1 2 E . So E ??  E . (b) Suppose v 2 (E1 _ E2 )? . Then v 2 E1? and v 2 E2? , so that ? ? v 2 E1 ^ E2 . This shows one ontainment. Conversely, suppose v 2 E1? ^ E2? . Then for any w1 2 E1 and w2 2 E2 we have hv; w1 + w2 i = hv; w1 i + hv; w2 i = 0; so v 2 (E1 _ E2 )? . This veri es the reverse ontainment. ( ) By (a) and (b), ?

E1

_

?

E2

= (E1? _ E2? )?? = (E1?? ^ E2?? )? = (E1 ^ E2 )? ;

as desired. It follows from this proposition that subspa es E1 and E2 . 1.3

E1

_

E2

= (E1? ^ E2? )? for any

Observables

The quantum me hani al version of an observable is a self-adjoint matrix. DEFINITION 1.3.1

omplex matrix A A

= A .

The

= [aij ℄

Hermitian transpose or adjoint of an n  n  = [ aji ℄. A is self-adjoint if

is the matrix A

A straightforward omputation shows that hAv; wi = hv; A wi for all 2 Cn . Conversely, applying this equality to the anoni al basis ve tors of Cn re overs the fa t that A = [aji ℄. In parti ular, it follows that A is self-adjoint if and only if hAv; wi = hv; Awi for all v and w. Now one might expe t that sin e lassi al observables are real-valued fun tions on lassi al phase spa e, a quantum me hani al observable should be some kind of real-valued fun tion on Cn , perhaps a linear fun tional. Understanding why it is instead self-adjoint matri es whi h play this role requires the spe tral theorem (Theorem 1.3.3). First we need the following lemma. Let I = In be the n  n matrix whose diagonal entries are all 1 and whose non-diagonal entries are all 0. We take it as known that the determinant of a matrix is a polynomial in the entries of the matrix and that det(AB ) = det(A)det(B ). v; w

© 2001 by Chapman & Hall/CRC

7 LEMMA 1.3.2

Every

n  n omplex matrix has at least one nonzero eigenve tor.

PROOF

Let A be an n  n matrix. The expression det(A I ) is a polynomial in , so it has at least one omplex root. For su h a root  the matrix A I has null determinant. Fix su h a  and set B = A I . We laim that ker(B ) is nonzero. Suppose not. Then B is invertible and we have det(B )det(B

1

) = det(BB

1

) = det(I ) = 1:

Thus det(B ) 6= 0, a ontradi tion. Therefore there is a nonzero ve tor v su h that (A I )v = Bv = 0, i.e., Av = v . THEOREM 1.3.3

Let

A be a self-adjoint n  n omplex matrix.

(a) every eigenvalue of

Then

A is real;

(b) distin t eigenspa es are orthogonal; and ( ) the eigenspa es of

A span Cn .

Conversely, if 1 ; : : : ; k are distin t real numbers and E1 ; : : : ; Ek are orthogonal subspa es whi h olle tively span n , then there is a unique self-adjoint

PROOF

(a) Let v

C

n  n omplex matrix with these eigenvalues and eigenspa es.

2 Cn be an eigenve tor with eigenvalue . Then hv; v i = hAv; v i = hv; Av i =  hv; v i;

 and hen e  must be real. so  =  (b) Let v; w and . Then

2 Cn be eigenve tors belonging to distin t eigenvalues  hv; wi = hAv; wi = hv; Awi = hv; wi:

Sin e  and  are real and distin t, this implies hv; wi = 0.

C

C

( ) Suppose the eigenspa es do not span n , and let E  n be the ortho omplement of their span. Observe that A(E )  E be ause if v 2 E and w is an eigenve tor with eigenvalue , we have hv; wi = 0, and hen e hAv; wi = hv; Awi = hv; wi = 0;

so hAv; wi = 0 also, and this shows that Av 2 E . Thus AjE is a linear operator on the omplex ve tor spa e E , and so AjE must have a nonzero

© 2001 by Chapman & Hall/CRC

8

Chapter 1: Quantum Me hani s

eigenve tor by the lemma, whi h means that A has a nonzero eigenve tor in E. This is a ontradi tion and we on lude that the eigenspa es of A must span Cn . To prove the onverse assertion, let 1 ; : : : ; k be distin t real numbers and let E1 ; : : : ; Ek be orthogonal subspa es whi h together span Cn . De ne a linear operator A by setting Av = i v for v 2 Ei and extending linearly. Then for any v; w 2 Ei we have hAv; wi = i hv; wi = hv; Awi, and for any v 2 Ei and w 2 Ej , i 6= j, we have hAv; wi = 0 = hv; Awi. Sin e the Ei span Cn , this implies that hAv; wi = hv; Awi for all v; w 2 Cn, so A is self-adjoint. We laim that the hara terization of self-adjoint matri es given in the pre eding theorem is the exa t quantum me hani al analog of a real-valued fun tion on a nite set. Why is this? Let X be a nite set and f : X ! R a real-valued fun tion. De ne an equivalen e relation on X by setting x  y if f(x) = f(y). Then the blo ks of this equivalen e relation partition X into disjoint subsets, and f labels ea h subset with a distin t real number i . This is the stru ture that a real-valued fun tion imparts to X. The quantum me hani al analog of a partition into disjoint subsets is a de omposition into orthogonal subspa es. Therefore, the analog of a real-valued fun tion should de ompose Cn into orthogonal subspa es and label ea h subspa e with a real number. By Theorem 1.3.3, this is pre isely what self-adjoint matri es do.

1

Figure 1.2

2

. . .

1 k

.

.

. k

Fun tions versus matri es

Of ourse, there are other ways of looking at real-valued fun tions, and it may be possible to nd another stru ture on Cn whi h is analogous to them in some dierent way. But the above analogy is the one whi h is physi ally orre t. In detail, here is the physi al interpretation of a self-adjoint matrix as an observable. Let A be a self-adjoint matrix and let E1 ; : : : ; Ek be its eigenspa es with orresponding eigenvalues 1 ; : : : ; k . Then the

© 2001 by Chapman & Hall/CRC

9 observable orresponding to A will take the value i on any state v 2 Ei , with ertainty. In general, for any v 2 Cn we an write v = v1 +


+ vk with vi 2 Ei , and the observable will then take the value i on vi with probability kvi k2 . P We de ne the expe tation value of A for v to be ki=1 i kvi k2 . That is, it is the average of all possible observations weighted by their likelihood. The following gives an alternative formula. PROPOSITION 1.3.4 Let for

v be a state and A an observable. v equals hAv; v i.

Then the expe tation value of

A

PROOF

Let E1 ; : : : ; Ek be the eigenspa es of A, with orresponding eigenve tors i , and write v = v1 +


+ vk with vi 2 Ei . Then



Av; v i = A(v1 +


+ vk ); (v1 +


+ vk ) = h1 v1 +


+ k vk ; v1 +


+ vk i = 1 kv1 k2 +


+ k kvk k2 ;

h

as desired. 1.4

Dynami s

The dynami s of a quantum system are des ribed using unitary matri es. DEFINITION 1.4.1

UU  = I .

An

n  n omplex matrix U

is

unitary if U  U =

In other words, U is invertible and U  = U 1 . Unitary matri es are isometries be ause

Uv k2 = hUv; Uv i = hU  Uv; v i = hv; v i = kv k2

k

for all v 2 Cn . Thus, they are the natural analog for Cn of permutations of a set, and for this reason they are used to model transformations in quantum me hani s. In parti ular, they are used to model time evolution. A omplete des ription of the dynami s of a quantum system involves a family of unitaries fUt : t 2 Rg, where the operator Ut takes a state ve tor v to the state Ut v that it will evolve into after t units of time. Obviously, we must have U0 = I , and it is also natural to require that Us+t = Us Ut for all s; t 2 R, on the grounds that allowing a system to evolve for su

essively t and then s units of time is equivalent to allowing

© 2001 by Chapman & Hall/CRC

Chapter 1: Quantum Me hani s

10

it to evolve for

s + t units of time all at on e.

There is also a ontinuity

requirement.

A ontinuous one-parameter unitary group on n  n matri es fUt : t 2 Rg su h that U0 = I , 2 R, and Us ! Ut as s ! t.

DEFINITION 1.4.2

n is a family of unitary Us Ut = Us+t for all s; t C

Here onvergen e of matri es is taken in the only reasonable sense, namely, pointwise onvergen e of ea h entry. Note that the above onditions imply

Ut = Ut 1 = U t

for all

t 2 R.

Example 1.4.3

Let A be a self-adjoint matrix and let E1 ; : : : ; Ek and 1 ; : : : ; k be its eigenspa es and eigenvalues. For t 2 R de ne Ut by setting Ut v = eitj v for all v 2 Ej (1  j  k) and extending linearly. Then fUt g is a ontinuous one-parameter unitary group. We write Ut = eitA . The notation

Ut = eitA

in the pre eding example is well-taken in light

of the following fa t.

PROPOSITION 1.4.4

Let be a self-adjoint n  n omplex matrix. Then the in nite sum P1A(itA k itA . ) =k ! onverges to the unitary operator e k=0 Let v Ak v = k v , and so

PROOF

m X

be an eigenve tor for

!

m k X itA)k (it) v= k! k! k=1 k=1 (

!1 PmitA k =k v ! eitA v

as (

. Sin e the eigenve tors of

(

)

!)

for all

v2

A

n C ,

!

A

with eigenvalue

.

Then

v ! eit v = eitA v n

span C , we an on lude that

whi h is enough.

Remarkably, Example 1.4.3 is ompletely general. This result requires the following generalization of the spe tral theorem.

LEMMA 1.4.5

Let A1 ; : : : ; Am be self-adjoint n  n omplex matri es whi h ommute in pairs. Then there is a sequen e of orthogonal subspa es E1 ; : : : ; Ek whi h span Cn , su h that ea h Ei is an interse tion of eigenspa es of the Aj .

© 2001 by Chapman & Hall/CRC

11

PROOF This is true for m = 1 by Theorem 1.3.3. To pass from m to m + 1, suppose E  Cn is an interse tion of eigenspa es of the Aj for 1  j  m. That is, there exist 1 ; : : : ; m 2 R su h that v 2 E if and only if Aj v = j v for 1  j  m. It follows that for any v 2 E and any 1  j  m we have Aj Am+1 v

= Am+1 Aj v = j Am+1 v;

so that Am+1 v 2 E as well. Applying Theorem 1.3.3 to Am+1 jE , we may therefore de ompose E into orthogonal eigenspa es of Am+1 jE . Ea h of these is then the interse tion of E with an eigenspa e of Am+1 . Sin e the subspa es at the mth step spanned Cn , it follows that the subspa es at the (m + 1)st step also span Cn . LEMMA 1.4.6 Let

fUtg

be a ontinuous one-parameter unitary group on

there is a sequen e of orthogonal subspa es su h that for every

t

ea h

Ei

E1 ; : : : ; Ek

Cn

.

Then

whi h span

is ontained in an eigenspa e of

Ut .

Cn

,

PROOF Note rst that UsUt = Us+t = UtUs for all s; t 2 R, so all of the unitaries ommute. Writing Re Ut = 12 (Ut + U t ) and Im Ut = 21i (Ut U t ), we have Ut = Re Ut + iIm Ut , and Re Ut and Im Ut are self-adjoint. The matri es Re Ut and Im Ut ommute pairwise. By Lemma 1.4.5, for any t1 ; : : : ; tm 2 R there is a sequen e of orthogonal subspa es E1 ; : : : ; Ek whi h span Cn , su h that ea h Ei is an interse tion of eigenspa es of the Re Utj and Im Utj . Sin e k must be less than or equal to n, there is a maximum value of k whi h an be a hieved in this way. For this k the subspa es E1 ; : : : ; Ek must ea h be

ontained in an eigenspa e of Re Ut and ImUt , for all t. It follows that ea h Ei is ontained in an eigenspa e of every Ut . THEOREM 1.4.7 Let

fUtg

be a ontinuous one-parameter unitary group on

there is a self-adjoint

n

n

PROOF

omplex matrix

A

su h that

Ut

Cn

.

Then

= eitA .

Let E1 ; : : : ; Ek be as in Lemma 1.4.6, and for ea h 1  j  k let j (t) be the eigenvalue of Ut to whi h Ej belongs. Observe that jj (t)j = 1 for all t (sin e Ut is unitary) and j (0) = 1. By ontinuity, there exists Æ > 0 su h that j (t) 6= 1 for t 2 [0; Æ ℄. Then for 0  t  Æ , de ne fj (t) = i ln(j (t)), taking the bran h of the logarithm whi h yields  < fj (t) <  . Sin e j (s + t) = j (s)j (t) for all s and t, it follows that fj (s + t) = fj (s) + fj (t), provided s; t  0 and s + t  Æ .

© 2001 by Chapman & Hall/CRC

12

Chapter 1: Quantum Me hani s

This weak form of linearity implies that if t = rÆ where r is a rational number between 0 and 1, then fj (t) = rfj (Æ ) = aj t where aj = fj (Æ )=Æ ; by ontinuity fj (t) = aj t for all t between 0 and Æ . Thus, for 0  t  Æ and v 2 Ej we have j (t) = eiaj t and Ut v = eiaj t v . Doing this for ea h j , it follows that for suÆ iently small Æ 0 we have Ut = eitA for all t 2 [0; Æ 0 ℄, where A is the self-adjoint operator su h that Av = aj v for v 2 Ej . By the group property of fUt g we on lude that the same formula holds for all t. The matrix iA is alled the generator of the unitary group fUt g. Sin e any initial state v = v (0) evolves in time a

ording to v (t) = Ut v = eitA v , we have d v (t) = iAv (t): dt

This is an abstra t form of S hrodinger's equation. It is a dierential version of the fa t that the dynami s of a quantum system are des ribed by a ontinuous one-parameter unitary group. There is an alternative approa h to dynami s, path integration, whi h is popular in the physi s literature. The basi idea is this. Let (ei ) be the standard basis of Cn ; we want to determine the inner produ t hUs ei ; ej i for given i and j . These \transition amplitudes" are the matrix entries of the operator Us of evolution by s units of time. For any v; w 2 Cn , we have hv; wi = k hv; ek ihek ; wi. Therefore

P

hU e ; e i = hX U 2e ; U  2e i = hU 2 e ; e ihe s

i

j

s=

X hU =

i

j

s=

s=

i

k

k

2 ei ; e k

s=

k

ihU

 e ; Us= 2 j

i i

2 ek ; ej :

s=

k

A similar omputation shows that

hU e ; e i = s

i

j

X hU

3 ei ; ek

s=

ihU

3 ek ; ek0

s=

ihU

i

3 ek0 ; ej ;

s=

k;k0

and so on. In general we may break up the total time interval [0; s℄ into m short subintervals and sum over all \paths" from ei to ej , i.e. all sequen es of basis ve tors of length m + 1 whose rst entry is ei and whose last entry is ej . The orresponding term in the sum is alled the amplitude of that path. The omputational value in doing this lies in the fa t that over short times Ut = eiAt is approximated by I + iAt. Thus one ould hope to take a limit as m ! 1 and work with the matrix A in pla e of Ut . One might also identify Cn with the omplex-valued fun tions on a set of n points in R3 , say, and then take n ! 1, allowing the points to be ome

© 2001 by Chapman & Hall/CRC

13 dense in R3 . Then the \paths" might really be thought of as physi al paths in spa e.

Figure 1.3

t

=s

t

=

t

=0

s

2

Summing over paths

The problem with this idea is that the sum whi h is used to evaluate tends to be ome a badly divergent integral in the limit. It is said to be \os illatory" be ause the produ t hUs ei ; ej i

( + iAt)ei ; ek ih(I + iAt)ek ; ek

h I

0i

( + iAt)ek(m) ; ej i




h I

is interpreted in the limit as an exponential fa tor, and the os illation of this fa tor is responsible for the divergen e of the path integral. It is still possible to develop workable versions of the path integral approa h in various rather ir ums ribed situations. However, it seems unlikely that there will ever be a truly general theory, in spite of the evident heuristi value of the idea. With this pessimisti observation we abandon path integrals. 1.5

Composite systems

Composite systems whi h are made up of intera ting subsystems an be modelled using tensor produ ts: if the Hilbert spa es Cm and Cn model two separate quantum systems then their tensor produ t Cm Cn  = Cmn models the two systems together. The basi properties of the tensor produ t are these. Corresponding to any states v 2 Cm and w 2 Cn there is an elementary tensor produ t state v w 2 Cmn whose oordinate entries are ai bj , if v = (a1 ; : : : ; am ) and w = (b1 ; : : : ; bn ). This state is regarded as modelling the situation where the rst subsystem is in the state v and the se ond is in the state n w . For any unit ve tor w 2 C , the map v 7! v w is an isometri embedding of Cm in Cmn , so the states of the omposite system for whi h the se ond system is in the state w are in one-to-one orresponden e

© 2001 by Chapman & Hall/CRC

14

Chapter 1: Quantum Me hani s

with the states of the rst system. A general element of a tensor produ t an be expressed as a linear

ombination of elementary tensor produ t states. Thus, one usually

annot say what the state of one subsystem is independently of the state of the other subsystem. Composite systems of identi al parti les require a minor variation on the pre eding onstru tion. For a system of k identi al parti les, ea h modelled on Cn , rather than the full tensor power Cn


Cn  = Cnk we use either the symmetri tensor power Cn s


s Cn or the antisymmetri tensor power Cn a


a Cn . These onsist of those ve tors in the full tensor produ t whose omponents satisfy, respe tively, ai1 :::ik = a(i1 ):::(ik ) or ai1 :::ik = ( 1)jj a(i1 ):::(ik ) for any permutation  of the index set f1; : : : ; k g, where j j is the parity of  . The appropriate

hoi e varies; parti les are alled Bosons or Fermions depending on whi h tensor produ t is used to model their multiple-parti le systems. Using the notion of a omposite system, we an now larify the naive interpretation of the quantum me hani al formalism given in Se tion 1.2. What makes a diagonally polarized photon abruptly be ome horizontally or verti ally polarized when measured, and how does it hoose whi h to be ome? The answer to these questions involves the fa t that we, the observers of the system, are ourselves quantum systems. Thus, when measurement issues su h as these arise, we must onsider the omposite Hilbert spa e Cm Cn where Cm is the Hilbert spa e of the observed system and Cn is the Hilbert spa e of the observer (assumed here, with apologies, to be nite dimensional). For example, say that C2 models the polarization of a photon, and let fe1 ; e2 g be the anoni al basis of C2. Then any state of the omposite Hilbert spa e C2 Cn an be expressed in the form a1 (e1 v1 ) + a2 (e2 v2 ) for some unit ve tors v1 ; v2 2 Cn and omplex oeÆ ients a1 ; a2 . Now suppose we are given an initial elementary tensor produ t state (a1 e1 + a2 e2 ) v , where the two subsystems are ea h in well-de ned individual states. Here the observer is oblivious to the state of the photon. As long as there is no physi al intera tion between the two subsystems, time evolution will preserve the property that the omposite system is in an elementary state. However, if the subsystems do intera t then after some passage of time the omposite system will be in a general state of the form b1 (e1 v1 ) + b2 (e2 v2 ). This overall state is now \entangled" in the sense that it is not possible to determine the state of one subsystem independently of the state of the other subsystem. The remarkable fa t at this point is that the total state is a omposite of the states e1 v1 and e2 v2 , whi h are orthogonal and will remain so at all future times due to the unitarity of time evolution. Thus the system

an be de omposed into two orthogonal states in whi h the state of the

© 2001 by Chapman & Hall/CRC

15 observer is respe tively orrelated with the two dierent possible polarizations of the photon. In ea h of the two states the observer apparently has the subje tive experien e of being in a universe in whi h the photon has the orresponding polarization. This analysis suggests the surprising possibility that there is in some sense more than one opy of the observer after the measurement has been made. In general, after any measurement of a nite state system we an de ompose the state of the omposite system into an orthogonal sum of states in ea h of whi h the observer sees the measured system in a parti ular eigenstate. Now in most elementary treatments of quantum me hani s it is assumed that one of the terms in this sum is \real" and the rest are not; this transition of the state of the system to a single term in the sum is the \ ollapse" referred to in Se tion 1.2. One then has to deal with the problem of what me hanism auses the system to

ollapse and how su h a me hanism might be triggered. Moreover, the in nite dimensional ase, in whi h there generally is no de omposition of the Hilbert spa e into a dis rete family of eigenspa es, presents the separate problem of what the end result of the ollapse should even be. Alternatively, it has been suggested that no ollapse takes pla e and an image of parallel universes has been put forward in its pla e. Thus, one is supposedly fa ed with the problem of de iding whether every one of a family of possible universes really exists, or if not how and when the pro ess of sele ting the real one o

urs. Arguments on either side of this question an be given in terms of the parsimony either of not requiring a ollapse or of not invoking the existen e of unobservable universes. However, it is possible that the whole issue is really what some philosophers might all a meaningless pseudo-problem. One ould argue that any assertion that a given universe does or does not exist, or that other terms of the sum are or are not \really there", is in prin iple unveri able, and hen e metaphysi al in the most literal sense. For example, suppose it was asserted after some parti ular measurement that only the even terms of the de omposition had given rise to existing universes, and that the odd terms had vanished. What a tual ontent would su h an assertion have? Ordinary assertions about the existen e of ordinary obje ts an be understood as indi ating the appearan e of the obje ts in the speaker's universe. In prin iple, they an always be veri ed or falsi ed. But this is exa tly what one annot do with statements about the existen e of other universes. It may be that one is simply tri ked into thinking that the latter sort of assertion is meaningful by a false analogy with the former sort. For reasons like these, one may want to question whether statements about when and how a \ ollapse" takes pla e have any genuine ontent

© 2001 by Chapman & Hall/CRC

16

Chapter 1: Quantum Me hani s

at all. 1.6

Quantum omputation

In theory, the fa t that quantum states an be superimposed should enable us to perform some omputations faster than is lassi ally possible. (A tually building a \quantum omputer" whi h ould do this presents serious engineering obsta les, but it seems likely that these will be over ome in time.) One an simulate any quantum omputation on a

lassi al omputer, and vi e versa, so there is no absolute problem whi h

an be solved on one but not the other; the issue is solely one of speed. To illustrate the way superposition an be used to speed up a omputation, we will des ribe an algorithm that sear hes n data points in p O( n) time. The problem is formulated as follows. Let Cn , n  3, model the phase spa e of some quantum system; its physi al onstitution is irrelevant. Let V be a unitary n  n matrix whi h is diagonal with respe t to the standard basis fe1 ; : : : ; en g, so V ei = i ei , and assume there exists i0 su h that i = 1 for i 6= i0 and i0 = 1. Suppose also that the transformation v 7! V v on Cn is physi ally realizable, so that in the ourse of our omputation we an apply V to the state of the system at will. We wish to nd the eigenve tor ei0 whose eigenvalue is 1. Classi ally, there is no way to nd this 1 eigenve tor with ertainty in fewer than n steps, and weakening the problem to ask only that we nd the eigenve tor with probability p shortens the omputation to pn steps, whi h is still O(n). In ontrast, the quantum me hani al algorithm that wepare about to present nds the 1 eigenve tor with probability p in O( n) steps, for any p < 1. (Finding it with ertainty still annot be done in fewer than n steps.) The omputation also requires that we be able to apply the matrix U = [uij ℄ to the system, where uij = 2n 1 for i 6= j and uii = 1+2n 1. Note that this matrix is also unitary. The algorithm is then to prepare the system in the state v = (n 1=2 ; : : : ; n 1=2 ) and apply the operators U and V alternately k times, that is, apply the operator (UV )k . The

laim is that after this pro ess the system will be in the 1 eigenstate with probability p su h that p in reases with k . To see this, let w = (1; : : : ; 1) ei0 and write the initial state of the system as v = n 1=2 w + n 1=2 ei0 . It is easy to he k that a state of the form aw + bei0 is taken into another state of this form by the operator UV . So say (UV )k v = ak w + bk ei0 . Then a0 = b0 = n 1=2 and we have the re ursion relations ak+1 = 2 k ak and bk+1 = 2 k + bk , where

k = nn 1 ak n1 bk . Sin e (n 1)a2k + b2k = 1, if bk lies between n 1=2 and 2 1=2 then ak

© 2001 by Chapman & Hall/CRC

17 lies between (2n 2) 1=2 and n 1=2 . In this ase k is bounded below by 2 1=2 ((n 1)1=2 1)n 1 . This shows that as long as n 1=2  bk  2 1=2 holds, bk will in rease at ea h step by at least 21=2 ((n 1)1=2 1)n 1  21=2 n 1=2 . So for large n, on the order of n1=2 steps are needed to ensure bk  2 1=2 . At this point a measurement of the system will pla e in it the state ei0 with probability at least 1=2. Running the same pro edure m times yields a probability of at least 1 2 m of rea hing the statepei0 in at least one trial, and for any xed value of m this still takes O( n) steps. This shows that the algorithm has the property that was laimed. One is sometimes told that the reason quantum algorithms an be faster than lassi al algorithms is be ause they are able to arry out multiple omputations simultaneously, due to superposition. However, examination of the above algorithm reveals no pre ise sense in whi h simultaneous omputations are being performed. Perhaps it is learer to say that the reason quantum algorithms an be faster than lassi al algorithms lies in the phemomenon of superposition together with the possibility of applying arbitrary unitary matri es. Using only permutation matri es is equivalent to omputing with a lassi al nite state ma hine. 1.7

Notes

The lassi introdu tions to quantum me hani s are [18℄ and [52℄. A modern approa h is given in [8℄. Our explanation of the orresponden e between real-valued fun tions and self-adjoint operators given in Se tion 1.3 is due in a slightly dierent form to Ma key [45℄. We will dis uss this further in Chapter 3. For more on path integrals see [27℄. The so- alled \many-worlds" interpretation dis ussed in Se tion 1.5 is expounded in detail in [25℄. In parti ular, a derivation of the probability interpretation of Se tion 1.2 is given there. Our suggestion that it is meaningless to ask whether parallel universes really exist is based on philosophi al grounds arti ulated in [5℄. The quantum sear h algorithm given in Se tion 1.6 is due to Grover [31℄. There is also a quantum algorithm for fa torization whi h is presumably (provided P 6= N P ) exponentially faster than any lassi al algorithm; it is due to Shor [65℄.

© 2001 by Chapman & Hall/CRC

Chapter 2

Hilbert Spa es 2.1 De nitions and examples In the last hapter we onsidered only nite dimensional Hilbert spa es. We now present the in nite dimensional theory, paying parti ular attention to the analogies between Hilbert spa es and sets. As we saw in Chapter 1, this analogy is grounded in physi al reality, and it will be the basis of all of our subsequent work. We begin with the formal de nition of Hilbert spa es. (It should be made lear, however, that readers who are not already familiar with other fun tional analysis topi s at a omparable level, whi h we do not review, will nd later hapters diÆ ult to follow.) We adopt the mathemati al onvention of making inner produ ts antilinear in the se ond variable; in the physi s literature they are usually linear in the se ond and antilinear in the rst. Let

DEFINITION 2.1.1

inner produ t (a) (b) ( )

on

H

is a map

H

be a omplex ve tor spa e. A

h
;
i

: HH !C

pseudo

satisfying

+ bv2 ; wi = ahv1 ; wi + bhv2 ; wi; = hw; vi; and hv; vi  0 hav1

hv; wi

for all

a; b 2

= 0 then Hilbert spa e

implies A

C and

v

v1 ; v2 ; v; w 2 H. h
;
i

is an

We write

inner produ t.

kvk

= hv; vi1=2 .

If

kvk

=0

is a omplex ve tor spa e equipped with an inner

produ t whose orresponding norm

k
k

is omplete.

This de nition of Hilbert spa es is ontingent on us verifying that k
k really is a norm, whi h we do immediately below, in Corollary 2.1.3. First we need the Cau hy-S hwarz inequality. 19 © 2001 by Chapman & Hall/CRC

20

Chapter 2:

PROPOSITION 2.1.2 Let

H

H

h

i ij  k kk k

be a omplex ve tor spa e, let

, and let v; w

PROOF

2H

. Then

jh

v; w

k k = k k = 0 then 0h h i h

If

v

Hilbert Spa es

;

be a pseudo inner produ t on

v

w .

w

ij2 hen e h i = 0. Otherwise, without loss of generality k k 6= 0; then  h i k k h i  = k k2 k k2 jh ij2 0 k k k k k k so again jh ij  k kk k. v

v; w w; v

i i = 2jh

v; w w

v; w

;

v; w

w

v; w

w v

w;

w

v; w

v

w v

v; w w

h

i ;

w

v

v; w

;

w

COROLLARY 2.1.3 If

w

is a pseudo inner produ t then

k
k

is an inner produ t then

k
k

is a pseudonorm, and if

h

i ;

is a norm.

PROOF Suppose h
;
i is a pseudo inner produ t. It is immediate from the de nition of k
k that kv k  0 for all v . Next, we have k k=h av

av; av

i1=2 = (j j2 h i)1=2 = j jk k a

v; v

a

v

for all a 2 C. Finally, by the Cau hy-S hwarz inequality we have Rehv; wi  jhv; wij  kv kkwk, and hen e

k

v

+ wk2 = hv + w; v + wi = kv k2 + 2Rehv; wi + kwk2  kvk2 + 2kvkkwk + kwk2 = (kv k + kwk)2 ;

yielding the triangle inequality. If h
;
i is an inner produ t then k
k is a norm.

v

6= 0 implies k k2 = h i 6= 0, so v

v; v

Our de nition of Hilbert spa es is now logi ally sound. Pseudo inner produ ts an be onverted into inner produ ts using the following te hnique. We all this onstru tion fa toring out null ve tors. PROPOSITION 2.1.4 Let

h

i H0 = f 2 H : h i = 0g H H0 ;

the set

be a pseudo inner produ t on a omplex ve tor spa e v

v; v

des ends to an inner produ t on

© 2001 by Chapman & Hall/CRC

is a linear subspa e of

=

.

H

H

. Then

, and

h

i ;

21 The fa t that H0 = fv : kvk = 0g is a linear subspa e follows from the fa t that k
k is a pseudonorm (Corollary 2.1.3). In parti ular, v; w 2 H0 implies v + w 2 H0 by the triangle inequality. To show that h
;
i des ends to H=H0 , let v; w 2 H and v0 ; w0 2 H0 . We must verify that hv + v0 ; w + w0 i = hv; wi. But

PROOF

hv0 ; wi

= hv; w0 i = hv0 ; w0 i = 0

by the Cau hy-S hwarz inequality, so the desired equality holds. It is now routine to he k that h
;
i de nes a pseudo inner produ t on H=H0 , and this is a tually an inner produ t be ause v 62 H0 implies hv; vi 6= 0. Having turned a pseudo inner produ t into an inner produ t in this way, we may still need to omplete the spa e. There is no obsta le to doing this; the proof is routine but tedious, so we omit it. PROPOSITION 2.1.5

Let H be a omplex ve tor spa e and let . Then the formula

h
;
i

be an inner produ t on

H

h

lim vn ; lim wn i = lim hvn ; wn i

de nes an inner produ t on the Cau hy ompletion of it a Hilbert spa e.

H

whi h makes

For the remainder of this se tion we dis uss examples. None of these a tually uses the pre eding te hniques of fa toring out null ve tors and

ompleting; these will be needed later. In parti ular, all of the examples we onsider now are omplete from the start, whi h is good be ause it means that we have a on rete representation of every ve tor in the Hilbert spa e. Example 2.1.6

C

n

X

is a Hilbert spa e, with inner produ t n

hv; w i

=



ai bi ;

i=1

where

v

= (a1 ; : : : ; an ) and

w

= (b1 ; : : : ; bn ).

The spa e Cn of omplex n-tuples an be identi ed with the set of fun tions from f1; : : : ; ng into C. In this way, the pre eding example

an be seen as a spe ial ase of the following more general onstru tion.

© 2001 by Chapman & Hall/CRC

22

Chapter 2:

Hilbert Spa es

Example 2.1.7

Let X be a set. For any fun tion f : X ! C, we write X f (x) = a (a 2 C) ifPfor every  > 0 there exists a nite subset S of X su h 0 that ja S 0 f (x)j <  for any nite S  X whi h ontains S . 2 De ne l (X ) to be the olle tion of all fun tions f : X ! C su h P that kf k2 = X jf (x)j2 is nite. For f; g 2 l2 (X ) we de ne hf; g i by P hf; gi = f (xP )g (x). To see that f (x)g (x) exists, given  > 0 nd a nite set S  X su h that kf jS k  kf k  and kg jS k  kg k . Then for any nite subset S 0  X whi h ontains S , the Cau hy-S hwarz inequality in 2 0 l (S ) yields P

X f (x)g (x)

S0 S

 f jS S kkgjS 0

0

2 S   :

hf; gi is indeed well-de ned. Finally, to show l2 (X ) is omplete, let (fn ) be a Cau hy sequen e. Then for any x 2 X the sequen e (fn (x)) satis es jfm (x) fn (x)j  kfm fn k and hen e is Cau hy in C. So we an de ne a limit fun tion f = lim fn : X ! C pointwise. This fun tion belongs to l2 (X ) be ause for any nite S  X we have kf jS k  sup kfn k. Finally, given  > 0 we an nd an integer N su h that m; n  N implies kfm fn k  ; then for any n  N and any nite S  X we have



(f fn )jS  ; fn )jS = lim (fm So

so fn

m

! f.

Next we onsider L2 spa es. Any l2 (X ) an be regarded as an L2 spa e by treating the set X as a measure spa e with ounting measure. However, this fa t is of little use to us be ause mu h of what we do measure theoreti ally in later hapters requires  - niteness and if X is un ountable then ounting measure does not have this property. So in general we have to treat sets and measure spa es separately. To simplify the presentation we are going to assume throughout that all measures are  - nite. We need the following lemma, whi h gives a useful riterion for determining whether a normed ve tor spa e is omplete. LEMMA 2.1.8 A normed ve tor spa e

onverges whenever

V

(vn )

onverges.

© 2001 by Chapman & Hall/CRC

is omplete if and only if the sum

is a sequen e of ve tors su h that

1 1 1kvnvnk

P

P

1

P

P

23

1v If V is omplete and 1 1 n 1 kvn k < 1 then the sum N

onverges be ause the sequen e of partial sums 1 vn is Cau hy. To prove the onverse, let (wn ) be a Cau hy sequen e in V ; we must show it onverges. Choose a subsequen e (wkn ) with the property that PROOF

kwkn

P

wkn+1 k  2 n

P

P

for all n. Then de ne v1 = wk1 and vn = wkn wkn 1 for n > 1. By hypothesis vn onverges, but the nth partial sum of vn equals wkn , so the sequen e (wkn ) onverges. This implies that (wn ) onverges. Example 2.1.9

Let  be a  - nite measure on a set X and let L2 (X ) = L2 (X; ) be the spa e of measurable fun tions f : X ! C, modulo fun tions supported on null sets, whi h satisfy kf k2 = jf j2 d < 1. Then for any f; g 2 L2 (X ) the inner produ t hf; g i = f gd is well-de ned be ause the inequality 2jf gj  jf j2 + jg j2 implies that f g is integrable. This is a genuine inner produ t be ause jf j2 d = 0 implies f = 0 almost everywhere. To see that L2 (X ) is omplete, let (fn )  L2 (X ) satisfy kfn k = C < 1. Let gN = N 1 jfn j and g = lim gN . Sin e kgN k  C for all N , the monotone onvergen e theorem implies that kg k  C as well. So g < 1 almost everywhere, whi h implies that fn onverges almost everywhere to a measurable fun tion f . Finally, jf j  g so N f k ! 0 by the dominated onvergen e f 2 L2 (X ), and kf 1 n N theorem sin e f f ! 0 almost everywhere. Thus L2 (X ) is n 1

omplete by Lemma 2.1.8.

R

R

P

P

P

R

P

P

For the sake of larity we o

asionally denote the L2 norm by k
k2 . A similar argument shows that Lp (X ) is omplete for 1  p  1. After p = 2 we will be most interested in the ases p = 1 and p = 1 (but we will not formally introdu e them). 2.2

Subspa es

We want to develop an analogy between sets and Hilbert spa es. The analog of an element of a set is a unit ve tor in a Hilbert spa e | for instan e, in l2 (X ) (Example 2.1.7) the element x 2 X orresponds to the fun tion x = fxg whi h takes the value 1 at x and is 0 elsewhere. Likewise, losed subspa es of a Hilbert spa e are analogous to subsets of a set. We now introdu e the operations _, ^, and ? on subspa es, whi h

orrespond to union, interse tion, and omplementation of subsets. In the nite dimensional ase these redu e to the operations de ned in Se tion 1.2.

© 2001 by Chapman & Hall/CRC

24

Chapter 2:

Hilbert Spa es

PROPOSITION 2.2.1 Let

fE :  2 J g spanT fE g

be a family of subspa es of a Hilbert spa e

losure of every

H

is the smallest losed subspa e of

H

E , and E is the largest losed subspa e of E . If E is a losed subspa e of then the

H

in every

fv 2 H : hv; wi = 0 H

is a losed subspa e of

w

for all

H

. Then the

whi h ontains

whi h is ontained set

2 Eg

.

For example, the last statement is proven as follows. It is lear that the given set of ve tors is a subspa e; to prove losure, suppose that vn ! v and that ea h vn belongs to the set. Then we have hv; wi = hvn v; wi ! 0 for any w 2 E by the Cau hy-S hwarz inequality, so v is also in the set. DEFINITION 2.2.2

fE :  2 J g spanfE g

Let

H

span

The

H

, denoted

and their meet is their interse tion ortho omplement of a subspa e E of H is the

E?

We write

v

= fv 2 H : hv; wi = 0 for

? w hv; wi = 0 v ? E if

,

if

v

and we say in the respe tive ases that are

W E join V E = T E

be a Hilbert spa e. The

of losed subspa es of

orthogonal.

all

2 E?

v

and

w

, and

w, v

of a family

, is their losed



.

losed subspa e

2 E g: E1 ? E2

and

E,

or

E1 E2? ; E1 and E2 ,

if



The operations _, ^, and ? a tually behave very mu h like [, \, and [ S = X , S \ S = ;, [ \ \ = S1 [ S2 , for any sub a Hilbert spa e. Before proving this we require two preliminary lemmas whi h are both of independent interest. The rst is the parallelogram law and the se ond says that ve tors of minimal norm exist in any losed onvex set.

( omplementation): the set theoreti laws S S

= S , (S1 S2 ) = S1 S2 , and (S1 S2 ) sets S; S1 ; S2 X , all transfer to subspa es of

LEMMA 2.2.3 Let

H

for all

be a Hilbert spa e. Then

kv + wk2 + kv

v; w

kwk  1

2H  > 0 k 21 (v + w)k  1 Æ .

For any

, and

k

w 2

= 2kv k2 + 2kwk2

there exists imply

kv

Æ > w

0

su h that

k

kv k  1

,

.

PROOF The parallelogram law is a straightforward al ulation with inner produ ts. For the se ond assertion, given  > 0 let Æ = =8. Then

© 2001 by Chapman & Hall/CRC

25

kvk  1, kwk  1, and k 21 (v + w)k  1 kv

k

w 2

= 2kv k2 + 2kwk2

Æ

together imply

kv + wk2  4

4(1

Æ )2

 ;

as desired. The se ond assertion of Lemma 2.2.3 is alled uniform onvexity. It and ompleteness are the only properties needed to prove the next lemma. LEMMA 2.2.4 Let

K

be a losed, onvex subset of a Hilbert spa e. Then there exists

a unique ve tor

v

2K

of minimal norm.

PROOF Let a = inf fkvk : v 2 K g. If a = 0 then there is a sequen e of ve tors in K whose norms onverge to zero, hen e 0 2 K , and we are done. Otherwise let vn be a sequen e in K su h that kvn k ! a. Given  > 0, hoose Æ as in Lemma 2.2.3 and nd N 2 N su h that n  N implies kvn k  a(1 + Æ )  b. Then for m; n  N we have kvm k; kvn k  b and 21 (vm + vn ) 2 K , and hen e k 21 (vm + vn )k  a  b(1 Æ ); so uniform

onvexity implies kvm vn k  b. This shows that the sequen e (vn ) is Cau hy, so it onverges to some v 2 K , and we have kv k = lim kvn k = a. For uniqueness, suppose w 2 K and kwk = kv k. Then 12 (v + w) also belongs to K , so by minimality of kv k we have k 12 (v + w)k  kv k = kwk. Uniform onvexity then implies that kv wk = 0, i.e., v = w. THEOREM 2.2.5 Let

E

_

E be E? =

a losed subspa e of a Hilbert spa e

H and E ^ E ? = f0g.

in the form

PROOF

v

= v1 + v2

with

2H

v1

Every

2E

v

and

H

2H v2 2 E ?

.

Then

E

+ E? =

an be expressed uniquely .

2

Let v . By Lemma 2.2.4 there exists a ve tor v0 v + E of minimal norm. We laim that v0 E ? . To verify this, hoose w E , w = 0. Then v0 + aw v0 for all a , by minimality of v0 . In parti ular, taking a = v0 ; w = w 2 and performing a short

2 k k

6

k

omputation, we get

2

kk k 2C h ik k

2 2

kv0 k2 jhvk0w; wk2ij =

v0 hkvw0 ;kw2i w

 kv0 k2 : Thus hv0 ; wi = 0, as laimed. We on lude that v0 2 E ? , and v (v v0 ) + v0 is the desired de omposition of v . Thus E + E ? = H.

© 2001 by Chapman & Hall/CRC

=

26

Chapter 2:

Hilbert Spa es

E ^ E = f0g be ause any ve tor simultaneously in E and E is orthogonal to itself. If v = v1 + v2 = v1 + v2 with v1 ; v1 2 E and v2 ; v2 2 E then v1 v1 = v2 v2 . Sin e the left side belongs to E and the right side belongs to E , and we know that E ^ E = f0g, we on lude that both sides are zero. Thus v1 = v1 and v2 = v2 . This proves that the de omposition v = v1 + v2 is unique. ?

?

0

0

0

0

0

?

0

?

?

0

0

The tri k used to prove v0 2 E ? in the pre eding theorem also appeared in the proof of the Cau hy-S hwarz inequality (Proposition 2.1.2). The point is that if hv; wi = hw; wi then kv wk2 = kv k2 kwk2 . That is, if v w is orthogonal to w then kwk2 + kv wk2 = kv k2 , whi h is just the Pythagorean theorem. Furthermore, given any v and w, w 6= 0, we

an ensure the hypothesis by s aling w by the fa tor hv; wi=kwk2 , i.e., repla ing w by the proje tion of v onto w. COROLLARY 2.2.6

Let

E

E ( 2 J ) be losed subspa es of a Hilbert

and

spa e

H. Then

E = E; W V (b) ( E ) = V W E ; and ( ) ( E ) = E . (a)

??

?

?

?

?

PROOF

(a) Let v 2 E . Then v ? E ? , so v 2 E ?? . This shows that E  E ?? . Suppose the ontainment is proper. Then by Theorem 2.2.5 there exists a nonzero w 2 E ?? orthogonal to E . But w is also orthogonal to E ? , and hen e w = 0, a ontradi tion. So E ?? = E . W V ? (b) Suppose v 2 ( E ) . Then v ? E for all , so v 2 E? . This shows one ontainment. V Conversely, suppose v 2 E? . Then v is orthogonal to every element of every E , and hen e it is orthogonal to the losed span of all of the E . This veri es the reverse ontainment. ( ) By (a) and (b),

_

E = ?

_

E

?



??

=

^

E

??



?

=

^

E



?

;

as desired. We have now shown that the basi identities obeyed by subsets of a set are also obeyed by subspa es of a Hilbert spa e. However, there is a slightly more ompli ated set theoreti identity whi h does not transfer to Hilbert spa es. It is alled the distributive law, and it asserts that

© 2001 by Chapman & Hall/CRC

27

S1 [ (S2 \ S3 ) = (S1 [ S2 ) \ (S1 [ S3 ) for any subsets S1 ; S2 ; S3 of a set. The following example falsi es this law for Hilbert subspa es. It also falsi es the dual law obtained by inter hanging [ and \, whi h is true for sets as well. Example 2.2.7

Take

H = C2 and

f(a; a) : a 2 Cg f(a; 0) : a 2 Cg E3 = f(0; a) : a 2 Cg: Then E1 _ (E2 ^ E3 ) = E1 and (E1 _ E2 ) ^ (E1 _ E3 ) = H. E1

=

E2

=

In the ase of l2 (X ), ea h subset S of X gives rise to a losed subspa e of l2 (X ) onsisting of those fun tions whose support is ontained in S . It is easy to see that this orrespenden e between subsets and subspa es takes [, \, and into _, ^, and ?. (Thus, Example 2.2.7 hinges on the fa t that there are other subspa es in l2 (f0; 1g) besides those of the form l2 (S ) for S  f0; 1g.) This observation an also be made in the ontext of L2 spa es, but in order to state it we need a lemma. This is our rst result that uses  - niteness. If S and S 0 are measurable sets, we say that S essentially ontains S 0 if S 0 S is null.

l2 (S )

LEMMA 2.2.8

Let (X; ) be a  - nite measure spa e and let fS :  2 J g be a family of measurable subsets of X . Then there is a measurable set S su h that (a) S essentially ontains ea h S and (b) any measurable set that essentially ontains ea h S also essentially

ontains S and a measurable set S 0 su h that (a0 ) S 0 is essentially ontained in ea h S and (b0 ) any measurable set that is essentially ontained in ea h S is also essentially ontained in S 0 . The sets S and S 0 are unique up to null sets. We prove the rst assertion. Assume rst that (X ) is nite. Sin e the family fS g is arbitrary, without loss of generality we an assume it is losed under nite unions. Now de ne a = sup2J (S ). Sin e (X ) is nite, so is a. PROOF

© 2001 by Chapman & Hall/CRC

28

Chapter 2:

Hilbert Spa es

S

Fix a sequen e (Sn ) su h that (Sn ) ! a and let S = Sn . Then S essentially ontains every S be ause if (S~ S ) > 0 for some ~ then (Sn [ S~ ) onverges to (S [ S~ ) > a, whi h ontradi ts the maximality of a. Also, if T is any set whi h essentially ontains ea h Sn then it essentially ontains S . Now let  be any - nite measure and let (Xn ) be a partition of X into nite measure subsets. By the pre eding argument, for ea h n we

an nd a set Tn  Xn whi h essentially ontains S \ Xn for all  2 J , and isSessentially ontained in any other set with this property. Then let S = Tn . Properties (a) and (b) hold be ause they hold on ea h Xn . Furthermore, S is unique up to null sets be ause any other set T with the same properties would both essentially ontain and be essentially

ontained in S . The existen e and uniqueness of S 0 an either be proven by analogy with the above, or as a onsequen e of it via omplementation. We all the sets S and S 0 of the pre eding proposition the essential union and the essential interse tion of the family fSg. A more sophisti ated proof of their existen e involves taking the weak* limit in L1 (X )  = L1 (X ) of the hara teristi fun tions S , assuming the family fS g is losed under nite unions or interse tions and is dire ted by in lusion or reverse in lusion, respe tively. Example 2.2.9

W

S

(a) Let X be a set. Then l2 (S ) is a losed subspa e of l2 (X ) and we have l2 (S )? = l2 (S ) for every S  X . Also l2 (S ) = l2 ( S ) and l2 (S ) = l2 ( S ) for any family of sets S  X . (b) Let (X; ) be a  - nite measure spa e. Then L2 (S ) is a losed subspa e of L2 (X ) and we have L2 (S )? = L2 (S ), for every measurable S  X . Also L2 (S ) = L2 (S ) and L2 (S ) = L2 (S 0 ) for any family of measurable sets S  X , where S and S 0 are respe tively the essential union and the essential interse tion of the S .

V

T

W

2.3

V

Orthonormal bases

In Example 2.1.7 we observed that l2 (X ) is a Hilbert spa e. The point of orthonormal bases is to allow us to reverse this onstru tion and realize any Hilbert spa e in the form l2 (X ). We start with the de nition of orthonormal bases and a proof that they always exist. DEFINITION 2.3.1

A subset

if

and

kvk = 1

for all

v

orthornormal basis

2

v

if in addition

© 2001 by Chapman & Hall/CRC



?w

of a Hilbert spa e for all distin t

span() = H

.

H orthonormal 2

v; w

is

.

It is an

29 We will say that  generates H if span() = H. This is generally more useful than the on ept of algebrai ally spanning H. Observe that orthonormal sets are always linearly independent. To see this suppose P  is orthonormal and v1 ; : : : ; v 2  obey a linear dependen e 1 a v = 0. Then for any 1  j  n we have n

n

i i

aj =

*n X i=1

+

ai vi ; vj = 0:

So  must be independent. PROPOSITION 2.3.2 Let



H

be an orthonormal set in a Hilbert spa e

. Then



extends to

an orthonormal basis.

PROOF By Zorn's lemma, we an nd a maximal orthonormal set 

ontaining . We laim that  generates H (meaning that span( ) = H; see above). Suppose not and let E = span( ). Theorem 2.2.5 implies the existen e of a nonzero ve tor v orthogonal to E , and v=kvk is then a ve tor of norm one that is orthogonal to every ve tor in  . This ontradi ts maximality of  , so we on lude that span( ) = H. Thus  is an orthonormal basis of H whi h ontains . 0

0

0

0

0

0

0

0

In parti ular, applying this proposition to the set  = ; shows that every Hilbert spa e has an orthonormal basis. Now we an reverse the

onstru tion of Example 2.1.7 and show that every Hilbert spa e is isometri ally isomorphi to an l2 spa e. THEOREM 2.3.3 Let

H

fex : x 2 X g. Then H l2 (X ) by a map whi h takes ex to x .

be a Hilbert spa e with orthonormal basis

is isometri ally isomorphi to

PROOF

De ne U : spanfe g ! l2 (X ) by U

X

x

 X ax ex = ax x :

(Both sums are nite.) This map satis es U (e ) =  , and it is learly linear. Moreover, it preserves inner produ ts be ause E DX E X DX X X a ; b : a e ; b e = a b = x

x x

x x

x x

x

x

x

In parti ular, it preserves norms, so it is an isometry.

© 2001 by Chapman & Hall/CRC

x

x

Chapter 2: Hilbert Spa es

30

U uniquely extends to span ex = by ontinuity. It is surje tive be ause its range is losed and ontains the orthonormal basis x of l2 (X ). f

g

H

f

g

COROLLARY 2.3.4

Let fe :  2 J g v 2 H. Then

be an orthonormal basis of a Hilbert spa e and let

v=

H

Xa e

and

 

2J

v

2

k k

where the oeÆ ients a are given by a =

=

Xa j

 2J

2

j

v; e i.

h

This orollary is an immediate onsequen e of Theorem 2.3.3 and the fa t that it is true of the standard basis in l2 (J ). We de ne the dimension of a Hilbert spa e to be the ardinality of an orthonormal basis of . This depends on the fa t that any two bases have the same ardinality, whi h is lear if either one is nite. Otherwise, let  and  be orthonormal bases of an in nite dimensional spa e and onsider the set D of all nite linear ombinations PHilbert a e where ea h a belongs to the ountable set Q + iQ C and ea h e belongs to . Now D is dense in and the open balls of radius 1= 2

entered at the elements of  are disjoint, so ard(D) ard( ). But

ard(D) = 0 ard() = ard(), so we on lude ard() ard( ). By symmetry, the two are equal. The following remarkable fa t is now a onsequen e of Theorem 2.3.3. H

H

0

H



p

H

0



0






0



COROLLARY 2.3.5

Any two Hilbert spa es of the same dimension are isometri ally isomorphi . In parti ular, any ountably in nite dimensional Hilbert spa e is isometri ally isomorphi to l2 (N). As an example we onsider L2 of the unit ir le. Example 2.3.6

T = R=2Z.

2Z

1=2 inx The fun tions e ~n = (2 ) e (n ) form an 2 orthonormal basis of L ( ): orthonormality is easy to he k, and the Let

T

fa t that their span is dense follows from its density in

C (T)

| a

onsequen e of the Stone-Weierstrass theorem | and the density of C ( ) in L2 ( ), whi h is standard measure theory. 2 Consider the standard orthonormal basis en = n (n ) of l ( ). 2 By Theorem 2.3.3 the map ~ : L ( ) l2 ( ) whi h takes e~n to en

T

T

© 2001 by Chapman & Hall/CRC

F

T! Z

2Z

Z

31

2 L2 (T) then the nth entry of F~ f

is an isometry. In general, if f R 2 f; e~n = (2 ) 1=2 0 fe inx dx.

h

i

is

The map F~ of the pre eding example is the Fourier transform on the

ir le and the values hf; e~n i are the Fourier oeÆ ients of f . In the ase of L2 (R) it is a little harder to nd a ni e orthonormal basis. There is one, however; see Se tion 4.3. 2.4

Duals and dire t sums

So far we have been working inside a single Hilbert spa e. In this se tion and the next we will look at three onstru tions whi h start with one or more Hilbert spa es and produ e a new spa e as a result. The rst

onstru tion, dualization, has no set theoreti analog, but the other two have. DEFINITION 2.4.1 Let H be a Hilbert spa e. Denote its dual spa e, the set of bounded linear fun tionals (i.e., bounded linear maps from H into C) by H . For ea h v 2 H the map v^ : w 7! hw; v i is bounded by the Cau hy-S hwarz inequality, and hen e belongs to H . This de nes a map v 7! v^ from H to H .

A map T : V

! W between ve tor spa es is antilinear if (

T av

+ bw) = aT v + bT w

for all v; w 2 V and a; b 2 C. It is an anti-isomorphism if it is antilinear and one-to-one. PROPOSITION 2.4.2

Any Hilbert spa e H is isometri ally anti-isomorphi to its dual via the map v 7! v^.

PROOF Verifying antilinearity is routine. To see that kv^k = observe that for any w 2 H we have j^( )j = jh v w

w; v

ij  k kk k w

v ;

and hen e kv^k  kv k, while

j^( )j = jh ij = k k2 and hen e k ^k  k k. This also shows that 7! ^ is isometri . v v

v

v

© 2001 by Chapman & Hall/CRC

v; v

v

v

;

v

k k, v

32

Chapter 2: Hilbert Spa es

For surje tivity, let ! : H ! C be any linear fun tional. The kernel of is a losed, odimension one subspa e of H, so by Theorem 2.2.5 we have H = ker(!) + E where E is the one-dimensional ortho omplement of ker(!). Say E = span(v0 ). Let v1 = av0 where a = !(v0 )=kv0 k2 ; we will show that v^1 = !. For any w 2 ker(!) = E ? we have hw; v1 i = 0, so that v^1 (w) = 0 = ! (w). Also, a dire t omputation shows that v^1 (v0 ) = ! (v0 ). Sin e H is spanned by ker(!) and v0 it follows that v^1 = !, as laimed. This proves surje tivity.

!

It follows that H is a Hilbert spa e in the sense that its norm is

ompatible with an inner produ t. Namely, for v; w 2 H de ne hv^; w^i = hw; vi. Sin e v 7! v^ is a bije tion this de nes an inner produ t on H . By the pre eding result we have in parti ular that

hv^; v^i = hv; vi = kvk2 = kv^k2 for all v 2 H, so this inner produ t does give rise to the original norm on H . Next we onsider the dire t sum onstru tion. DEFINITION 2.4.3 LLet fH :  2 J g be a family of Hilbert spa es. De ne the dire t sum H to be the set of allP sequen es (v ), also L denoted v , su h that v 2 H for all  and kv k2 < 1. Give it the inner produ t

DM

v ;

M

w

E

=

X

2J

hv ; w i:

This sum onverges be ause jhv; wij  kvkkwk  12 (kvk2 + kwk2 ), and an easy al ulation shows that it does de ne an inner produ t. The stru ture of dire t sums is exhibited in the next result. PROPOSITION 2.4.4

L

Let fH :  2 J g be a family of Hilbert spa es. Then H is a Hilbert spa e. Ea h H naturally embeds in the dire t S sum, and if  is an orthonormal basis of H for ea h  2 J then  is an orthonormal  L basis of H .

Completeness of the dire t sum is veri ed by an argument similar to the one used in Example 2.1.7. This is no a

ident; the dire t sum

onstru tion redu es to l2(J ) when ea h H is one-dimensional. The rest of the proposition is an easy onsequen e of Theorem 2.3.3 in the

© 2001 by Chapman & Hall/CRC

33 following way. For ea h  let  be an orthonormal basis of H . Then we may identify H with l2( ), and an elementary L S al ulation shows that H is orrespondingly identi ed with l2( ). Evidently dire t sums of Hilbert spa es orrespond to disjoint unions of sets. The same is true for measure spa es: Example 2.4.5

L

(a) Let (X ) be a family of sets and let X be their disjoint union. Then l2 (X )  = l2 (X ). (b) Let (Xn ; n ) be a ountable family of  - nite measure spa es and let (X; ) be their disjoint union. Then  is also  - nite, and L2 (X )  = L2 (Xn ).

L

There is also a measurable version of dire t summation. The situation here involves a family of Hilbert spa es fHx : x 2 X g where X is a  nite measure spa e. If ea h Hx is separable (i.e., of either nite or

ountably in nite dimension) then we an partition X into a ountable family of subsets Xn (0  n  1) on whi h dim(Hx ) = n, perform the

onstru tion on ea h Xn separately, and dire t sum the result. This redu es the problem to the ase where the dimension of the spa es Hx is onstant. Sin e Hilbert spa es of the same dimension are isomorphi , we need only onsider the problem of taking a \measurable dire t sum" of a single Hilbert spa e H over a measure spa e X . The following de nition indi ates how this is done. Let  be a  - nite measure on a set X and

DEFINITION 2.4.6

H be a separable Hilbert spa e. A fun tion f : X ! H is weakly measurable if the fun tion x 7! hf (x); vi is measurable for ea h v 2 H. Then L2 (X ; H) is the set of all weakly measurable fun tions f : X ! H

let

su h that

kf k = 2

Z

X

kf (x)k2 < 1;

modulo fun tions whi h are zero almost everywhere. An inner produ t on L2 (X ; H) is given by

hf; gi =

Z

X

hf (x); g(x)i:

The pre eding integral exists be ause 2jhf (x); g(x)ij  2kf (x)kkg(x)k  kf (x)k2 + kg(x)k2 pointwise almost everywhere, and ompleteness of L2(X ; H) an be veri ed using Lemma 2.1.8, just as for ordinary L2 spa es ( f. Example 2.1.9).

© 2001 by Chapman & Hall/CRC

34

Chapter 2: Hilbert Spa es

The following proposition gives an alternative hara terization of the spa e L2 (X ; H). Yet another hara terization will be given in Example 2.5.4 ( ). PROPOSITION 2.4.7

Let (X; ) be a  - nite measure spa e and let H be a separable Hilbert spa e of dimension n (1  n  1). Then L2 (X ; H) is isomorphi to L n L2(X ). i=1

PROOF We assume n = 1; the nite dimensional ase is similar but easier. L 2 2 First we de ne a linear isometry U : 1 1 L (X ) ! L (X ; H). Let fei : i 2 Ng be an orthonormal basis of H. For any sequen e of fun tions fi 2 L2(X ), all but nitely many of whi h are zero, de ne U

M 

fi (x) =

1 X

fi (x)
ei :

i=1

The right side is learly measurable, and we have

X



f
e 2 = X kf k2 = M f 2 ;

i i i i so U Lis an isometry on its domain of de nition. But this domain is dense in L2 (X ), so U extends by ontinuity to the entire spa e. 2 For surje tivity, P let f 2 L (X ; H) and de ne fi (x) = hf (x); ei i for ea h i; then f = fi
ei and ea h term of the sum belongs to the range of U . But sin e U is an isometry, its range is losed. This shows that U is surje tive. We an now formalize the notion of a measurable dire t sum. This

onstru tion will be entral to Chapter 3, and it will remain important in Chapters 5 and 6. We will develop it further in Se tion 9.2. Let X be a  - nite measure spa e. A (separable) measurable Hilbert bundle over X is a disjoint union [ X = ( Xn  H n )

DEFINITION 2.4.8

where fXn g is a measurable partition of X and Hn is a Hilbert spa e of dimension n, 0  n  1. Hilbert spa e of L2 se tions of X is the dire t sum L2 (X ; X ) = LThe L2 (Xn ; Hn ). Equivalently, it is the set of weakly measurable fun tions

© 2001 by Chapman & Hall/CRC

35 S

fR : X ! Hn with the properties that f (x) 2 kf (x)k2 < 1.

Hn for all x 2 Xn and

We will dis uss ontinuous Hilbert bundles in Se tion 9.1. If the ber spa es Hn in De nition 2.4.8 are not assumed to be separable, then the measure theoreti issues be ome somewhat murky. In this setting the bundle onstru tion seems to break down, but there is a

orresponding module approa h whi h is still workable; see Chapter 9, espe ially Theorem 9.4.11, whi h has no separability assumptions. In any ase, we will always take our measurable Hilbert bundles to be separable. 2.5

Tensor produ ts

Our last Hilbert spa e onstru tion, the tensor produ t, is the most sophisti ated. Tensor produ ts of two (or nitely many) Hilbert spa es work like this. For any v 2 H and w 2 K there is a orresponding ve tor v w 2 H K, and ve tors of this form generate H K. The inner produ t of two su h ve tors is given by hv w; v w i = hv; v ihw; w i. In parti ular, the norm of v w is kvkkwk. If feg and fe~~ g are orthonormal bases of H and K, respe tively, then fe e~~ g is an orthonormal basis of H K. In the ase of in nitely many fa tors there are two ompeting de nitions of tensor produ ts, both of whi h redu e to the pre eding N onstru tion when the number of fa torsNis nite. The rst takes H to be generated v where Q by ve tors of the form Nea h v 2 H and the produ t kv k2 | whi h will be the norm ofN v | onverges. The result is a Hilbert spa e for whi h the ve tors v with v 2  for all  form an orthonormal basis, if ea h  is an orthonormal basis of H . This tensor produ t has had little appli ation, probably be ause it is essentially never separable. The other de nition, whi h we adopt, requires the hoi e of a distinguished unit ve tor u in ea h H ; this ve tor plays the role of a sort of zero or \ground state." 0

0

0

0

Let fH :  2 J g be a family of Hilbert spa es and for ea h  x a unit ve tor u 2 H . Consider the sequen es (v ) su h that ea h v belongs to H and v = u for all but nitely many values of . Let V be a ve tor spa e with a basis fe(v ) g indexed by all su h sequen es (v ). Give V the pseudo inner produ t de ned by

DEFINITION 2.5.1



e(v ) ; e(w )

© 2001 by Chapman & Hall/CRC



=

Y

hv ; w i;

36

Chapter 2: Hilbert Spa es

N

extending linearly. (All but nitely many terms of this produ t are 1, so it is well-de ned.) Then the tensor produ t H is the Hilbert spa e formed by fa toring out null ve tors and ompleting, as in Propositions 2.1.4 and 2.1.5.

N

N

We write vN  for the N equivalen e Q lass of the element e(v) in H. Observe that h v ; w i = hv ; w i. Also noti e that if the index set J is nite, the unit ve tors u are irrelevant. The following two propositions give the basi properties and essential stru ture of tensor produ ts. PROPOSITION 2.5.2

H and K be Hilbert spa es. Then (a) av w = v aw = a(v w) and (b) (v + v ) w = v w + v w for all a 2 C, v; v ; v 2 H, and w 2 K. Let

1

2

1

1

2

2

PROOF

(a) By dire t omputation, kav w a(v w)k2 = 2kavk2kwk2 2Re ahav; vihw; wi = 0: Thus av w = a(v w), and v aw = a(v w) similarly. (b) Taking the inner produ t of (v1 + v2 ) w (v1 w + v2 w) with itself yields h(v1 + v2 ) v1 v2 ; (v1 + v2 ) v1 v2 ihw; wi = 0: So (v1 + v2 ) w = v1 w + v2 w. PROPOSITION 2.5.3

N

Let fH :  2 J g be a family of Hilbert spa es with distinguished unit ve tors u and let H = H be their tensor produ t. For ea h  let  be an orthonormal basis of H whi h ontains u . Then the set

=f

O e : ea h e 2 ; and e = u

is an orthonormal basis of

for all but nitely many g

N H.

PROOF N It is easy to he k that Nis orthonormal. To show that it generates

H , we must verify that

© 2001 by Chapman & Hall/CRC

v

is in the losure of its span

37 whenever (v ) is a sequen e su h that v 2 H for all  and v = u for all but nitely many . This redu es the problem to the ase of a tensor produ t of nitely many Hilbert spa es, and by indu tion we redu e to the ase of two Hilbert spa es. Thus let H and K be Hilbert spa es and let fe g and fe~~ g be respe tive orthonormal bases. Let v w 2 H K and given  > 0 hoose v 0 2 spanfe g and w0 2 spanfe~~ g su h that kv v 0 k; kw w0 k  . Then

kv w

w k  kv w v wk + kv w v w k = kv v kkwk + kv kkw w k  (kv k + kwk)   (kvk + kwk + ) : This shows that v w is in the losure of spanfe e~ g, and it follows that spanfe e~ g is dense in H K, as desired. v

0

0

0

0

0

0

0

0

0

0

~

~

N produ t H . In this ase the natural embedding H0 into the tensor N is the map v 7! v where v0 = v and v = u for  6=  . The set theoreti analog of a tensor produ t of two Hilbert spa es is a  l (X  Y ). This

artesian produ t of two sets. Indeed, l (X ) l (Y ) = follows from the last proposition sin e the natural basis of l (X  Y ) an As with the dire t sum (Proposition 2.4.4), we an embed ea h fa tor 0

2

2

2

2

be identi ed with the produ t of the natural bases of l (X ) and We an also ompute some more ompli ated examples. 2

l2 (Y ).

Example 2.5.4

(a) Let X and Y be sets. Then l2 (X  Y )  = l2 (X ) l2 (Y ). (b) Let X and Y be  - nite measure spa es. Then L2 (X  Y ) is isometri ally isomorphi to L2 (X ) L2 (Y ) via the identi ation of f g 2 L2 (X ) L2 (Y ) with the fun tion f (x)g (y ) 2 L2 (X  Y ). ( ) Let X be a  - nite measure spa e and let H be a Hilbert spa e. Then L2 (X ; H) is naturally isomorphi to L2 (X ) H. This follows from Propositions 2.4.7 and 2.5.3.

Tensor powers | tensor produ ts in whi h every fa tor is the same | an be symmetrized or antisymmetrized, and we turn to this topi now. First we give the abstra t de nitions, and then we des ribe their interpretation in terms of an orthonormal basis. Let H be a Hilbert spa e, let n  1, and let H


H be the n-fold tensor power of H. (a) De ne the symmetrization of v


vn 2 H n to be the ve tor 1 X [v


vn ℄s = v n; n!

DEFINITION 2.5.5

H

n=





1

1

© 2001 by Chapman & Hall/CRC



(1)


( )

38

Chapter 2: Hilbert Spa es

where the sum is taken over all permutations  of the set f1; : : : ; ng. The n-fold symmetri tensor power of H is the losed linear span, in H n , of the ve tors [v1


vn ℄s with v1 ; : : : ; vn 2 H. It is denoted Hs n or H s


s H. (b) De ne the antisymmetrization of v1


vn 2 H n to be the ve tor [v1


vn ℄a =

1 X ( 1)jj v(1)


(n) ; n! 

where j j is the parity of  and the sum is again taken over all permutations of the set f1; : : : ; ng. The n-fold antisymmetri tensor power of H is the losed linear span, in H n , of the ve tors [v1


vn ℄a with v1 ; : : : ; vn 2 H. It is denoted Ha n or H a


a H. PROPOSITION 2.5.6

Let H be a separable Hilbert spa e, let fei g be an orthonormal basis of H, and let n  1. Then (a) the ve tors

[ei1




ein ℄s

with i1 


 in form an orthonormal basis of Hs n without repetition, and (b) the ve tors [ei1


ein ℄a

with i1 <


< in form an orthonormal basis of Ha n without repetition. For example, if H = C2 and fe1 ; e2 g is the standard basis, then H s H has a basis onsisting of the three ve tors [e1 e1 ℄s , [e1 e2 ℄s , and [e2 e2 ℄s , while H a H has a basis onsisting of the single ve tor [e1 e2 ℄a . (Note: in this ase H 2 = Hs 2  Ha 2 , but the analogous statement is not true if dim(H) > 2.) The proof of Proposition 2.5.6 is left to the reader. If dim(H) = m is
nite, a little ombinatori s shows that the dimension of Hs n is m+nn 1
m

n and the dimension of Ha is n . In general there is no good way to de ne measurable tensor produ ts. But in the spe ial ases H = l2 (N) and H = C2 , we an use the pre eding onstru tions to express dis rete tensor powers in a way whi h suggests a de nition of measurable tensor powers. Example 2.5.7

X be a ountable set, let Hx = l2 (N) for x 2 X2 , and take ux = e0 for all x, N where fek g is the anoni al basis of l (N). Then

(a) Let

the tensor power

Hx

© 2001 by Chapman & Hall/CRC

has a natural basis onsisting of ve tors of

39

N

the form ekx with ea h kx 2 N and kx = 0 for all but nitely many x. For su h a sequen e k = (kx ), write jkj = kx . Then the set of all sequen es with jkj = n may be identi ed with the set of all n-element subsets of X , allowing multipli ity. These are naturally in one-to-one orresponden e with the orthonormal basis of the n-fold symmetri tensor power of l2 (X ) des ribed in Proposition 2.5.6, so we have a natural isomorphism between x2X Hx and the dire t sum, over n 2 N, of the n-fold symmetri tensor powers of l2 (X ). (b) Let X be a ountable set, let Kx = C2 for x 2 X , and let fe1 ; e2 g be the anoni al basis of C2 . Taking ux = e1 for all x, the tensor power Kx has a basis onsisting of ve tors of the form vx with vx = e2 for nitely many x 2 X and vx = e1 for all other x. There is a natural bije tion between this basis of Kx and the

olle tion of all nite subsets of X , not allowing multipli ity. But the set of n-element subsets of X is naturally in one-to-one

orresponden e with the orthonormal basis of the n-fold antisymmetri tensor power of l2 (X ) des ribed in Proposition 2.5.6. Thus, there is a natural isomorphism between x2X Kx and the dire t sum, over 2 n 2 N, of the n-fold antisymmetri tensor powers of l (X ).

P

N

N

N

N

N

The requirement that X be ountable in the pre eding example is not really ne essary. This example motivates the following de nition. DEFINITION 2.5.8

(a) The

Let

be a Hilbert spa e.

symmetri (Boson) Fo k spa e over H Fs H

of the

H

is the dire t sum

= C  H  Hs 2 




n-fold symmetri tensor powers of H, for n 2 N. antisymmetri (Fermion) Fo k spa e over H is the dire t sum

(b) The

Fa H

of the

= C  H  Ha 2 




n-fold antisymmetri tensor powers of H, for n 2 N.

In light of Example 2.5.7, if X is a  - nite measure spa e then we regard Fs L2 (X ) as a measurable tensor produ t of the Hilbert spa es l2 (N) over the index set X , and we regard Fa L2 (X ) as a measurable tensor produ t of the Hilbert spa es C2 over the index set X . These spa es will be used in Se tion 7.1. \Boson" and \Fermion" are physi al terms; their signi an e was indi ated in Se tion 1.5.

© 2001 by Chapman & Hall/CRC

40 2.6

Chapter 2: Hilbert Spa es Quantum logi

Quantum logi was the rst fully arti ulated quantum version of a lassi al mathemati al subje t. Classi al propositional logi an be understood in the following way. Let X be a set, whi h we think of as the set of all possible states of some (not ne essarily physi al) system. Any de nite proposition p about the system will be true in some states and not in others, so we have a oneto-one orresponden e between the subsets of X and the propositions about the system, up to equivalent propositions. Under this orresponden e the logi al operators _, ^, and : (and, or, and not) orrespond to the set theoreti operators [, \, and (union, interse tion, omplement). For instan e, the states in whi h p ^ q holds are pre isely the states in whi h p and q both hold, whi h is to say the interse tion of the set of states whi h satisfy p and the set of states whi h satisfy q . In this way we an onvert propositional logi | the logi of propositions | into set theory. The quantum me hani al analog of this formulation of propositional logi repla es the set X with a Hilbert spa e H and models propositions by losed subspa es of H. This purely formal analog has a natural physi al interpretation if H a tually is the Hilbert spa e of some quantum system. Then any losed subspa e E of H represents a proposition about the system, namely, the proposition that the state ve tor lies in E . In

ontrast to the lassi al ase, one annot in general say with ertainty whether a given state ve tor satis es a given proposition (as one an say whether an element lies in a subset), but an only give the probability that it will do so. Conversely, to any proposition p about the system one

an asso iate the losed subspa e generated by the state ve tors whi h satisfy p with full probability. The logi al interpretation of : and ^ is straightforward. A unit ve tor v belongs to E ? pre isely if, with full probability, v fails to satisfy the proposition asso iated to E . And v 2 E1 ^ E2 pre isely if v , with full probability, satis es both the proposition asso iated to E1 and the proposition asso iated to E2 . The disjun tion _ is not so easily interpreted, and is best understood as a non-primitive symbol whi h is de ned in terms of : and ^ by the law p _ q = :(:p ^ :q ). Quantum logi , in the above sense, is usually understood as a manyvalued logi , with the possible truth-values being the losed subspa es of H. But the intuition behind this interpretation is un lear. It makes more sense to think of quantum logi as a many-valued logi for whi h the truth-values are probabilities, values in the interval [0; 1℄. As in the

lassi al ase, although it is the ase that every proposition p orresponds to a subspa e E of H, the truth-value of p only makes sense relative to a given state v of the underlying system, and the only sensible value this

© 2001 by Chapman & Hall/CRC

41

an be is the probability that an experiment testing p will produ e a positive result for the state v . On the basis of this point of view it is possible to formulate a quantum version of the predi ate al ulus. This is the logi al setting that involves variables, relations, and quanti ation, in whi h all ordinary mathemati s is done. In the predi ate al ulus, a well-formed formula (wf.) is built up from atomi wfs, whi h are of the form r(x1 ; : : : ; xn ). Here r stands for some relation and x1 ; : : : ; xn are variables (not ne essarily distin t). If A and B are wfs then so are :A, A _ B , A ^ B , and (9x)A and (8x)A for any variable x. Every wf. an be built up from atomi wfs in this way. Classi ally, the truth of a given wf. an be determined only when the relation and variable symbols are given some on rete interpretation. This means that a set X is xed, and to ea h variable symbol x we must assign an element of X and to ea h relation symbol r whi h takes n variables we must assign a subset of X n . The truth-value of the wf. an then be determined indu tively in a straightforward way. The quantum analog of the predi ate al ulus involves the same language, and the same notion of a well-formed formula, but interprets these formulas in a Hilbert spa e H rather than a set X . Now the variable symbols x are interpreted as unit ve tors in H, and the n-pla e relation symbols r are interpreted as losed subspa es of H n . In order to determine the truth-value of a wf. in a given Hilbert spa e interpretation, we initially ignore the unit ve tors orresponding to the variables and indu tively asso iate a losed subspa e of a tensor power of H to ea h subformula of the wf. At the end of this indu tive pro ess, when we have a losed subspa e E of H n whi h orresponds to the entire wf., we de ne its truth-value in the given interpretation to be (in the language of Se tion 1.2) the probability that v1


vn satis es the event E , where vi is the unit ve tor in H whi h orresponds to the variable xi . For the most part this pro edure an be arried out without diÆ ulty. There are two problemati points, however. The rst is the issue of repeated variables. If r(x; y ) is a two-pla e relation then it is interpreted as a losed subspa e E of H H. How then to interpret r(x; x)? The most natural solution is to interpret it as E \ (H s H). Thus, we treat repeated variables as distin t, but use the symmetri tensor produ t rather than the full tensor produ t. The se ond issue is how to interpret quanti ation. Suppose r orresponds to the losed subspa e E of H n = H H0 , where H0 = H (n 1) , and the variable x orresponds to the rst opy of H. Then we let (8x)r

orrespond to the set of ve tors w 2 H0 su h that v w 2 E for all v 2 H. This is indeed a losed subspa e. Having dealt with universal quanti ation, we an then handle existential quanti ation by means

© 2001 by Chapman & Hall/CRC

42

Chapter 2:

Hilbert Spa es

of the equation (9x)r = :(8x):r. Alternatively, we an interpret (9x)r independently and prove the pre eding equality as a theorem. This is worth dis ussing be ause intuitively, existential quanti ation involves proje tion from a tensor produ t onto one of its fa tors. In general there is no natural proje tion map from H K onto H; however, given a subspa e E of H K there is a subspa e  (E ) of K whi h an naturally be onsidered its proje tion. Namely, there exists a losed subspa e of K whose tensor produ t with H ontains E and whi h is the smallest subspa e with this property. Then if r orresponds to the losed subspa e E of H n , we let (9x)r

orrespond to  (E ). To prove the pre eding assertion, for ea h u 2 K de ne a map u : H K ! H by setting u (v w) = hw; uiv, and for ea h u 2 H de ne u0 : H K ! K similarly. LEMMA 2.6.1

H

Let

K

and

be Hilbert spa es and let

losed subspa es

E1

 E2 K

and

with

of

H

 E20 v0 2 E2 E20

and

E10

E1

and

E10

whenever

of

K

E2

v0

with

and

2 H K v0 2 E1 E10

E20

.

Then there exist and su h that

are losed subspa es of

H

.

Let E1 be the losure of fu (v0 ) : u 2 Kg and let E10 be the losure of fu0 (v0 ) : u 2 Hg. Then E1 and E10 are losed subspa es of H and K, respe tively. Choose an orthonormal basis (e ) of H whi h ontains a basis of E1 and an orthonormal basis (~e~ ) of K whi h ontains a basis of E10 . Write v0 = ;~ a~ (e e~~ ). Then, onsidering the map e~~ : v0 7!  a~ e , we see that a~ = 0 whenever e 62 E1 . Similarly, a~ = 0 whenever e~~ 62 E10 . Thus v0 2 E1 E10 . Now suppose E2 and E20 are losed subspa es of H and K su h that v0 2 E2 E20 . For any u 2 K we laim that u (E2 E20 )  E2 . It is

lear that u (v w) = hw; uiv 2 E2 for all v 2 E2 and w 2 E20 , and sin e ve tors of the form v w generate E2 E20 , the laim follows by linearity and ontinuity of u . Thus sin e v0 2 E2 E20 , it follows that u (v0 ) 2 E2 for all u 2 K, and hen e E1  E2 . Similarly, E10  E20 .

PROOF

P

P

PROPOSITION 2.6.2

E be a losed subspa e of H K. E1 of H and E10 of K with E  E1 E10 0 0 0 and su h that E1  E2 and E1  E2 whenever E2 and E2 are losed 0 subspa es of H and K with E  E2 E2 . Let

H

and

K

be Hilbert spa es and let

Then there exist losed subspa es

© 2001 by Chapman & Hall/CRC

43 For ea h v0 2 E let Ev0 and Ev0 0 be the subspa es des ribed by the lemma, i.e., whi h are minimal among those whose tensor produ t

ontains v0 . Then simply let E1 and E10 be the joins of the Ev0 and Ev0 0 , respe tively, as v0 ranges over all elements of E .

PROOF

We observe that if H = l2 (X ) and K = l2 (Y ) for some sets X and Y , and if R  X  Y , then we have  (l2 (R)) = l2 (~  (R)), where  is the proje tion of subspa es into H de ned above and ~ : X  Y ! X is the usual proje tion of sets. To verify , observe that l2 (~  (R)) K ontains l2 (R). To verify  note that for any (x; y ) 2 R we have ey (ex ey ) = ex , i.e., every element in a basis for l2 (~ (R)) is of the form u (v ) for some u 2 K and v 2 l2 (R). Thus we get  by the onstru tion of the proje tion of subspa es. This shows that our quantum me hani al interpretation of existential quanti ation is a valid analog of lassi al existential quanti ation. The above is only a sket h of how to determine the truth-value of a wf. relative to a Hilbert spa e interpretation, but it ontains all of the essential ideas. One an show that the l2 version of the lassi al interpretation of any wf. will give rise, by this pro edure, to the same truth-value as in the lassi al ase. However, there will generally be other Hilbert spa e interpretations whi h do not arise from lassi al stru tures in this way. 2.7

Notes

Most of the material in this hapter an be found in many standard texts; [8℄, [13℄, [34℄, and [58℄ are all good for further reading. The quantum propositional al ulus was introdu ed in [7℄. See also [37℄. The quantum predi ate al ulus will be treated more thoroughly in [75℄.

© 2001 by Chapman & Hall/CRC

Chapter 3

Operators 3.1

Unitaries and proje tions

In this hapter we dis uss linear operators on Hilbert spa es. The entral result is the spe tral theorem, whi h ompletely des ribes the stru ture of the most important kinds of operators. Throughout the hapter, H will be a Hilbert spa e; in Se tions 3.5 { 3.8 it will be separable. DEFINITION 3.1.1 Let H be a Hilbert spa e. B (H) will denote the spa e of all bounded linear operators A : H ! H.

The norm of an operator is given by kAk = supfkAv k : kv k  1g. It is easy to he k that the operator norm is omplete; thus B (H) is a Bana h spa e. If dim(H) = n is nite, then we an identify B (H) with the spa e Mn (C) of n  n omplex matri es, as in Chapter 1. However, this singles out a preferred basis, whi h may only ompli ate matters. In addition to adding and s aling operators, we an multiply them in the sense of omposition and also take their adjoints. In the ontext of Mn (C) the adjoint is the Hermitian transpose matrix, but it an be de ned for general Hilbert spa es in a basis-independent manner. We do this rst, and then prove it agrees with the Hermitian transpose of matri es in Corollary 3.1.4. Any bounded linear map A : V ! W between Bana h spa es has a natural \adjoint" map going from W  to V  . The adjoint of a Hilbert spa e operator is essentially this map, but we also identify H with H via the map v 7! v^ of De nition 2.4.1. For larity we use the notation A# for the Bana h spa e adjoint and A for the Hilbert spa e adjoint. DEFINITION 3.1.2 Let A 2 B (H). De ne a bounded linear map A# : H ! H by setting A# (!) = ! Æ A for ! 2 H . Then de ne A : H ! H by A v = (A# v^), where  is the inverse of ^. A is the

45 © 2001 by Chapman & Hall/CRC

46

Chapter 3: Operators

adjoint of A.

H

A

  A# (! ) 

-

H

H

6

 R ?

- H

^

!

C

Figure 3.1

A#

H

_

A

-

?

H

A# and A

More generally, if H and K are Hilbert spa es, one an de ne the adjoint of any bounded map A : H ! K by the same method. It will be a bounded map from K to H. The fundamental property of the adjoint is given in the next proposition. This an also be used as a de nition. PROPOSITION 3.1.3

Let A 2 B (H). Then A 2 B (H) and for all v; w 2 H.

hAv; wi = hv; A wi

PROOF A is linear be ause it is a omposition of two antilinear maps and one linear map, and it is bounded be ause v 7! v^ is isometri and kA# k = kAk. For the se ond assertion, let v; w 2 H; then hv; A wi = hv; (A# w^)i = (A# w^)(v) = w^(Av) = hAv; wi;

as desired. It is generally easier to ompute adjoints using the pre eding result rather than De nition 3.1.2. This is possible be ause the inner produ ts hv; A wi ompletely determine (A w)^, and hen e they ompletely determine A w, for all w 2 H. COROLLARY 3.1.4

Let A 2 B (H) and suppose H  = Cn is a nite dimensional Hilbert spa e. Then B (H)  M ( C ). If A is represented by the matrix [aij ℄ = n then A is represented by its Hermitian transpose [aji ℄.

© 2001 by Chapman & Hall/CRC

47

PROOF

The rst assertion is trivial. For the se ond, let fe1 ; : : : ; en g be the standard orthonormal basis of Cn . Then

h



A ei ; ej

i=h



ej ; A ei

i=h

Aej ; ei

i=

aji :

Thus the (i; j ) entry of the matrix whi h represents A is aji . Now we list the basi laws satis ed by adjoints. PROPOSITION 3.1.5

Let

A; B

2 (H) and B

a; b

2 C. Then

(aA + bB ) = aA + bB  ; (b) (AB ) = B  A ; ( ) A = A; (d) kA k = kAk; and (e) kA Ak = kAk2 .

(a)

Properties (a) { ( ) follow from similar laws for A# , and property (d) is a onsequen e of the fa t that the map v 7! v^ is an isometry. To verify (e), note rst that kA Ak  kA kkAk = kAk2 (using property (d)). Conversely, for any unit ve tor v 2 H we have

k k2 = h Av

Av; Av

i=h



k; yields k k2  k

A Av; v

ik



A A

 taking the supremum over all unit ve tors v A A Ak. This proves (e). The most important lasses of operators on a Hilbert spa e are de ned in terms of adjoints; we identify them now. DEFINITION 3.1.6

(a)

A

(b)

N

( )

U

(d)

P

2 2 2 2

(H (H B (H B (H B

B

Denote the identity map on

H by

I.

) is self-adjoint if A = A . ) is normal if N N  = N  N . ) is unitary if U U  = U  U = I . ) is a proje tion if P = P  = P .

2

Proje tions are sometimes alled \orthogonal proje tions" to distinguish them from operators whi h only satisfy P 2 = P . But we will not need to onsider the latter type of operators. Observe that normality is implied by the other three onditions. Normal operators are important be ause they are pre isely the operators for whi h one an prove a spe tral theorem (see Theorem 3.5.3). Unitaries and proje tions have simple geometri properties, as we will see shortly,

© 2001 by Chapman & Hall/CRC

48

Chapter 3:

Operators

and self-adjoint operators are on eptually fundamental be ause they are the Hilbert spa e analogs of real-valued fun tions on a set. Before des ribing the stru ture of unitaries and proje tions, we rst give some examples. Example 3.1.7

Let (X; ) be a  - nite measure spa e and let H = L2 (X ). For f 2 L1 (X ) de ne the multipli ation operator Mf 2 B (H) by setting Mf g = fg for g 2 L2 (X ). The norm of Mf is kf k1 . Its adjoint is Mf, the operator of multipli ation by the pointwise omplex onjugate of f . It follows that Mf is normal. Furthermore, if f is real-valued then Mf is self-adjoint; if jf (x)j = 1 for almost every x then Mf is unitary; and if f (x) = 0 or 1 almost everywhere then Mf is a proje tion. Exa tly the same statements hold if H = L2 (X ; X ), where X is a measurable Hilbert bundle over X (De nition 2.4.8). Multipli ation by f 2 L1 (X ) still makes sense.

The pre eding example is general; see Theorem 3.5.3. But there is also a very dierent intuition for unitary operators, illustrated in the following example. Example 3.1.8

Let (X; ) be a  - nite measure spa e and let f : X ! X be a measure preserving bije tion. De ne the omposition operator Cf : L2 (X ) ! L2 (X ) by Cf g = g Æ f . Then Cf g = g Æ f 1 , and so Cf is unitary.

Now we pro eed to a general geometri des ription of unitaries and proje tions. The only tool we need is the following polarization identity, whose proof is a routine omputation. It is useful be ause it allows us to re over any inner produ t hAv; wi, and hen e omplete knowledge of the operator A, from inner produ ts of the form hAv; v i. LEMMA 3.1.9 Let A

2 B (H).

Then

hAv; wi = for all v; w

2 H.

3 1X k A(v + ik w); (v + ik w) i 4 k=0

PROPOSITION 3.1.10 Let U

2 B (H).

The following are equivalent:

© 2001 by Chapman & Hall/CRC

49 (a) U is unitary; (b) U is invertible and U 1 = U  ; and ( ) U is an surje tive isometry.

PROOF

(a) ) (b). Trivial. (b) ) ( ). Suppose

1 = U  . Then hU v; U wi = hU  U v; wi = hv; wi U

for all v; w 2 H. In parti ular, kU v k2 = kv k2 for all isometry. It is surje tive be ause it is invertible. ( ) ) (a). Suppose U is an isometry. Then

h  i=h 2 H. Thus h(  U

U v; v

v

2 H, so

U

is an

i = k k2 = k k2 = h i ) i = 0 for all , and it follows by

U v; U v

Uv

v

v; v

for all v U U I v; v v polarization that U  U I = 0, i.e. U  U = I . If U is surje tive, then U 1 exists and is also an isometry. Sin e  1 = U  ; so U U  = I as well. U U = I , it follows that U

In ontrast to the nite dimensional ase, isometries need not be surje tive in general. The simplest ounterexample is the unilateral shift + on l2(N) de ned by S S

+ (a0 ; a1 ; : : :) = (0; a0 ; : : :):

Its adjoint, the ba kward shift S

S

, satis es

(a0 ; a1 ; : : :) = (a1 ; a2 ; : : :);

and from this one an verify that S Now we turn to proje tions.

S

+ = I but S +S = 6 I.

Let be a losed subspa e of H. For any 2 H let = 1 + 2 be the unique de omposition with 1 2 and 2 2 ? , as in Theorem 2.2.5. Then de ne E : H ! H, the orthogonal

DEFINITION 3.1.11 v v

v

v

E

v

E

proje tion onto E , by PE v = v1 .

v

E

P

PROPOSITION 3.1.12

The proje tions in B (H) are pre isely the orthogonal proje tions onto

losed subspa es of H. First, x a losed subspa e E  H. We verify that PE is a proje tion in the sense of De nition 3.1.6. It is linear be ause if

PROOF

© 2001 by Chapman & Hall/CRC

50

Chapter 3:

v = v1 + v2 and w = w1 + w2 with v1 ; w1 av + bw = (av1 + bw1 ) + (av2 + bw2 ), so

Operators

2 E and v2 ; w2 2 E ? then

PE (av + bw) = av1 + bw1 = aPE v + bPE w:

kv1 k2  kv1 k2 + kv2 k2 = kvk2 . Also hPE v; wi = hv1 ; w1 + w2 i = hv1 ; w1 i = hv1 + v2 ; w1 i = hv; PE wi;

It is bounded be ause

so that in

PE

B (H).

=

PE ,

and

PE2

=

PE

is lear. Thus every

PE

is a proje tion

Conversely, suppose P is a proje tion and let E = P (H) be its range. E is losed be ause if (vn )  E and vn ! v then vn = P vn ! P v implies P v = v , and thus v 2 E . Now hoose v 2 H and write v = v1 + v2 where v1 2 E and v2 2 E ? . Then P v1 = v1 and

kP v2 k2 = hP v2 ; P v2 i = hP  P v2 ; v2 i = hP v2 ; v2 i = 0;

so P v2 = 0. Thus P v = v1 = PE v , and we on lude P = an orthogonal proje tion onto a losed subspa e of .

H

3.2

PE .

So

P

is

Continuous fun tional al ulus

In this and the following two se tions we develop the tools needed for the spe tral theorem. We begin with the spe trum.

DEFINITION 3.2.1 the set of

Let

 2 C su h that A

A 2 B (H). The spe trum of A, I is not invertible in B (H).

sp(A),

Example 3.2.2

(a) Let A be a omplex n  n matrix. Then A I is invertible if and only if its kernel is zero. That is, it fails to be invertible if and only if there exists a nonzero ve tor v 2 Cn su h that (A I )v = 0, i.e., Av = v . So sp(A) is exa tly the set of eigenvalues of A. (b) Now let H = L2 (X ) for X a  - nite measure spa e and let f 2 1 1 L (X ). The essential range of f is the set of  2 C su h that f (O ) has positive measure for every open neighborhood O of . Let Mf be the operator of multipli ation by f , as in Example 3.1.7. If  does not belong to the essential range of f , then for some  > 0 we have jf (x) j   for almost every x 2 X . Thus k(f ) 1 k1  1=, and multipli ation by (f ) 1 is an inverse of Mf I = Mf  . So sp(Mf ) is ontained in the essential range of f . Conversely, suppose  belongs to the essential range of f and let 2  > 0. Find a fun tion g 2 L (X ) of norm one whi h is supported on 1 f (O ) where O is the ball of radius  about . Then

k(Mf

© 2001 by Chapman & Hall/CRC

Z

k2 = j(f

I )(g )

)g

Z

j2  2 jgj2 = 2

is

51

j

j

sin e f

2 H k k



 on the support of g . Suppose A B ( ) were I . Then the above implies that A 1= for all positive , whi h is absurd, so we on lude that Mf I has no inverse in B ( ). Thus sp(Mf ) is exa tly the essential range of f . 2 2 The same on lusions hold if L (X ) is repla ed by L (X ; ) for any measurable Hilbert bundle over X ( f. Example 3.1.7).

an inverse of Mf

H

X

X

If A is a diagonal n  n omplex matrix, a short omputation shows that the norm of A, as an operator, equals the modulus of its largest eigenvalue. In light of the previous example, we an say that kAk = maxf :  2 sp(A)g. This is not true for all matri es, but it does still hold for self-adjoint operators in in nite dimensions. This is the main fa t we need about the spe trum, and we now pro eed to prove it. The following lemma gives a simple suÆ ient ondition for invertibility in B (H). LEMMA 3.2.3

Let

A 2 B (H )

PROOF

and suppose

kAk < 1. Then I

A is invertible.

Note that kAn k  kAkn . It follows that the in nite series

I + A + A2 +




onverges in B (H). A trivial al ulation shows that the produ t of this operator and I A, in either order, equals the identity. Therefore I A is invertible. PROPOSITION 3.2.4

Let

A 2 B (H )

be self-adjoint. Then

kAk = supfjhAv; vij : kvk = 1g = supfjj :  2 sp(A)g: PROOF For the rst equality, let C = supfjhAv; vij : kvk = 1g. Then jhAv; v ij  C kv k2 for all v 2 H. Also note that hAv; v i = hv; Av i = hAv; vi, so that hAv; vi is always real. So if v and w are any unit ve tors in H, then polarization and the parallelogram law (Lemmas 3.1.9 and 2.2.3) yield

jRehAv; wij = 41 jhA(v + w); (v + w)i hA(v  C4 kv + wk2 + kv

© 2001 by Chapman & Hall/CRC

w k2




w ); (v

w)ij

52

Chapter 3:

Operators

C (2kv k2 + 2kwk2 ) 4 = C:

=

More generally, we have

jRe ahAv; wij = jRehAv; awij  C for any a 2 C with jaj = 1; in parti ular, taking a = jhAv; wij=hAv; wi yields jhAv; wij  C , and as this is true for all unit ve tors v and w we

on lude that kAk  C . Conversely, C  kAk by the Cau hy-S hwarz

inequality. So we have proven the rst equality. Now let C 0 = supfjj :  2 sp(A)g. Choose  2 C su h that jj > kAk. Then k 1 Ak < 1, so I 1 A is invertible by the lemma. Hen e A I is invertible, and so  62 sp(A). This shows that C 0  kAk. To prove kAk  C 0 , we use the rst equality and the fa t that hAv; v i is real for all v . It follows that we an nd a sequen e of unit ve tors vn su h that hAvn ; vn i ! kAk. Let  = lim hAvn ; vn i = kAk; then

kAvn

vn k2 = kAvn k2 2hAvn ; vn i + kvn k2  2kAk2 2hAvn ; vn i:

The last expression onverges to zero as n ! 1, so (A I )vn also

onverges to zero. But kvn k = 1 for all n, so A I annot be invertible. We on lude that  = kAk 2 sp(A), so that kAk  C 0 . COROLLARY 3.2.5

The spe trum of any self-adjoint operator

A 2 B (H) is a ompa t subset

of the real line.

PROOF Boundedness of the spe trum follows immediately from the theorem. To prove ompa tness suppose  62 sp(A) and let B = (A I ) 1 . Then for any  2 C with j j < kB k 1 we have

kI (A I )B k = [(A I ) (A I )℄ B = k( )B k < 1; by Lemma 3.2.3 this implies that (A I )B is invertible, so A I is right invertible, and a similar argument shows that it is left invertible. Thus  62 sp(A). This shows that the omplement of sp(A) is open. So the spe trum of A is ompa t. It remains to prove that sp(A)  R. Let a + ib 2 C with a and b real and b 6= 0. We must show that A (a + ib)I is invertible. Let A0 = 1b (A aI ); then A0 is also self-adjoint and it will suÆ e to show that A0 iI is invertible.

© 2001 by Chapman & Hall/CRC

53 Observe that

k(A0  iI)vk2 = hA0 v; A0 vi  hA0 v; ivi  hiv; A0 vi + hv; vi = kA0 v k2 + kv k2 : In parti ular, k(A0 iI)v k  kv k and hen e A0 iI is an isomorphism onto its range. So we need only show that its range is dense in H. But if w ? ran(A0 iI) then 0 = hw; (A0 iI)(A0 + iI)wi = k(A0 + iI)wk2 = kA0 wk2 + kwk2 ; so w = 0. Thus ran(A0 iI) is indeed dense in H, and we on lude that A0 iI is invertible in B(H), as desired. Now we turn to fun tional al uli for a self-adjoint operator. For any A 2 B(H) we an form polynomials in A just using basi algebra of operators. The idea of fun tional al ulus is to generalize this to fun tions other than polynomials. We rst treat ontinuous fun tions, and then in the next se tion we will pass to bounded Borel fun tions. Let p(z) = a0 +a1 z+


+an z n be a polynomial with omplex oeÆ ients and let A 2 B(H). We de ne p(A) to be the operator a0 I + a1 A +


+ an An . DEFINITION 3.2.6

LEMMA 3.2.7

Let A 2 B(H) be self-adjoint and let p(z) be a polynomial. Then sp(p(A)) = p(sp(A)).

PROOF Let  2 C and write p(z)  = a(z b1)


(z bn). The

omplex numbers b1 ; : : : ; bn are the roots of the equation p(z) = . Now p(A)

I = a(A

b1 I)


(A

bn I)

is a produ t of ommuting operators, so it is invertible if and only if ea h fa tor A bi I is invertible, i.e., ea h bi 62 sp(A), or equivalently, p(z) 6=  for all z 2 sp(A). Thus  is not in sp(p(A)) if and only if it is not in p(sp(A)). The following theorem des ribes the ontinuous fun tional al ulus for a self-adjoint operator. Here we use the notation C(X) for the Bana h spa e of ontinuous omplex-valued fun tions on the ompa t Hausdor spa e X, with norm kf k1 = supfjf(x)j : x 2 X g. We also use the following terminology. A linear map T from a onjugation- losed alge = T (f) and bra of fun tions into B(H) is a -homomorphism if T (f)

© 2001 by Chapman & Hall/CRC

54

Chapter 3:

Operators

T (fg ) = T (f )T (g ) for all f and g . It is unital if T (1X ) = I , and it is a -isomorphism if it is isometri . (We will generalize this terminology in De nition 5.1.1.) Here 1X is the fun tion whi h is onstantly 1 on X . THEOREM 3.2.8

Let A 2 B (H) be self-adjoint. Then the map whi h takes the polynomial p(x) 2 C (sp(A)) to the operator p(A) 2 B (H) extends uniquely to a unital -isomorphism from C (sp(A)) into B (H). PROOF First suppose the oeÆ ients of p are real. Then p(A) is self-adjoint, and it follows from Proposition 3.2.4 and Lemma 3.2.7 that

kp(A)k = supfjp()j :  2 sp(A)g = kpk1 ;

where the sup norm of p is taken in C (sp(A)). For an arbitrary omplex polynomial p, apply the above to the real polynomial jpj2 = pp; this yields kp(A)p(A) k = kpk21 . By Proposition 3.1.5 (e), it follows that kp(A)k = kpk1 . Thus the map taking p(x) to p(A) is an isometry. By the Stone-Weierstrass theorem, the polynomials are dense in C (sp(A)), so this map uniquely extends to an isometry from C (sp(A)) into B (H) whi h is learly unital. It is trivial that (ap + bq )(A) = ap(A) + bq (A), (pq )(A) = p(A)q (A), and p(A) = p(A) for any polynomials p(x) and q (x) and any a; b 2 C. The same properties hold on all of C (sp(A)) by ontinuity. For any f 2 C (sp(A)), we write f (A) for the orresponding operator. Thus f (A) = lim pn (A) where (pn ) is any sequen e of polynomials whi h

onverges to f uniformly on sp(A). Example 3.2.9

f 2 L1 (X ) and let Mf be the orresponding multipli ation 2 2 operator on L (X ) (or on L (X ; X )), as in Example 3.1.7. Then for any polynomial p(x) it is easy to see that p(Mf ) = MpÆf . By

ontinuity it follows that for any g 2 C (sp(Mf )) we have g (Mf ) = MgÆf .

Let

3.3

Borel fun tional al ulus

We now want to extend the ontinuous fun tional al ulus of the previous se tion to Borel fun tions. In parti ular, this will allow us to de ne operators of the form S (A) where S is the hara teristi fun tion of a Borel subset of sp(A). These operators are ru ial to understanding the

© 2001 by Chapman & Hall/CRC

55 stru ture of the operator A. The method of de ning f (A) for a Borel fun tion f is again extension by ontinuity, but now with respe t to dierent topologies. DEFINITION 3.3.1 The weak operator topology on B (H) is the weakest topology su h that for every v; w 2 H the map A 7! hAv; wi is ontinuous. In terms of nets, we have A ! A weak operator if and only if hA v; wi ! hAv; wi for all v; w 2 H.

We also need the following generally useful fa t. A sesquilinear form is a map f
;
g : H  H ! C whi h is linear in the rst variable and antilinear in the se ond. It is bounded if there exists C > 0 su h that jfv; w gj  C kv kkw k for all v; w 2 H. LEMMA 3.3.2

Let f
;
g : H  H ! C be a bounded sesquilinear form. Then there is a unique operator A 2 B (H) su h that fv; wg = hAv; wi for all v; w 2 H. Conversely, any bounded operator A de nes a bounded sesquilinear form by this equation.

PROOF

Fix v 2 H and onsider the map w 7! fv; wg. This is a bounded linear fun tional, so by Proposition 2.4.2 there exists a unique v0 2 H su h that fv; wg = hw; v0 i; that is, fv; wg = hv 0 ; wi for all w 2 H. De ne Av = v 0 . Then A is linear be ause h

A(av1 + bv2 ); wi = fav1 + bv2 ; wg = afv1 ; wg + bfv2 ; wg = ahAv1 ; wi + bhAv2 ; wi = haAv1 + bAv2 ; wi

for all w, and it is bounded be ause jh

Av; wij = jfv; wgj  C kv kkwk:

Uniqueness is lear, as is the onverse. We an now de ne the Borel fun tional al ulus. Re all the Riesz representation theorem: for any ompa t Hausdor spa e X , the dual spa e C (X ) an be identi ed with the spa e M (X ) of regular omplex Borel measures on X . ThisRidenti ation asso iates the measure  with the linear fun tional f 7! f d. We write Bor(X ) for the spa e of bounded

© 2001 by Chapman & Hall/CRC

56

Chapter 3:

Operators

Borel fun tions on X , and we de ne the -topology on Bor(X ) to be the weakest topology for whi h integration against any R regular RBorel fun tion is ontinuous. Thus f ! f () if and only if f d ! f d for all  2 M (X ). THEOREM 3.3.3

Let A 2 B (H) be self-adjoint. There is a unique unital -homomorphism Bor(sp(A)) into B (H) whi h extends the ontinuous fun tional al ulus and is ontinuous from the

-topology to the weak operator topology.

PROOF

Let T : C (sp(A)) ! B (H) be the -homomorphism given by the ontinuous fun tional al ulus for A. Then the double (Bana h spa e) adjoint map T ## takes C (sp(A))  = M (sp(A)) weak* ontin uously into B (H) . R Now for any bounded Borel fun tion f on sp(A), the map !f :  7! f d is a bounded linear fun tional on M (sp(A)) and hen e belongs to C (sp(A)) . For v; w 2 H let v;w 2 B (H ) be the linear fun tional A 7! hAv; wi. Then fv; wg = T ##!f (v;w ) is a bounded sesquilinear form and so the lemma implies that fv; wg = hf (A)v; wi for some bounded operator f (A). Note that if f 2 C (sp(A)) then this de nition agrees with the ontinuous fun tional al ulus be ause then hf (A)v; wi = T ##!f (v;w ) = v;w (T f ) = hf (A)v; wi :

Borel

ontinuous

Clearly, for any net (f ) of Borel fun tions on sp(A) we have f ! f () if and only if !f ! !f weak*. It then follows from weak* ontinuity of T ## that T ##!f (v;w ) ! T ##!f (v;w ) for all v; w 2 H, and hen e that f (A) ! f (A) weak operator. The map f 7! f (A) is a -homomorphism on C (sp(A)), and the unit ball of C (sp(A)) is weak* dense in the unit ball of C (sp(A)) . The fa t that f(A) = f (A) therefore follows from ontinuity, and fg (A) = f (A)g (A) an be proven by a two step ontinuity argument, where rst we assume f is ontinuous and approximate g by ontinuous fun tions, and then we let f be arbitrary and approximate it by ontinuous fun tions. Thus the map f 7! f (A) is a -homomorphism on the set of all bounded Borel fun tions. It is unique by weak* density of C (sp(A)) in C (sp(A)) . The passage from C (sp(A)) to C (sp(A)) in this theorem is an instan e of a more general fa t: if V is any Bana h spa e then any bounded linear map from V into B (H) has an extension to V  whi h is weak* to

© 2001 by Chapman & Hall/CRC

57 weak operator ontinuous. The proof of this fa t is substantially the same as the proof for C (sp(A)). The real reason why this works is that B (H) is a dual Bana h spa e, and its weak* topology agrees with the weak operator topology on bounded sets (see Theorem 6.3.8). This is why the weak operator topology is so important, at least on bounded sets. Example 3.3.4

f 2 L1 (X ) be real-valued and let Mf be the orresponding mul2 2 tipli ation operator on L (X ) (or L (X ; X )). Then for any bounded Borel fun tion g on sp(Mf ) we have g (Mf ) = Mg Æf ( f. Example 3.2.9). In parti ular, if S is a Borel subset of R then S (Mf ) = MS Æf = Mf 1 (S) . Let

3.4

Spe tral measures

Let A 2 B (H) be self-adjoint and let S be a Borel subset of R. Consider the hara teristi fun tion S : it is bounded and its restri tion to sp(A) is Borel measurable. So we an form the operator S (A). Sin e S = S = 2S , the operator S (A) also has these properties, and therefore it is an orthogonal proje tion onto a losed subspa e of H by Proposition 3.1.12. It is alled a spe tral proje tion of A. Example 3.3.4 shows that if A is a multipli ation operator then its spe tral proje tions are also multipli ation operators. Spe tral proje tions help explain why self-adjoint operators are the Hilbert spa e analog of real-valued fun tions. Let f : X ! R be a measurable fun tion on a  - nite measure spa e (X; ). In Se tion 1.3 we viewed real-valued fun tions on a nite set as partitioning the set into subsets and tagging these subsets with real numbers; for general X we need some measurable version of a partition. If f is only de ned up to null sets, the sets f 1 (a) (a 2 R) may well all be null and hen e individually arry no information about f . Instead, onsider the nested family of sets S( 1;a℄ = f 1 ( 1; a℄ for a 2 R. Sin e the intervals ( 1; a℄ generate the Borel sets, and


f 1 (S ) = f 1 (S ) [ f 1 S = f 1 (S )

[

f

1

\

n

S

n



=

\

n

f (S ) 1

n

for any Borel sets S; S  R, it follows that the sets S( 1 f 1 (S ) for every Borel subset S of R, up to null sets. n

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;a℄

determine

58

Chapter 3:

Operators

6R

X

f

-

S( 1;a℄ S( 1;b℄

a b

??

? A \measurable partition"

Figure 3.2

The analogous Hilbert spa e onstru tion is a family of losed subspa es su h that E( 1 ℄  E( 1 ℄ if a  b and E( 1 ℄ = T EE( 1 . ℄The E( 1 ℄ determine a losed subspa e E (S ) of H for ( 1 ℄ every Borel subset S of R in the same way as in the set theoreti ase, repla ing [, \, and with _, ^, and ?. Su h a map E from Borel subsets of R to losed subspa es of H is alled a \spe tral measure" on R, and this is the Hilbert spa e version of a real-valued fun tion. The spe tral theorem states that spe tral measures on R orrespond pre isely to self-adjoint operators on H. The following is the formal de nition of a spe tral measure. Let a measurable spa e be a set X together with a -algebra of subsets . ;a

;b

b>a

;a

;b

;a

;a

Let (X; ) be a measurable spa e. An Hvalued spe tral measure on X is a map E from into the set of losed

DEFINITION 3.4.1

H su h that the proje tions P 2 and (a) E (;) = 0 and E (X ) = H; subspa es of

all

S; T

E (S )

and

PE (T )

ommute for

E (S S ) = E (W S )? ; and T V ( ) E ( Sn ) = E (Sn ) and E ( Sn ) = E (Sn )

(b)

for all

S; Sn 2 .

There is a simple geometri interpretation of the ondition that P ( ) and P ( ) ommute. This is the ase if and only if E (S ) = E1  E2 and E (T ) = E1  E3 for some mutually orthogonal subspa es E1 , E2 , and E3 . We leave the proof of this fa t as an exer ise. We also omit the proof that if X = R then De nition 3.4.1 is equivalent to the previous des ription of spe tral measures in terms of the sets E( 1 ℄ . E S

E T

;a

© 2001 by Chapman & Hall/CRC

59 Spe tral measures an be diÆ ult to work with if one has no mental pi ture of them. This is the value of the following result; in the separable

ase, the pi ture be omes espe ially simple (Corollary 3.4.3). THEOREM 3.4.2

H-valued spe tral measure on a measurable spa e X . Then f g on X and an isometri L isomorphism U from  L (X;  ) onto H su h that Let

E

be an

there is a family of probability measures 2

E (S ) = U for every measurable

M 

!

L2 (S;  jS )

S  X.

For any v 2 H let Ev be the losed subspa e generated by the ve tors PE (S ) v as S ranges over all measurable subsets of X . Let fv g be a maximal family of unit ve tors in H su h that the subspa es Ev are orthogonal. Then the Ev generate H: if this were false, we

ould nd a unit ve tor v 2 H su h that v ? Ev for all . But then hPE(S) v; PE(T ) v i = hv; PE(S\T ) v i = 0 for all S; T  X and all , whi h shows that Ev ? Ev for all , ontradi ting maximality. For ea h  de ne  (S ) = kPE (S ) v k2 . It is straightforward to verify that these are probability measures on X . Then for any measurable set S  X , de ne U (S ) = PE (S ) v . Extend linearly to the span of the

hara teristi fun tions in L2 (X;  ); the result is an isometri map into H be ause

PROOF

D

U

X

X E X  aibj hPE (Si ) v ; PE (Sj ) v i bj Sj = ai Si ; U i;j X = aibj (Si \ Sj ) i;j DX E X = bj Sj : ai Si ;

ThisP omputation also P shows that U is well-de ned, sin e it implies that isometri emU ( ai Si ) = 0 if ai Si = 0. Thus U extends to an W bedding of L2 (X;  ) into H, and its L range is Ev . Sin e Ev = H, it follows Lthat the dire t sum map U = U is an isometri isomorphism from  L2 (X;  ) onto H. Fix S  X . For any  and any S  S we have U (S ) = PE (S ) v 2 E (S ); it follows by linearityLand ontinuity that U (L2 (S;  jS ))  E (S ) for all , and therefore U ( L2 (S;  jS ))  E (S ). The reverse ontainment follows by taking omplements. 0

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0

0

60

Chapter 3:

Operators

COROLLARY 3.4.3 Let

E

be an

suppose

H

H

-valued spe tral measure on a measurable spa e

is separable.

a measurable Hilbert bundle

L2 (X ; X ) onto measurable S  X . U

from

H

X

su h



X

and

X, X , and an isometri isomorphism 2 that U (L (S ; XjS )) = E (S ) for every

Then there is a probability measure

on

over

PROOF Let fk g be probability measures satisfying the on lusion of Theorem 3.4.2. Sin e H is separable, this family is ountable. k De ne  = 1 1 2 k . Then  is also a probability measure, and k is absolutely ontinuous with respe t to  for ea h k . Thus k = fk  for some fk 2 L1 (X; ), fk  0. For ea h k let Sk = fx 2 X : fk (x) > 0g, and for 0  n  1 let

P

Xn = fx 2 X : x 2 Sk for exa tly n values of kg:

S

Then fXn g is a measurable partition of X . For ea h n let Hn be an ndimensional Hilbert spa e, and let X = (Xn  Hn ) be the orresponding measurable Hilbert bundle. By Proposition 2.4.7 we an identify L2 (X ; X ) with L2(Sk ; jSk ). For ea h pk the map Vk : L2 (Sk ; jSk ) ! L2 (Sk ; k jSk ) de ned by Vk (g) = g= fk is an isometri isomorphism, and this gives rise to an isometri isomorphism

L

M L (Sk ; jS ) M M L (X ; k): ! L (Sk ; k jS )  =

 V : L2(X ; X ) =

2

2

k

k

2

Composing V with the map U from Theorem 3.4.2 yields the desired result. Theorem 3.4.2 is highly non anoni al: a dierent hoi e of ve tors v would lead to a dierent family of measures. Some degree of anoni ality is regained in Corollary 3.4.3; here  ould be repla ed by any measure with whi h it is mutually absolutely ontinuous, but otherwise the onstru tion is rigid. We say that the onstru tion only depends on the measure lass of . The inability of Corollary 3.4.3 to generalize to the nonseparable ase is illustrated by the following example. Let H = L2 [0; 1℄  l2 [0; 1℄, using Lebesgue measure on the rst summand and ounting measure on the se ond. De ne an H-valued spe tral measure on X = [0; 1℄ by setting E (S ) = L2 (S )  l2 (S ) for any Borel set S  [0; 1℄. In some sense H is a bundle over [0; 1℄ with two-dimensional bers, but it annot be identi ed

© 2001 by Chapman & Hall/CRC

61 with L2 [0; 1℄ C2 for any measure on [0; 1℄. Be ause of this diÆ ulty, and the fa t that spe tral measures are su h a entral topi , it will generally be onvenient for us to work only with separable Hilbert spa es (just as we work only with  - nite measure spa es). Still, for most purposes Theorem 3.4.2 is suÆ ient for work in the nonseparable ase, if one needs to do this. 3.5

The bounded spe tral theorem

For the remainder of the hapter H will be a xed separable Hilbert spa e. We now have enough ma hinery to give a qui k proof of a strong form of the spe tral theorem for bounded self-adjoint operators. Given su h an operator A, the idea is rst to obtain a spe tral measure on sp(A) by applying the Borel fun tional al ulus to hara teristi fun tions, and then to use Corollary 3.4.3 to realize H as the L2 se tions of a measurable Hilbert bundle. Finally, we observe that this identi ation asso iates A to the operator of multipli ation by x. THEOREM 3.5.1 Let on

A 2 B (H )

sp(A),

be self-adjoint.

Then there is a probability measure

a measurable Hilbert bundle

isomorphism

X

sp(A), and A = UMxU

over

U : L2 (sp(A); X )  = H su h that



an isometri

1.

PROOF

Re all that sp(A)  R by Corollary 3.2.5. Now for any Borel set S  sp(A) let E (S ) be the range of the spe tral proje tion S (A) of A for S . The fa t that E is a spe tral measure follows immediately from the properties of the Borel fun tional al ulus. Let , X , and U be as in Corollary 3.4.3. It is straightforward to verify that the map f 7! UMf U 1 is a unital -homomorphism from Bor(sp(A)) into B (H) whi h is ontinuous from the -topology to the weak operator topology. As it agrees with the Borel fun tional al ulus on hara teristi fun tions, ontinuity implies that the two must agree on all bounded Borel fun tions. In parti ular, taking f (x) = x, we obtain A = f (A) = UMx U 1 .

Evidently there is a one-to-one orresponden e between bounded selfadjoint operators on H and spe tral measures on ompa t subsets of R. Given su h an operator one an use its spe tral proje tions to de ne a spe tral measure on its spe trum, and onversely, given su h a spe tral measure one an use Corollary 3.4.3 to transfer Mx to UMx U 1 2 B (H). This is still true when H is nonseparable. Although we annot apply

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62

Chapter 3:

Operators

Corollary 3.4.3 in this ase, Theorem 3.4.2 still holds, and it an be used to establish a weaker version of the realization of A as a multipli ation operator. Namely, let f g be probability measures on sp(A) and U an isometri isomorphism from L2 (sp(A);  ) onto H as in Theorem 3.4.2. For ea h  let X be sp(A) equipped with the measure  , and let X be the disjoint union of the X . Then L2 (sp(A);  ) an be identi ed with L2 (X), and we have A = UMf U 1 where f(x) = x on ea h X . Of ourse, X is not - nite in this ase. It is worth noting the extent to whi h the onstru tion of Theorem 3.5.1 is anoni al. If A and A0 are unitarily equivalent self-adjoint operators, meaning that there is a unitary U su h that A = U 1 A0 U, then sp(A) = sp(A0 ) and a ontinuity argument shows that f(A) = U 1 f(A0 )U for all bounded Borel fun tions f. Thus their spe tral measures are unitarily equivalent. Then the measures  and 0 from Corollary 3.4.3 must be mutually absolutely ontinuous, and X and X 0 are

hara terized by the sequen es (Xn ) and (Xn0 ) of De nition 2.4.8, whi h must be the same up to -null sets. Thus sp(A),  (up to measure

lass), and (Xn ) (up to null sets) onstitute a set of invariants for A. Conversely, if B is any other self-adjoint operator with the same invariants then Theorem 3.5.1 gives rise to a unitary equivalen e between A and B. So sp(A), , and (Xn ) are a omplete set of invariants. Next we want to apply the self-adjoint theory to prove a version of the spe tral theorem for normal operators. This an be done in the following way. If N 2 B(H) is normal, that is, N  N = NN  , then the self-adjoint operators Re N = 12 (N + N  ) and Im N = 21i (N N  ) ommute. So N = Re N + iIm N is a linear ombination of ommuting self-adjoint operators. Thus, the next result is all we need.

L

L

LEMMA 3.5.2

A1 ; A2 2 B(H) be ommuting self-adjoint operators and let E1 and sp(A1 ) and sp(A2 ). Then there is a spe tral measure E on sp(A1 )  sp(A2 ) su h that E(S1  R) = E1 (S1 ) and E(R  S2 ) = E2 (S2 ) for all Borel sets S1  sp(A1 ) and S2  sp(A2 ). Let

E2

be the asso iated spe tral measures on

This lemma an be proven by mimi king the onstru tion of ordinary produ t measures: rst de ne E on produ t sets by setting E(S1  S2 ) = E1 (S1 ) \ E2 (S2 ), then extend to nite unions of produ t sets, and nally extend to all Borel sets by taking limits. The proof that this indeed produ es a spe tral measure is not essentially dierent from the realvalued ase. Alternatively, the lemma an be established in an algebrai manner; a more general result will be proven in this way in Lemma 5.3.4, so we omit the details here.

© 2001 by Chapman & Hall/CRC

63 THEOREM 3.5.3

Let N 2 B (H) be normal. Then there is a probability measure  on sp(N ), a measurable Hilbert bundle X over sp(N ), and an isometri 2 isomorphism U : L (sp(N ); X )  = H su h that N = UMz U 1 .

PROOF Let A1 = Re N and A2 = Im N , and let E be the spe tral measure on sp(A1 )  sp(A2 )  R2 given by the lemma. Let , X , and U be as in Corollary 3.4.3. We have A1 = UMx U 1 and A2 = UMy U 1 just as in the proof of Theorem 3.5.1. Thus N = A1 + iA2 = UMz U 1 where z = x + iy . Evidently Mz  = U 1 (A I )U is invertible for any  62 sp(N ). This implies that  is supported on sp(N ). Just as for self-adjoint operators, in the nonseparable ase we an still prove that N is equivalent to a multipli ation operator by using Theorem 3.4.2 in pla e of Corollary 3.4.3. See the omment following Theorem 3.5.1. Also, the observation made there that sp(A), , and (Xn ) are a

omplete set of invariants applies here too. Sin e unitary operators are normal, this theorem applies to them. But a multipli ation operator Mf is unitary if and only if jf j = 1 almost everywhere. It follows that a normal operator is unitary if and only if its spe trum is ontained in fz : jz j = 1g. Essentially, Theorem 3.5.3 generalizes Theorem 3.5.1 to a pair of ommuting self-adjoint operators. This will be subsumed under a far greater generalization when we dis uss abelian C*-algebras in Se tion 5.3. 3.6

Unbounded operators

An operator is bounded if and only if it is ontinuous. For this reason, one generally prefers to work with bounded operators. However, there are ases where we must onsider unbounded operators; in parti ular, some of the most basi operators in quantum me hani s are unbounded (see Se tion 4.1). This is related to the fa t that the generators of one-parameter unitary groups are in general unbounded self-adjoint operators, as we will prove in Theorem 3.8.6. Fortunately, the unbounded Hilbert spa e operators whi h arise in pra ti e usually have a ertain form whi h renders them moderately tra table. Namely, they are de ned only on a dense (un losed) subspa e of the Hilbert spa e and on this domain they satisfy a kind of ontinuity property. DEFINITION 3.6.1

© 2001 by Chapman & Hall/CRC

An

unbounded operator on

H

is a linear map

64

Chapter 3:

( ) of H into H. It is losed if (A) = fv  Av : v 2 D(A)g is a losed subspa e of H  H.

A from a dense subspa e D A

Operators

its graph

O

asionally one may have to deal with operators whi h are not densely de ned. But sin e all of our unbounded operators will have dense domains, we in orporate this property into the de nition. Closed unbounded operators are still ontinuous in the sense that if (vn ) is a sequen e in D(A) and v; w 2 H, then vn ! v and Avn ! w imply that v 2 D (A) and Av = w . In parti ular, if A is losed and D(A) = H then A must be ontinuous and hen e bounded. Thus, not requiring unbounded operators to be de ned everywhere has the advantage that it allows us to formulate a reasonable ontinuity property, namely, graph losure. On the other hand, it has the disadvantage that in general one an neither add nor ompose two unbounded operators (although one an always add or ompose a bounded operator with an unbounded one). There is no general resolution of this diÆ ulty; it must be dealt with ase by ase. Often one rst onstru ts an unbounded operator A whi h is not

losed, and then de nes A to be the losed unbounded operator whose graph is (A). The operator A is alled the losure of A. This pro edure does not automati ally work: one must he k that (A) really is the graph of an operator, whi h amounts to verifying that it ontains no element of the form 0  v with v 6= 0. We all a dense subspa e D  D(A) a ore of A if (A) = (AjD ). Thus, in the last paragraph's onstru tion D(A) would be a ore of A. This is the unbounded version of a dense subspa e of the domain. This

on ept will be useful in Chapter 4. Next we de ne the adjoint of a losed unbounded operator. The sli kest way to do this is in terms of the map V : H  H ! H  H given by V (v  w) = w  v. First we need to prove that the de nition will make sense. PROPOSITION 3.6.2 Let A be a losed unbounded operator on

H

.

Then

(V (A))?

is the

graph of a losed unbounded operator B. Its domain onsists of pre isely

2H i=h i

those ve tors w have

h

Av; w

for whi h the map v

v; Bw

for all v

2

7! h

Av; w

i

is bounded, and we

( ) and w 2 D(B ).

D A

PROOF Let E = (V (A))? . We begin by showing that the proje tion of E onto the rst summand is one-to-one. It is enough to show that 0  w 2 E implies w = 0. But if (0  w) ? ( Av  v) for all v 2 D (A) then w ? v for all v 2 D (A), whi h implies w = 0 as desired.

© 2001 by Chapman & Hall/CRC

65 For all v  w 2 E de ne Bv = w; this is possible by the last paragraph. To show that D(B ) is dense, let w 2 D(B )? . Then (0  w) ? ( Bv  v) for all v 2 D(B ), so that 0  w 2 (V (E ))? . But sin e V is unitary it

ommutes with ?, so (V (E ))? = V ((V (A))? )? = ( (A))?? = (A): Thus w = 0, and we on lude that D(B ) is dense in H. As E = (B ) is

learly losed, B is a losed unbounded operator. If w 2 D(B ) then (w  Bw) ? ( Av  v) for all v 2 D(A), and hen e hAv; wi = hv; Bwi for all v 2 D(A). This implies that the map v 7! hAv; wi = hv; Bwi is bounded. Conversely, if the map v 7! hAv; wi is bounded then it is given by taking the inner produ t with some w0 2 H. Then hAv; wi = hv; w0 i for all v 2 D(A), and hen e (w w0 ) ? ( Av v). So w 2 D(B ). This ompletes the proof. DEFINITION 3.6.3



Then A A is

Let A be a losed unbounded operator on

is the losed unbounded operator whose graph is

self-adjoint if A = A .

H

.

(V (A))? .

It follows from Proposition 3.6.2 that De nition 3.6.3 generalizes the de nition of adjoints of bounded operators. It is important to note that in the unbounded ase the de nition of self-adjoint operators in ludes the provision that D(A) = D(A ). If A is losed, then A = A; this follows immediately from the de nition of A and the fa t that V and ? ommute. We an hara terize A more simply as follows: v1 2 D(A ) and  A v1 = v2 if and only if hv1 ; Aw i = hv2 ; w i for all w 2 D (A). We indi ate the unbounded version of multipli ation operators. Example 3.6.4

Let X be a  - nite measure spa e and let f : X ! C be any measurable fun tion. De ne Mf : L2 (X ) ! L2 (X ) by Mf g = fg , with domain the set of fun tions g 2 L2 (X ) su h that fg 2 L2 (X ). To see that D(Mf ) is dense de ne Xk = fx 2 X : jf (x)j  kg, so 2 2 that X = 1 1 Xk . Then we have fg 2 L (Xk )  L (X ) whenever 2 2 g 2 L (Xk ), so L (Xk )  D(Mf ) for all k. This is enough. The graph of Mf is losed be ause if gn ! g and fgn ! h, both in L2 (X ), then we an pass to a subsequen e for whi h both onverge pointwise almost everywhere. This implies fg = h, so g 2 D(Mf ) and Mf g = h. We have Mf = Mf where f is the pointwise omplex onjugate of f ; this an be veri ed dire tly from Proposition 3.6.2. In parti ular, D(Mf ) = D(Mf ). Also, Mf is self-adjoint if and only if f is real almost everywhere.

S

© 2001 by Chapman & Hall/CRC

66

Chapter 3:

Operators

L2 (X ) repla ed by L2 (X ; X ) for X a measurable Hilbert bundle over X . Exa tly the same results are true with

3.7

The unbounded spe tral theorem

From the point of view of spe tral theory, unbounded self-adjoint operators are quite well-behaved. They arise from spe tral measures on the real line in the same way as bounded self-adjoint operators, the only differen e being that in the unbounded ase the relevant spe tral measure is not supported on a ompa t set. We will prove this result by redu ing to the bounded normal ase via fun tional al ulus: the fun tion f (x) = (x + i) 1 is bounded on the real line, so if A is an unbounded self-adjoint operator we expe t 1 to be a bounded normal operator. Having shown this, f (A) = (A + i) we will then invoke Theorem 3.5.3 to obtain a multipli ation operator realization of (A + i) 1, and nally we will re over the original operator 1 . We pro eed through a series of A by applying the fun tion (x i) lemmas. LEMMA 3.7.1 Let A be an unbounded self-adjoint operator. Then A inverse in the sense that there exists B and

(A  iI )Bv = v for all v 2 H, and B

2 (H ) (  ) B

A



iI has a bounded

2 D(A) = v for all v 2 D(A).

su h that Bv

iI v

Sin e A is losed, it is easy to he k that A  iI (with the same domain as A) is also losed. As in the proof of Corollary 3.2.5 we have k(A  iI )vk2 = kAvk2 + kvk2

PROOF

for all v 2 D(A), so k(A  iI )vk  kvk. Thus proje ting (A  iI ) into the se ond summand yields an isomorphism between (A  iI ) and the range of A  iI . So ran(A  iI ) is losed, and we need only show that it is dense; this will imply that ran(A  iI ) = H, so that B = (A  iI ) 1 is de ned everywhere. B will then be bounded, and in fa t a ontra tion, sin e k(A  iI )vk  kvk. Thus, suppose hw; (A  iI )vi = 0 for all v 2 D(A) = D(A  iI ). Then i = hiw; vi for all v 2 D(A), and hen e w 2 D(A ) = D(A) and  Aw = A w = iw . But then

h

w; Av

h

i w; w

i=h

Aw; w

i=h

w; Aw

i= h

i

i w; w ;

so that w = 0. We on lude that ran(A  iI ) is indeed dense in H.

© 2001 by Chapman & Hall/CRC

67 In the next lemma we let D(A2 ) be the set of ve tors v 2 H su h that 2 is densely v 2 D (A) and Av 2 D (A). It is not a priori lear that A de ned, but we do not need this fa t here. (It will follow from Theorem 3.7.4, however.) LEMMA 3.7.2

+ iI arries D(A2 ) iI D (A + iI ). We have 2 iI )(A + iI )v for all v 2 D (A ).

Let A be an unbounded self-adjoint operator. Then A

( (A + iI )(A

into D A

iI

)

2

arries D (A ) into

and A

) = (A

iI v

Suppose v 2 D(A2 ). Then v and Av both belong to D(A), so that (A + iI )v = Av + iv 2 D(A) = D(A iI ). Similarly, we have (A iI )v 2 D(A + iI ).

PROOF

For v 2 D(A2 ) we have (A + iI )(A so A + iI and A

) = (A + iI )(Av iv) = A2 v + v = (A iI )(Av + iv) = (A iI )(A + iI )v;

iI v

iI

ommute on D(A2 ).

LEMMA 3.7.3

Let A be an unbounded self-adjoint operator. Then

(A

is normal and its adjoint is

iI

)

1.

(A + iI )

1

2 (H ) B

PROOF

The inverses exist by Lemma 3.7.1. Note that (B 1 ) is the transpose of (B ) whenever B is invertible. This, together with the easy fa t that (A + iI ) = A iI , implies that the adjoint of (A + iI ) 1 is (A iI ) 1. To prove normality, let u 2 H. Then u = (A + iI )v for some v 2 D (A), and v = (A iI )w for some w 2 D (A). Sin e v and w are both in D(A) it follows that Aw = v + iw 2 D(A), so that w 2 D(A2 ). By Lemma 3.7.2 we then have u

so

= (A + iI )(A

(A + iI ) 1 (A

Thus (A + iI )

1

iI

and (A

) iI

) = (A

iI

= w = (A

iI

iI w

1

)

u

1

)(A + iI )w; ) 1 (A + iI )

1

u:

ommute.

Now we an prove the spe tral theorem for unbounded self-adjoint operators.

© 2001 by Chapman & Hall/CRC

68

Chapter 3:

THEOREM 3.7.4

A

Let

be an unbounded self-adjoint operator on

H

.

Operators

Then there is a

 on R, a measurable Hilbert bundle X over R and 2 an isometri isomorphism U : L (R; X )  = H su h that A = UMx U 1 . probability measure

PROOF By Lemma 3.7.3 the bounded operator (A + iI ) 1 is normal. Let X = sp(A+iI ) 1 and nd , X , and U as in Theorem 3.5.3 su h that (A + iI ) 1 = UMz U 1 . Then A = UMf U 1 where f (z ) = i + 1=z ; sin e A is self-adjoint, it follows that f (X )  R, so we an use f to identify X with a subset of R. The result follows.

As usual, in the nonseparable ase we an still prove that A is equivalent to a multipli ation operator by using Theorem 3.4.2 in pla e of Corollary 3.4.3. 3.8

Stone's theorem

The time evolution of a quantum me hani al system is modelled by a one-parameter group of unitary operators with a ertain ontinuity property ( f. Se tion 1.4.) These groups also arise in purely mathemati al settings, of ourse. Their general de nition is as follows. DEFINITION 3.8.1

group

is a family

A

weakly ontinuous one-parameter unitary

fUt : t 2 Rg

U0 = I , Us Ut = Us+t whenever s ! t in R.

that

of unitary operators with the properties

for all

s; t 2 R,

and

Us

! Ut

weak operator

This ontinuity requirement is equivalent to the apparently stronger

ondition that s ! t implies Us v ! Ut v for all v 2 H. That is be ause weak operator ontinuity implies hUs t v; v i ! hv; v i = kv k2 ; so

Ut v k2 = 2kvk2

kUs v

2RehUs v; Ut v i ! 0

as s ! t. The reverse impli ation is trivial. Example 3.8.2

A be an unbounded self-adjoint operator on H. Without loss H = L2 (R; X ) and A = Mx . For ea h t 2 R itA . This is a weakly ontinuous iAt and let Ut = e write e = Me

Let

of generality suppose itx

one-parameter unitary group.

We now want to prove Stone's theorem, whi h states that onversely to Example 3.8.2, every weakly ontinuous one-parameter unitary group

© 2001 by Chapman & Hall/CRC

69 is of the form Ut = eitA for some unbounded self-adjoint operator A. The main tool we need is the Fourier transform on R. Let C 1 (R) be the spa e of ompa tlyR supported C 1 fun tions on R. For f 2 C 1 (R) we de ne f^(t) = p12 f (x)e itx dx. We will use the following properties of f^: (a) f 2 C 1 (R) implies f^ 2 L1 (R), (b) (e itx f )^(s) = f^(s + t), and R

(t) = p1 f^(s)^g (t s) ds. ( ) fg 2 We will also need to integrate operator-valued fun tions. This an be done in the following way. DEFINITION 3.8.3 Let X be a  - nite measure spa e. Suppose A : X ! B (H) has the Rproperty that the fun tion x 7! hA(x)v; wi is measurable and satis es jhA(x)v; wij  C for allR v; w 2 H. Then the weak integral R of A is the bounded operator A = A(x) whi h satis es hAv; wi = hA(x)v; wi for all v; w 2 H.

Weak integrals exist by Lemma 3.3.2. Now x a weakly ontinuous one-parameter unitary group fUt g on H. De ne T : C 1 (R) ! B (H) by

T (f ) =

p1

Z

2

f^(t)Ut dt:

Here we are integrating the operator-valued fun tion t 7! f^(t)Ut a

ording to De nition 3.8.3. This uses property (a) of the Fourier transform. LEMMA 3.8.4

The map T is a -homomorphism.

PROOF

Let f

2 C 1 (R). Then f^(t) = f^ (

hT (f)v; wi = p1

Z

t), so

f^ (t)hUt v; wi dt 2 Z 1 f^ (t)hU t v; wi dt =p 2 = hv; T (f )wi:

Thus T (f) = T (f ) .

© 2001 by Chapman & Hall/CRC

70

Chapter 3:

(t) = p1 Now let f; g 2 C 1 (R). Sin e fg 2 ( ) of the Fourier transform), we have

hT (fg)v; wi = p1

Z

2Z Z

R

f^(s)^g (t

Operators

s) ds (property

(t)hUt v; wi dt fg

1 f^(s)^g (t s)hUt s v; U s wi dsdt 2 Z Z 1 ^ = f (s) g^(t)hUt v; U s wi dtds 2 Z 1 =p f^(s)hT (g )v; U s wi dt 2 = hT (f )T (g )v; wi

=

for all v; w 2 H. The appli ation of Fubini's theorem is legitimate be ause f^(s)^g (t s) 2 L1 (R2 ). We on lude that T (fg ) = T (f )T (g ). As T is learly linear, this ompletes the proof. LEMMA 3.8.5

T

extends to a unital

-homomorphism T~ from Bor(R) into B (H) whi h

is ontinuous from the

-topology to the weak operator topology.

PROOF We prove this by rst extending T to C0(R) by norm ontinuity, and then passing to bounded Borel fun tions by the te hnique of Theorem 3.3.3. The main novelty is the veri ation that T is ontinuous in sup norm. To see this, let f 2 C 1 (R) be real-valued and hoose  2 C su h that jj > kf k1 . Then g = (f ) 1 +  1 2 C 1 (R) and we have fg g  1 f = 0. Sin e T is multipli ative it follows that (T (f )

I )(T (g )

 1 I ) = T (fg

g

 1 f ) + I = I:

Thus  62 sp(T (f )). By Proposition 3.2.4 we on lude that kT (f )k  kf k1. For omplex-valued f 2 C (R) we then have kT (f )k2 = kT (f )T (f )k = kT (jf j2)k  kjf j2k1 = kf k21; so T is a ontra tion. Continuous extension of T to the set of bounded Borel fun tions on R is established by an argument similar to the one given in the proof of Theorem 3.3.3. The fa t that T~ is unital follows from the fa t p that 2the 2 fun tions fn (x) = e x =n - onverge to 1R : we have f^n (t) = n2 e nt =4 , and T (fn ) ! I weak operator is a straightforward omputation. This implies that T~(1R ) = I .

© 2001 by Chapman & Hall/CRC

71 We an now prove Stone's theorem. THEOREM 3.8.6

Let

fUtg be a weakly ontinuous one-parameter unitary group on H. A on H su h that

Then there is an unbounded self-adjoint operator

Ut

= eitA .

PROOF By Lemma 3.8.5, T~ de nes an H-valued spe tral measure E on R su h that PE (S ) = T~(S ) for S  R. Find , X , and U as in Corollary 3.4.3, identify H with L2 (R; X ), and let A = Mx . We must

verify that Ut = eitA . It is enough to show that heitA v; Ut wi = hv; wi for all v; w 2 L2 (K ; X ) where K is an arbitrary ompa t interval in R. Fix K , v , and w and let f 2 C 1 (R) satisfy f (x) = eitx for x 2 K . Then e itx f = 1 on K and we have T (e itx f )v = Me itx f v = v , so

hT (f )v; Ut wi = p1

Z

f^(s)hUs v; Ut wi ds 2 Z 1 f^(s + t)hUs v; wi ds =p 2

 = T (e itx f )v; w

= hv; wi: But T (f )v = eitA v , so heitA v; Ut wi = hv; wi, as desired. The operator A is alled the generator of the unitary group fUt g be ause iA is the derivative at t = 0 of the fun tion t 7! Ut in the following sense. By Theorem 3.7.4, assume H = L2 (R; X ) and A = Mx . Then g 2 L2 (R; X ) belongs to D(A) if and only if xg 2 L2 (R; X ). Now 1 lim (Ut g

t!0 t

U0 g )

1 = lim (eitx

t!0 t

1)g:

Sin e kxg k < 1, for any  > 0 we an nd N > 0 su h that g = g1 + g2 where g1 is supported on [ N; N ℄ and kxg2 k  . But (eitx 1)=t ! ix uniformly on [ N; N ℄, while the elementary estimate j(eitx 1)=tj  jxj implies that k 1t (eitx 1)g2 k  . From this it follows that iAg

1 = ixg = lim (Ut g t!0 t

U0 g ):

Thinking of Ut as a multipli ation operator also helps explain our

onstru tion of A. Informally, for any Borel set S  R we have T (S ) =

© 2001 by Chapman & Hall/CRC

72

Chapter 3: Operators R

R

p12 ^S (t)Ut dt, and p12 ^S (t)eita dt = S (a) by Fourier inversion.

So if Ut = Meitx then we ought to have T (S ) = MS . This nonrigorous argument shows why we expe t T to give rise to the spe tral measure asso iated with A = Mx . 3.9

Notes

Similar topi s are overed in [13℄. A shorter proof of a slightly weaker version of the spe tral theorem for bounded operators is given in [58℄. Our approa h to spe tral and multipli ity theory via Hilbert bundles seems to be new. Our proof of Stone's theorem is based on an idea of Ed Eros.

© 2001 by Chapman & Hall/CRC

Chapter 4

The Quantum Plane 4.1

Position and momentum

We now have the ne essary ma hinery to treat what is probably the most fundamental example in quantum me hani s: the one-dimensional parti le. Its lassi al analog was mentioned in Se tion 1.1, and there we pointed out that the phase spa e of this lassi al system an be identi ed with the real plane R2 . The phase spa e of the orresponding quantum system is modelled on the Hilbert spa e L2 (R), and this spa e, together with some asso iated stru ture, plays the role of a \quantum" plane. As we will see in this and later hapters, the omplex, topologi al, measure theoreti , metri , and dierentiable stru tures of the ordinary plane all have quantum analogs. In this se tion we introdu e \ oordinates" and \translations" on the quantum plane. We will begin with the position and momentum operators, whi h orrespond to the oordinate fun tions on R2 (see Se tion 1.1). These operators are best understood in terms of the RFourier transform. Re all that for f C 1 (R) we de ne f^(t) = p12 f (x)e itx dx. We also R require the Fourier inversion formula f (x) = p12 f^(t)eitx dt.

2

2

PROPOSITION 4.1.1

R R f; g C 1 (R). Then f^g^ dt = f g dx. The : f f^ extends to a unitary operator on L2 (R).

F 7! Let

PROOF

We have Z

f^(t)g^(t) dt =

=

p12

Z

Z Z

Fourier transform

f^(t)g (x)eitx dxdt

f (x)g (x) dx:

Fubini's theorem applies be ause the fun tion f^(t)g (x) is in L1 (R2 ). 73 © 2001 by Chapman & Hall/CRC

74

Chapter 4:

The Quantum Plane

This shows that F takes C 1 (R) isometri ally into L2 (R), and it extends to L2 (R) by ontinuity. To see that the extension is surje tive, let f 2 C 1 (R). Then dierentiating under the integral sign (this is legitimate) yields

Z

1 f^0 (t) = p ixf (x)e 2 = i(xf )^(t);

itx

dx

^ n = ( i)n (xn f )^, and hen e f^ 2 C 1 (R). indu tively, we obtain dn f=dt 1 ^ As we also have f 2 L (R) \ L2 (R), it follows that there is a sequen e (gn )  C 1 (R) su h that gn ! f^ in both L1 and L2 norms. Let g~n (x) = gn ( x). Then (~gn )^(x) ! (f^)(x) = f (x) pointwise (sin e gn ! f^ in L1 norm) and (~gn )^ onverges in L2 (R) (sin e g~n onverges in L2 norm), so we must have F g~n ! f in L2 (R). This shows that the range of the extension of F ontains C 1 (R), and together with the fa t that F is isometri this implies F is unitary. We use the same symbol F for the extension of the Fourier transform to L2 (R). As we are now treating F as a map from L2 (R) to itself, the variables x and t an be freely inter hanged. Fix a real number

DEFINITION 4.1.2

h > 0.

The

position operator

Q = Mx on L2 (R). The momentum operator is the unbounded self-adjoint operator P = h F 1 Mx F . is the unbounded self-adjoint operator

The number h is Plan k's onstant. For us its signi an e is that as it approa hes zero the quantum system be omes lassi al. We will return to this point in the next se tion. If f 2 C 1 (R) then integration by parts yields

F



i



p1

Z

df i e dx 2 Z 1 tf (x)e =p 2 = tf^(t);

df (t) = dx

itx

dx

itx

dx

d so F ( i(df=dx)) = Mx F (f ). Thus it makes sense to de ne i dx to be the unbounded self-adjoint operator F 1 Mx F . We an then write P = ih dxd . Physi ally, Q and P are interpreted in the following way. If S is a Borel subset of R then L2 (S )  L2 (R) models the event that the

© 2001 by Chapman & Hall/CRC

75 parti le lies in the region S . That is, a state ve tor f 2 L2 (R) des ribes a parti le whi h de nitely belongs to S pre isely if f is supported on S . Correspondingly, the spe tral proje tion of Q = Mx for the fun tion S is the proje tion operator S (Mx ) = MS with range L2 (S ), and on any state supported in L2 (S ) the observable Q will take values in S with

ertainty. The momentum operator has exa tly the same stru ture, up to the s aling fa tor h  , if one works in the Fourier transform pi ture. This means that the event that the parti le's momentum lies in h  S is modelled by the subspa e F 1 (L2 (S )), and hen e that a state ve tor f des ribes a parti le whose momentum de nitely belongs to h  S pre isely if f^ is supported on S . Re all that self-adjoint operators arise as generators of weakly ontinuous one-parameter unitary groups (Theorem 3.8.6). Sin e bounded operators are more tra table than unbounded operators, it is often easier to deal with the unitaries eisQ and eitP than with Q and P themselves. We have immediately that eisQ = Meisx and eitP = F 1 Meih tx F . But also, if f 2 L2 (R) then

Meih tx F (f ) = eih txF (f ) = F (T h t f ); where Tt is the translation operator Tt f (x) = f (x t). Thus eitP = T h t . Likewise, eitQ = F 1 Tt F . Next we examine the failure of Q and P , and eisQ and eitP , to ommute.

THEOREM 4.1.3

(b)

1

f 2 C (R). Then QP f eisQ eitP = e ih st eitP eisQ for

(a) Let

PQf = ihf . s; t 2 R.

all

PROOF

d (xf ) = i d f ). d f and PQf = i h dx h(f + x dx (a) We have QP f = ihx dx Thus QP f PQf = ihf . (b) If f 2 L2 (R) then eisQ eitP f (x) = eisx f (x + h  t) and eitP eisQ f (x) = is ( x + h t ) is Q it P i h  st it P is e f (x + ht). Thus e e = e e e Q.

Usually in physi s one just says that QP PQ = ihI . This and the fa t that Q and P ea h ommute with themselves are alled the

anoni al ommutation relations (CCRs). Now sin e P and Q are both unbounded, and are not de ned on all of L2 (R), there is a question of what is meant by the expression QP PQ. Here we have dealt with this issue by restri ting ourselves to ve tors f whi h have the property that f 2 D(Q) \ D(P ), Qf 2 D(P ), and P f 2 D(Q). We will dis uss

ommutators of unbounded operators further in Se tion 4.4.

© 2001 by Chapman & Hall/CRC

76

Chapter 4: The Quantum Plane

The existen e of this kind of question shows why it is generally preferrable to work with the bounded operators eitQ and eitP whenever possible. The equation in Theorem 4.1.3 (b), whi h an be thought of as an integral version of the equation in part (a), is alled the Weyl form of the anoni al ommutation relations. These exponentiated operators are signi ant in another way. Observe that if f 2 L2 (R) is supported on a set S | intuitively, its position \ oordinate" lies in S | then e isP =h f = Ts f is supported on S + s. Thus e isP =h plays the quantum plane role of translation in the position variable. Likewise, eitQ=h plays the role of translation in the momentum variable. Just as the operators Q and P a t like oordinate fun tions on the quantum plane, polynomials in Q and P (that is, operators of the form i j aij Q P ) a t like polynomials on the quantum plane. These expressions require some interpretation, sin e Q and P are unbounded.

P

P

PROPOSITION 4.1.4

Let

i j

aij x y

P

P

be a omplex polynomial in two variables, and for f 2 a ij P j Qi f . Let (A) =

1 i j C (R) de ne Af = aij Q P f and Bf = 1 ff  Af : f 2 C (R)g.

(a) We have hAf; g i = hf; Bg i for all f; g 2 C 1 (R). (b) The losure of (A) in L2 (R)  L2 (R) is the graph of an unbounded operator.

PROOF

(a) This follows from self-adjointness of Q and P . (b) The issue is whether the losure of (A) de nes a single-valued operator on L2 (R). It will suÆ e to show that (A) ontains no element of the form 0  f for f 6= 0. Thus, suppose that (fn )  L2(R), fn ! 0, and Afn ! f 2 L2 (R). Then for any g 2 C 1 (R) we have

h

i=h n i!0 i = 0 for all 2 n ! , this means that h

Sin e Af f = 0.

Afn ; g

f

P

f ; Bg

f; g

:

g

1 (R), and hen e

C

In light of part (b) of this proposition, it makes sense to de ne the operator aij Qi P j to be the losure of A. Thus, C 1 (R) is a ore (in the sense of Se tion 3.6) for every operator of the form aij Qi P j . Moreover, it has the pleasant property of being invariant in the sense that Qi P j (C 1 (R))  C 1 (R). Thus, polynomials in Q and P an be added and multiplied in a straightforward manner on C 1 (R). The

© 2001 by Chapman & Hall/CRC

P

77 produ t of any two polynomials an be omputed using the ommutation relation of Theorem 4.1.3 (a). 4.2

The tra ial representation

The realization of Q and P as multipli ation and dierentiation operators on L2 (R) is the most basi , but there are other models whi h bring out ertain features of these operators more learly. For instan e, by

hoosing a suitable orthonormal basis of L2 (R) we an use Theorem 2.3.3 to establish an isometri isomorphism between L2 (R) and l2 (N), in su h a way that a ertain aspe t of the stru ture of Q and P be omes transparent. We will do this in Se tion 4.3. In the present se tion we do something slightly dierent: we produ e a unitary operator W on L2 (R2 )  = L2 (R) L2 (R) su h that the operators 1 1 W (Q I )W and W (P I )W have a spe ial symmetri al form for whi h there is a reasonable limit as h ! 0. These operators are equivalent not to Q and P , but rather to Q I and P I . There is no physi al signi an e in this distin tion; in ee t we are treating a

omposite system where one subsystem is trivial. Our reasons for doing this are purely mathemati al. Loosely speaking, this representation is a symmetri al ombination of the ordinary representation of Q and P and the Fourier transform of the ordinary representation.

R2 )

De ne unitary operators Us and Vt on L2 (

DEFINITION 4.2.1

(s; t 2

R) by

1 hs) 2

(

) = eisx1 f (x1 ; x2

(

1 ) = eitx2 f (x1 + h t; x2 ): 2

U s f x 1 ; x2

and Vt f x1 ; x2

For (t1 ; t2 ) 2 R2 de ne the translation operator Tt1 ;t2 on L2 (R2 ) by (

Tt1 ;t2 f x1 ; x2

) = f (x1

t1 ; x2

)

t2 :

Then in terms of multipli ation and translation operators we an write Us = Meisx1 T 1 h and Vt = Meitx2 T 21 h ti , where i = (1; 0) and j = (0; 1) 2  sj are the standard basis ve tors in R2 . PROPOSITION 4.2.2

R

There is a unitary operator W on L2 ( 2 ) su h that for all have Us = W 1 (eisQ I )W and Vt = W 1 (eitP I )W .

© 2001 by Chapman & Hall/CRC

s; t

2 R we

78

Chapter 4: The Quantum Plane

Note that eisQ I and eitP I are operators on L2 (R2 ) whi h a t trivially on the se ond variable. De ne a unitary W1 on 2 2 L (R ) by x x x + x  1 p 2 ; 1p 2 : W1 f (x1 ; x2 ) = f 2 2 Then

PROOF

W1

1

(eisQ I )W1 = W1

and

1

W1

(eitP



)

I W1

1

= W1

Meisx1 W1

1

h  ti W1

T

= Meis(x1 +x2 )=p2 =T

p

h  t(i+j)= 2 :

Next, let W2 be the inverse Fourier transform in the se ond variable; that is, W2 = I F 1 . Then W2

and W

2

1

1

p

h  t(i+j)= 2 W2

T

Finally, de ne a unitary

W3

by

r

(W3 f )(x1 ; x2 ) = then W3

and W3

1

1

= Meisx1 p2 Tsj=p2

p

eis(x1 +x2 )= 2 W2

M

p

p

= Meih tx2 =p2 T

 h f 2

eisx1 = 2 Tsj= 2 W3

M

p

p



2

x1 ;

p1

2



hx2 ;

= Meisx1 T 21 h sj = Us

 ti= 2 W3 eih tx2 = 2 T h

M

p1

p

h  ti= 2 :

= Meitx2 T

1  ti 2h

= Vt :

So W = W1 W2 W3 is the desired unitary. It follows that W 1 (Q I )W and W 1 (P I )W are the in nitesimal generators of the unitary groups fUs g and fVt g, respe tively. We an des ribe these operators more on retely: the rst is Mx1 + i h2 x 2 and the se ond is Mx2 i h2 x 1 . This, then, is the tra ial representation, so- alled for reasons to be explained in Se tion 5.6: the position operator is modelled by Mx1 + h   , and the h   , the momentum operator is modelled by M i i x2 2 x2 2 x1 unitary groups they generate are fUs g and fVt g. We have assumed throughout that h  > 0, but let us now take h = 0. Then the position and momentum operators be ome Mx1 and Mx2 , multipli ation by the

oordinate fun tions, whi h are the lassi al position and momentum observables. This is the sense in whi h Q and P approa h the lassi al

oordinate variables as h  ! 0. In the representation on L2 (R) of Se tion 4.1 nothing similar happens and the h  ! 0 limit is rather mysterious.

© 2001 by Chapman & Hall/CRC

79 Re all that the unitary operators e isP =h and eitQ=h play the role of translation in the position and momentum variables. However, the operators W 1 e isP =h W and W 1 eitQ=h W do not have limits as h ! 0. We will resolve this diÆ ulty in Se tion 5.4 by showing that there is a sense in whi h e isP =h and eitQ=h are equivalent to the ordinary

oordinate translations Tsi and Ttj on L2 ( 2 ).

R

4.3

Bargmann-Segal spa e

R

There is a ni e orthonormal basis of L2 ( ) that we an use to establish an isometri isomorphism with l2 ( ) whi h puts Q + iP and Q iP , the operators analogous to the lassi al omplex variables z = x + iy and z = x iy , in a simple form. It is expressed in terms of the Hermite polynomials 2 2 dn Hn (x) = ( 1)n ex n e x : dx The rst few Hermite polynomials are H0 (x) = 1, H1 (x) = 2x, H2 = 4x2 2. The Hermite fun tions are the fun tions

N

p

hn (x) = Nn Hn (x= h)e

x2 =2h

;

where Nn is a normalizing fa tor hosen so that khn k = 1. Spe i ally,

Nn =

1 : ( h)1=2 2n n!

p

Moreover, the hn are orthogonal; this an be proven in the following way. First, verify that the Hn satisfy 

d e dx

x2 dHn

dx



= 2ne

x2

Hn :

Then a short omputation shows that

d  e dx

x2



(Hm Hn0

0 Hn ) = (2m Hm

and from this it follows that Z

(2m

2n)

1

1

hm hn dx = =

2n)e

Z 1 p hNm Nn (2m 1 p x2

hNm Nn e

x2

2n)e

(Hm Hn0

Hm Hn ;

x2 H

m Hn dx

1

0 Hn ) = 0: Hm 1

So hhm ; hn i = 0 for m 6= n. (Note that hn is real, hen e the absen e of a omplex onjugate in our omputation of the inner produ t.)

© 2001 by Chapman & Hall/CRC

80

Chapter 4:

The Quantum Plane

Sin e ea h Hn is a polynomial of degree exa tly n, any polynomial an be written as a linear ombination of the Hn . Therefore any f 2 L2 (R) 2 whi h is orthogonal to every hn is a tually orthogonal to p(x)e x =2h for every polynomial p(x). But then (e

x2 =2 h

f )^ =

p1

2 1 =p 2

Z XZ e

n

x2 =2h

e

f (x)e

x2 =2h

itx

f (x)

dx

( itx)n dx = 0: n!

2

From this it follows that e x =2h f = 0, and hen e f = 0, almost everywhere. We on lude that the fun tions hn onstitute an orthonormal basis of L2 (R). Thus, there is a natural isomorphism between L2 (R) and l2 (N) whi h takes (hn ) to the standard basis (en ) of l2 (N). We re ord this fa t. PROPOSITION 4.3.1 The Hermite fun tions

hn l2 (N). map

7! en

(hn )

L2 (R). 2 from L (R)

are an orthonormal basis of

extends to an isometri isomorphism

The onto

The higher-dimensional ase is worth mentioning here. In general, the 2n-dimensional version of the quantum plane is simply obtained by taking nth tensor powers. For the L2 (R) model this would result in a model on L2 (Rn ). The nth tensor power of l2 (N) was des ribed in Example 2.5.7 (a): it is the dire t sum of the k -fold symmetri tensor powers of l2 (f1; : : : ; ng)  = Cn . In other words, it is a symmetri Fo k n spa e over C , as in De nition 2.5.8. For this reason, we all l2 (N) the Fo k spa e model of the quantum plane. The Fo k spa e model gives a parti ularly ni e representation of the operators Q iP and Q + iP . Ignoring domain issues for the moment, we have d (Q iP )hn = xhn h hn = 2h(n + 1)hn+1 dx and p d (Q + iP )hn = xhn + h hn = 2hnhn 1 dx (with (Q + iP )h0 = 0). This an be shown using the re ursion relations Hn0 (x) = 2nHn 1 (x) = 2xHn (x) Hn+1 (x). Thus, Q iP and Q + iP resemble the unilateral shift and the ba kward shift mentioned in Se tion p p 3.1; but they are unbounded be ause of the fa tors of n + 1 and n. Before dis ussing these operators further, we introdu e another representation in whi h they be ome even simpler.

p

© 2001 by Chapman & Hall/CRC

81

Let  be 21h e jzj =2h times Lebesgue measure on C. This is a probability measure. The Bargmann-Segal spa e is the spa e BS of all analyti fun tions in L2 (C; ). That is, BS onsists of all entire analyti fun tions f (z ) on the omplex plane su h that 2

DEFINITION 4.3.2

kf k

2

=

1

Z

2  h

jf (z )j2 e

jzj2=2h dz

is nite. The Bargmann-Segal spa e is also alled Fis her spa e or Fo k spa e, but the latter is a slight misnomer; as we noted earlier, l2 ( ) is the a tual Fo k spa e here (although the two are naturally isomorphi , as we will now show).

N

THEOREM 4.3.3

BS

is a Hilbert spa e. It has an orthonormal basis onsisting of the fun tions n n (z ) =

z

p

(2 h)n n!

(n = 0; 1; : : :).

PROOF

C

Sin e BS is a subspa e of L2 ( ; ), to prove that it is a Hilbert spa e we need only show that it is losed. Let f be an analyti fun tion on . By standard omplex analysis f (z ) is the average value of f on any disk entered at z , so using the Cau hy-S hwarz inequality we have

C

jf (z )j =

Z R2 B (z;R)

1

1

f (w) dw

Z

jf (w)j2 e  R2 B (z;R) p  R2h e(jzj+R)2 =4h kf k;

jw j

2 =2 h

!1=2

dw

Z

e jw j

2 =2 h

B (z;R)

!1=2

dw

where B (z ; R) is the disk of radius R entered at z . It follows that any sequen e of analyti fun tions whi h onverges in L2 norm also

onverges uniformly on ompa t sets, and therefore its limit (whi h we already know is in L2 ( ; )) is also analyti . So BS is omplete. The fun tions z n are orthogonal, and a short omputation shows that kz nk2 = (2h)n n!, so the fun tions n are orthonormal. In fa t, for any R > 0 the restri tions of the fun tions z n to B (0; R) are orthogonal.

C

© 2001 by Chapman & Hall/CRC

82

Chapter 4: The Quantum Plane

P Now if f 2 BS has a Taylor expansion f (z ) = an z n , then this series

onverges to f uniformly on B (0; R), and hen e it also onverges in L2 (B (0; R); jB (0;R) ). Thus X kf jB(0;R)k2 = jan j2 kz njB(0;R) k2 : P 2 n2 2 Taking the limit P as nR ! 1 yields kf k = jan j kz k . This shows that the sum an z onverges in BS , and sin e it onverges uniformly on ompa t sets to f , it must onverge to f in BS . Thus, we have shown that the fun tions z n generate BS , and we on lude that (n ) is an orthonormal basis.

Having identi ed a natural orthonormal basis for BS , we an now de ne an isomorphism between BS and L2 (R) in the same way that we de ned an isomorphism between L2 (R) and l2 (N) earlier. Namely, let B : L2(R) ! BS be the surje tive isometri isomorphism whi h takes hn to n . This map is alled the Bargmann transform. Now we return to the operators Q iP and Q + iP . More pre isely, we want to look at A+ = p12h (Q iP ) and A = p12h (Q + iP ), but rst we need to de ne them rigorously. This ould be done by the pres ription given following Proposition 4.1.4, but it is more onvenient for us to take a dierent (though equivalent) approa h. p Our heuristi

omputation made earlier yielded A+ hn = n + 1hn+1 p and A hn = nhn 1 ; thus the operators B A+ B 1 and B A B 1 on BS should satisfy p BA+ B 1z n = p(2h)n n!BA+B 1n 1 = (2h)n (n + 1)!n+1 = p z n+1 2h and p BA B 1z n = p(2h)n n!BA B 1np = (2h)n n
n!n 1 = 2hnz n 1 :

p d . It is easy That is, they should just be the operators p12h Mz and 2h dz to de ne these operators rigorously. Let Mz

DEFINITION 4.3.4

domain

d and let dz

2 B (BS ) be multipli ation by z , with

= ff 2 BS : zf 2 BSg 2 B (BS ) be dierentiation, with domain D(Mz )

D

d dz

© 2001 by Chapman & Hall/CRC

= ff

2 BS : f 0 2 BSg:

83

p

d B. These are, We also de ne A+ = p12h B 1 Mz B and A = 2hB 1 dz respe tively, the reation and annihilation operators on L2 (R). d ) both ontain all polynomials in z , so these Clearly, D(Mz ) and D( dz operators are densely de ned. Moreover, C 1 (R) is a ore for both A+ and A , so this de nition is onsistent with the omment made after Proposition 4.1.4. This an be seen by nding, for ea h n, a sequen e (fi )  C 1 (R) su h that fi ! hn and A fi ! A hn . By passing to the l2 (N) model (see the proof of the next result) it is easy to see that the span of the hn is a ore for both A+ and A , whi h implies that C 1 (R) is also a ore. d Evidently we have B (Q iP )B 1 = Mz and B (Q + iP )B 1 = 2h dz . It 1 is also possible to write B (Q+iP )B = P Mz, where P is the orthogonal proje tion of L2 (C; ) onto BS and z = x iy is the onjugate variable to z . Thus, the operators Q iP and Q + iP have parti ularly elegant representations in the Bargmann-Segal pi ture. We will develop this point in the next se tion. Let us observe here, though, that there is a slight mismat h in that we would expe t Q + iP to orrespond to z = x + iy and Q iP to orrespond to z = x iy . This would have happened if we de ned BS to be the spa e of antianalyti fun tions in L2 (C; ). Doing so leads to the formulas B (Q + iP )B 1 = P 0 Mz and B (Q iP )B 1 = Mz where P 0 = I P . However, we have hoosen instead to adopt the traditional de nition of BS and a

ept the resulting minor disparity. The meaning of the terms \ reation" and \annihilation" will be ome more lear in Se tion 7.1, when we interpret the summands of Fo k spa e as orresponding to dierent numbers of parti les. The terms should not be taken too literally, however, as these are not physi al operations whi h

an a tually be performed; they are not unitary.

PROPOSITION 4.3.5

d and dz are losed operators, they have the same domain, and p p12h Mz = 2h dzd .

Both

Mz

PROOF

The easiest way to prove this is by passing to l2 (N) via the orresponden e n $ p en . The operator A on l2 (N) orresponding 1 to p2h Mz satis es Aen = n + 1en+1 , with domain D(A)

and the operator

= f(an ) 2 l2 (N) : B

X

orresponding to

© 2001 by Chapman & Hall/CRC

(n + 1)jan j2 < 1g;

p

d 2h dz satis es

Ben

=

pne

n 1

,

84

Chapter 4: The Quantum Plane

with domain D (B )

= f(an ) 2 l2(N) :

X nja j n

2


N yields N 1 hAaN ; bi = pnaNn 1bn = njbn j2 = kaN k2 ;

X

X

n=1

n=1

and sin e k k ! 1 as N ! 1, this shows that the map a 7! hAa; bi is not bounded. Thus D(A ) = D(B ), and for b 2 D(B ) we have 1 hAa; bi = pnan 1bn = ha; Bbi: aN

X

n=1

p

So A = B , as laimed. It follows that p12h Mz = 2h dzd . We on lude our treatment of Bargmann-Segal spa e with a brief dis 2 h . For any w 2 C,

ussion of the reprodu ing kernels ew (z ) = ezw= 2 1 j w j
j z j = 2 h we have jew (z )j  e , so the integral 2h jew (z )j2 e jzj =2h dz

onverges, and hen e ew 2 BS . Now (z w)n w n n (z ); ew ( z ) = = (2h)n n! (2h)n n!

R

X Xp P P p(2h)nn!ann in BS we have so for any f = an z n = X hf; e i = a wn = f (w): w

n

This is the meaning of the term \reprodu ing kernel": taking the inner produ t of a fun tion in BS with ew reprodu es its value at w.  w . So even though the ve tors ew are not We have 2h dzd ew = we orthogonal, they are eigenve tors of the (non-normal) operator 2h dzd . Moreover, for any f; g 2 BS we have

Z

hf; ew ihew ; gi d(w) =

© 2001 by Chapman & Hall/CRC

Z

f (w)g (w) d(w)

= hf; gi:

85 That is,

f=

Z

hf; ew iew d(w);

where the integral is taken in the weak sense (meaning just what was said in the previous line). So the family (ew ) behaves rather like a basis; f. the rst assertion of Corollary 2.3.4, repla ing the sum with an integral. The fun tions ew might be thought of as the \points" of the quantum plane, an idea that will gain more support in the next se tion. We re ord these properties in the following result. PROPOSITION 4.3.6

 2 h . Then e 2 BS and w 2 C let ew (z ) = ezw= w (a) f (w ) = hf; ew i for all f 2 BS and w 2 C; d e = we (b) 2 h dz  w for all w 2 C; and Rw ( ) f = hf; ew iew d(w ) for all f 2 BS , where the integral is For ea h

taken in

the weak sense.

In the L2 (R) pi ture, the eigenfun tions of Q + iP are the fun tions 2 e (x ) =2h for  2 C. They are alled oherent states. A short omputation shows that 2 2 (Q + iP )e (x ) =2h = e (x ) =2h ;

d , this shows so the eigenvalue is . Re alling that B (Q + iP )B 1 = 2h dx that up to normalization the Bargmann transform takes the oherent states to the reprodu ing kernel fun tions ew with w =  . 4.4

Quantum omplex analysis

If bounded self-adjoint operators on L2 (R) orrespond to bounded measurable real-valued fun tions on the plane, then general bounded operators on L2 (R) should orrespond to bounded measurable omplex-valued fun tions on the plane, be ause any bounded operator A an be written as A = Re A + iIm A. Re all that the unitary groups e isP =h and eitQ=h play the role of

oordinate translations on the quantum plane. We an use these groups to take derivatives of operators on L2 (R) in the following way. For any bounded operator A on L2 (R), the operators e isP =h AeisP =h and eitQ=h Ae itQ=h are thought of as its shifts by s and t units in the horizontal and verti al dire tions, respe tively. Therefore the \derivative of A in the horizontal dire tion" should be the operator

e isP =h AeisP =h s!0 s lim

© 2001 by Chapman & Hall/CRC

A

86

Chapter 4:

The Quantum Plane

and its \derivative in the verti al dire tion" should be lim

t!0

e

itQ=h  Ae itQ=h 

A

t

:

Heuristi ally, sin e eB = I + B +


, we expe t that lim

s!0

e

isP =h  AeisP =h 

A

s

s!0 i

= where [P ; A℄ = P A lim

t!0

A

e

h

i h s

(I

= lim

P)

( + hi sP )

A I

s

[P ; A℄

P is the ommutator of P and

itQ=h  Ae itQ=h 

A

t

A

, and similarly

A

= [Q; A℄:  h i

(As usual, there are interpretational issues here due to the fa t that P and Q are not de ned on all of L2 (R).) We now want to determine whi h operators on L2 (R) orrespond to analyti fun tions. Classi ally, a C 1 fun tion f : C ! C is analyti if f y

=i

f x

;

this is a form of the Cau hy-Riemann equations. A

ording to the last paragraph, the quantum analog of this ondition is i

h

1 [Q; A℄ = [P ; A℄;  h

or equivalently [Q + iP ; A℄ = 0. Thus, operators on L2 (R) whi h ommute with the annihilation operator an be viewed as satisfying a quantum version of the Cau hy-Riemann equations. The rst thing we need to do is to make this ondition more rigorous. Sin e we are going to allow A to be unbounded, we must say what we mean by the ommutator of two unbounded operators. The appropriate de nition here is the following. DEFINITION 4.4.1

Let D be a dense subspa e of a Hilbert spa e

and let A and B be unbounded operators on





H

in the domains of A, A , B, and B . Then we say that A and B

relative to

D

h



Av; B w

for all v; w

2

D.

© 2001 by Chapman & Hall/CRC

i=h



H

. Suppose D is ontained

Bv; A w

i

ommute

87 If A and B are both bounded, ommuting relative to D is equivalent to the ondition h(BA AB )v; wi = 0; sin e we assume D is dense, it follows that A and B ommute in the usual sense, and onversely. Also, note that A and B ommute relative to D if and only if A and  B ommute relative to D. Thus, the quantum analyti ity ondition is morally equivalent to the ondition [Q iP ; A℄ = 0, whi h we interpret as saying that A is antianalyti . Let us now pass to the Bargmann-Segal spa e BS and repla e Q + iP d . Now d ertainly ought to ommute with itself, or moreover with 2h dz dz d , or indeed any fun tion of d to the extent with any polynomial in dz dz that su h an operator an be de ned. In fa t, it turns out that the only d in a reasonable sense are operators operators whi h ommute with dz whi h arise from it by a kind of fun tional al ulus. The next result des ribes the relevant lass of fun tions. Re all the reprodu ing kernels  2 h introdu ed at the end of the last se tion. ew (z ) = ezw= PROPOSITION 4.4.2 Let

 be an entire analyti fun tion on C. Then ew 2 BS for all w 2 C j(z )j = O(ejzj2 =4h N jzj ) for all N > 0.

if and only if

PROOF

Suppose j(z )j = O(ejzj =4h N jzj ). Then Z kew k2 = 21h j(z )ew j2 e jzj2=2h dz Z  C (ejzj2=4h N jzjejwjjzj=2h)2 e jzj2=2h dz Z = C e(jwj=h 2N )jzj dz: 2

We an ensure that this last integral is nite by taking N > jwj=2h. This shows that ew 2 BS . Conversely, suppose ew 2 BS for all w 2 C. Fix N > 0 and let wk = p 2h 2Ne2ik=4 for k = 0; 1; 2; 3. Then the Cau hy-S hwarz inequality yields j(z )ezwk =2h j = jhewk ; ez ij  kewk kejzj2=4h

sin e kez k = hez ; ez i1=2 = ejzj =4h . Also, for any z 2 C one of the wk satis es jArg(wk ) Arg(z )j  =4, and for this wk we have 2

Re z wk Thus

 p1 jz wk j = 2hN jz j: 2

j(z )ezwk =2h j = j(z )jeRe zwk =2h  j(z )jeN jzj;

© 2001 by Chapman & Hall/CRC

88

Chapter 4:

The Quantum Plane

and ombining this with the earlier estimate on j(z )ezwk =2h j yields

j(z )j  Cejzj2 =4h with C = maxfkewk k : 0  k  3g.

N jzj

Thus, for any  satisfying the stated growth ondition, the multipli ation operator M ontains all of the fun tions ew in its domain. (Naturally, we take D(M ) to be ff 2 BS : f 2 BSg.) We all su h  Bargmann-Segal multipliers. We an regard M as the operator obtained by applying  to Mz by a sort of fun tional al ulus, and its adjoint M as obtained by applying  to 2h dzd . Let R = spanfew : w 2 Cg. LEMMA 4.4.3

2  be an analyti fun tion on C su h that j(z )j = O(ejzj =4h N jzj ). Then R is a ore for M and M ew = (w )ew for all w 2 C.

Let

PROOF

Let w 2 C. Then for any f 2 D(M ) we have

hM f; ew i = (w)f (w) = hf; (w)ew i: This shows that ew 2 D(M ) and M ew = (w)ew . To show that R is a ore for M , let A be the losure of M jR . For any f 2 D(A ), if g = A f then g (w) = hg; ew i = hf; Aew i = (w)hf; ew i = (w)f (w)

for all w 2 C. This shows that f 2 D(M ) and A f = M f . So (A )  (M ), and it follows from the de nition of adjoints that (M )  (A ) = (A). But it is lear that (A)  (M ), so A = M , and this shows that R is a ore for M . THEOREM 4.4.4

Let for

A be a losed, unbounded operator on BS and suppose R is a ore A and R  D(A ). Then the following are equivalent:

d ommute relative to R; A and dz (b) ew is an eigenve tor of A for all w 2 C; and ( ) A = M for some entire analyti fun tion  on C satisfying j(z )j = 2 O(ejzj =4h N jzj ). (a)

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89

PROOF d (a) ) (b): Suppose A and dz v; w 2 C we have

ommute relative to

R.

Then for any

hAev ; zew i = h2h dzd ev ; A ew i = hvev ; A ew i = hvAev ; ew i:

d ) and 2 d Ae = vAe . That This shows that Aev 2 D(Mz ) = D(2h dz h dz v v d with eigenvalue v, and it follows that is, Aev is an eigenve tor of 2h dz Aev is a onstant multiple of ev . So ev is an eigenve tor of A. (b) ) ( ): Let  = A e0 and say Aew = ew . Then

hev ; Aew i = hMev e0; ew i = he0 ; Mev ew i =  he0 ; ev (w)ew i = ev (w)he0 ; Aew i = (w)ev (w) = hev ; ew i:

This shows that A ev = M ev , and so j(z )j = O(e(jzj =2 N jzj)=2h ) by Proposition 4.4.2. It also follows that M = A on R, and sin e R is a ore for both operators (by assumption and the lemma), we on lude that A = M . d om( ) ) (a): This follows from the fa t that M and Mz = 2h dz mute relative to R. 2

It is unknown whether the ondition that R be a ore for A an be repla ed by the weaker requirement that R  D(A). In any ase, subje t to reasonable domain onditions, the unbounded operators whi h satisfy the quantum version of the Cau hy-Riemann equations are pre isely those of the form M where  is an analyti fun tion satisfying the stated growth ondition. We note again that the slightly unnatural appearan e of an adjoint in this on lusion ould be remedied by modifying the de nition of BS ; see the omment following De nition 4.3.4. The next result, whi h is a quantum version of Liouville's theorem, follows from Theorem 4.4.4 and the lassi al Liouville theorem. COROLLARY 4.4.5 Let

A

the sense that Then

4.5

L2 (R) and suppose [A ; A℄ = 0 in i = hA f; A gi for all f; g 2 D(A+ ) = D(A ).

be a bounded operator on

h

Af; A+ g

A is a s alar multiple of the identity. Notes

The main ideas in Se tion 4.1 an be found in any standard book on quantum me hani s; for instan e, see [63℄ and [8℄ for physi al and mathemati al perspe tives, respe tively. Another onstru tion whi h is also

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90

Chapter 4: The Quantum Plane

alled the \quantum plane" is given in [47℄. The tra ial representation des ribed in Se tion 4.2 is based on a more general onstru tion in [60℄. See e.g. [28℄ for a more thorough treatment of the Fourier transform. Se tion 4.3 follows the treatment of [29℄. For more on oherent states see [1℄, and for more on the property des ribed in Proposition 4.3.6 ( ) see [15℄. Se tion 4.4 is based on [61℄. Proposition 4.4.2 is from [54℄; see also [55℄.

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Chapter 5

C*-algebras 5.1

The algebras

C (X )

We have determined that bounded real-valued fun tions on a set orrespond to bounded self-adjoint operators on a Hilbert spa e. Sin e any bounded operator an be written in the form A + iB with A and B self-adjoint, there is a sense in whi h bounded omplex-valued fun tions on a set orrespond to general bounded operators. Thus, any stru ture on a set X whi h an be des ribed in terms of real- or omplex-valued fun tions is likely to have a Hilbert spa e analog de ned the same way in terms of operators. This is the idea that we will use, in this hapter and the next, to de ne quantum versions of topologies and measures. The natural fun tional analyti obje t orresponding to a ompa t Hausdor topologi al spa e X is the spa e of ontinuous omplex-valued fun tions C (X ). The goal of this se tion is to relate topologi al properties of X to algebrai properties of C (X ). As we will see, there is a very

omplete orresponden e between the two. We begin with a de nition of the relevant algebrai on epts, given here in a general form whi h will also a

omodate the non ommutative setting that we dis uss later. DEFINITION 5.1.1 Let A and B be Bana h spa es equipped with a produ t and an involution. We say they are unital if they possess multipli ative units, whi h we then denote IA and IB . B is a C*-subalgebra of A if it is a losed subspa e of A and ; 2 B implies  ;  2 B. If A is unital then B is a unital C*-subalgebra if it is a C*-subalgebra and ontains IA . B is a C*-ideal of A if it is a C*-subalgebra of A with the property that  2 A and 2 B imply  ;  2 B. A bounded linear map  : A ! B is a -homomorphism if  ( ) = ()( ) and ( ) = () for all ; 2 A. It is unital if (IA ) = IB .

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Chapter 5: C*-algebras

(This only makes sense if A and B are both unital.) A -isomorphism is an isometri -homomorphism. If A is unital, we de ne its spe trum to be the set sp(A) of all unital -homomorphisms from A into C, endowed with the weak* topology it inherits from A . More generally, we will use the term \(unital) -subalgebra" for a possibly un losed subspa e whi h ( ontains IA and) is stable under formation of produ ts and adjoints. We will also use the term \homomorphism" in this more general ontext. In the present se tion A and B will be C(X) spa es, with produ t being the pointwise produ t of fun tions and involution being the pointwise

omplex onjugate. Every C(X) has a unit IC (X ) = 1X , the fun tion whi h is onstantly 1 on X. In the ase of C(X) spa es, the quali ation that -homomorphisms be bounded is super uous. In fa t, every unital -homomorphism  : C(X) ! C(Y ) is automati ally a ontra tion. To see this, let f 2 C(X) and hoose  2 C su h that jj > kf k1 ; then f 
1X is invertible in C(X), so (f 
1X ) = (f) 
1Y is invertible in C(Y ). Thus (f)(y) 6=  for all y 2 Y , and we on lude that k(f)k1  kf k1. C*-subalgebras of C(X) are related to quotients of X in the following way. Example 5.1.2

Let X and Y be ompa t Hausdor spa es and let  : X ! Y be a ontinuous surje tion. Then A = ff Æ  : f 2 C (Y )g is a unital C*-subalgebra of C (X ) and the omposition map C : f 7! f Æ  is a -isomorphism of C (Y ) onto A.

This onstru tion is general, as the following proposition shows. PROPOSITION 5.1.3

Let X be a ompa t Hausdor spa e and let A be a unital C*-subalgebra of C(X). Then there exists a ompa t Hausdor spa e Y and a ontinuous surje tion  : X ! Y su h that A = C (C(Y )). For x; y 2 X set x  y if f(x) = f(y) for all f 2 A. Let Y be the quotient spa e with the quotient topology, and let  : X ! Y be the quotient map. Then Y is ompa t be ause it is a ontinuous image of a ompa t spa e. To see that it is Hausdor let [x℄; [y℄ 2 Y be distin t; then there exists f 2 A su h that f(x) 6= f(y). Hen e there exist disjoint open sets O; O0  C su h that f(x) 2 O and f(y) 2 O0 , and f 1 (O) and

PROOF

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93

f 1 (O0 ) give rise to disjoint open sets in Y whi h respe tively ontain [x℄ and [y ℄. De ne C : C (Y ) ! C (X ) by C f = f Æ . Then C is a -

isomorphism, and its range ontains A. (Any fun tion in A lifts to a well-de ned fun tion on Y , whi h is ontinuous by the de nition of the quotient topology.) Thus C 1 maps A isometri ally into a C*subalgebra of C (Y ) whi h separates points. By the Stone-Weierstrass theorem, C 1 takes A onto C (Y ), and hen e C maps C (Y ) onto A. The requirement that A  C (X ) be unital is not essential. If 1X we an work instead with the unital algebra

62 A,

A+ = ff + a
1X : f 2 A and a 2 Cg: Then Proposition 5.1.3 implies that A+  = C (Y ) for some quotient Y of X . Sin e A  A+ separates points in Y , the Stone-Weierstrass theorem

implies that it orresponds to the set of ontinuous fun tions on Y whi h vanish at some distinguished point y0 . Thus, A  = C0 (Y fy0 g), the ontinuous fun tions whi h vanish at in nity on the lo ally ompa t spa e Y fy0 g. Indeed, removing a point from a ompa t spa e always leaves a lo ally ompa t spa e, and

onversely, every lo ally ompa t spa e has a one-point ompa ti ation. Nonunital C*-subalgebras orrespond to lo ally ompa t spa es in the same way that unital C*-subalgebras orrespond to ompa t spa es. We regard the nonunital ase as an illiary to the unital ase in this way. Next, we relate C*-ideals of C (X ) to losed subsets of X . Example 5.1.4

Let K be a losed subset of a ompa t Hausdor spa e X and let I = ff 2 C (X ) : f jK = 0g. Then I is a C*-ideal of C (X ). PROPOSITION 5.1.5

Let I be a C*-ideal of C (X ). Then there exists a losed subset K of X su h that I = ff 2 C (X ) : f jK = 0g.

For x; y 2 X set x  y if f (x) = f (y ) for all f 2 I . Observe that any point x at whi h some f 2 I is nonzero annot be equivalent to any other point. For if y 6= x then there exists g 2 C (X ) su h that g (x) = 1 and g (y ) = 0, so fg (x) 6= fg (y ), and thus x and y are not equivalent. Let I + = ff +a
1X : f 2 I and a 2 Cg. This is a unital C*-subalgebra of C (X ). De ne Y to be the quotient spa e as in Proposition 5.1.3 and PROOF

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Chapter 5: C*-algebras

let C : C (Y ) ! I + be the orresponding -isomorphism. Also let K = fx 2 X : f (x) = 0 for all f 2 Ig. It is lear that every f 2 I satis es f jK = 0. Conversely, if f jK = 0 then f lifts to a ontinuous fun tion on Y , and hen e f 2 C (C (Y )) = I + . But sin e f jK = 0 we must have f 2 I , so we on lude that I = ff 2 C (X ) : f jK = 0g. The pre eding result allows us to ha terize C*-quotients of C (X ). PROPOSITION 5.1.6

Let K be a losed subset of X and let I = ff C (X )=I is -isomorphi to C (K ).

2 C (Y ) : f jK = 0g. Then

PROOF Let  : C (X ) ! C (K ) be the restri tion map. It is lear that  is a -homomorphism and has kernel equal to I . For any f 2 C (K ), by the Tietze extension theorem there exists g 2 C (X ) with kgk1 = kf k1 and g = gjK = f . This shows that  is isometri and onto. Now we turn to the spe trum. This is the tool that is used to show that the spa e X an a tually be re overed from the algebra C (X ), and thus to establish that C (X ) ontains exa tly as mu h information as X . For x 2 X let x^ : f 7! f (x) be the evaluation map on C (X ). PROPOSITION 5.1.7

Every ompa t Hausdor spa e X is homeomorphi to sp(C (X )) via the

orresponden e x $ x^.

PROOF

Let X be a ompa t Hausdor spa e. For any x 2 X the map x^ : f 7! f (x) is a unital -homomorphism from C (X ) to C. Conversely, if ! : C (X ) ! C is any unital -homomorphism then its kernel is a odimension one C*-ideal, and hen e by Proposition 5.1.5 there exists x 2 X su h that ker(! ) = ff 2 C (X ) : f (x) = 0g. (The set K in Proposition 5.1.5 must onsist of exa tly one point in order for ker(! ) to have odimension one. This follows from Proposition 5.1.6.) Then for any f 2 C (X ), letting a = f (x) we have ! (f ) = ! (f

a
1X + a
1X ) = ! (f

a
1X ) + a = a = f (x)

sin e f a
1X vanishes at x. Thus ! = x^. We have established a bije tion between X and sp(C (X )). Suppose x is a net in X whi h onverges to x 2 X . Then f (x ) ! f (x) for all f 2 C (X ), and hen e x^ ! x^ weak*. This shows that the natural map

© 2001 by Chapman & Hall/CRC

95 from X onto sp(C (X )) is ontinuous. Sin e X is ompa t and sp(C (X )) is Hausdor, it follows that this map is a homeomorphism. Finally, we establish a orresponden e between ontinuous fun tions and -homomorphisms. Example 5.1.8

Let X and Y be ompa t Hausdor spa es and let  : Y ! X be

ontinuous. Then C is a unital -homomorphism from C (X ) into C (Y ). PROPOSITION 5.1.9

Let  : C (X ) ! C (Y ) be a unital -homomorphism. Then there is a

ontinuous map  : Y ! X su h that  = C . PROOF For any y 2 Y the map y^ is a unital -homomorphism from C (Y ) into C, so y^Æ  is a unital -homomorphism from C (X ) into C. By Proposition 5.1.7 there exists a unique point x 2 X su h that y^ Æ  = x^. De ne (y ) = x. For any f 2 C (X ) and any y 2 Y we then have

C f (y ) = f ((y )) = (y )^(f ) = y^(f ) = f (y ):

Thus  = C . To see that  is ontinuous, let (y ) be a net in Y whi h onverges to y 2 Y . Suppose (y ) 6! (y ); then there exists an open set O about (y ) su h that (y ) is not eventually in O. Find f 2 C (X ) su h that f ((y )) = 1 and f jX O = 0. Then g = f has the property that g (y ) = 1 and g (y ) = 0 frequently. But this ontradi ts the fa t that y ! y . 5.2

Topologies from fun tions

Given any topologi al spa e X , we an form the spa e Cb (X ) of bounded

ontinuous fun tions from X into C. This is a unital C*-subalgebra of l1 (X ). Conversely, one might hope that every unital C*-subalgebra of l1 (X ) arises in this way from a topology on X , but that is not exa tly true: for example, onsider the set of fun tions f 2 l1 (0; 1) whi h extend to ontinuous fun tions f~ on [0; 1℄ su h that f~(0) = f~(1). There is no topology on (0; 1) for whi h these are pre isely the bounded ontinuous fun tions. Still, we have the following result. PROPOSITION 5.2.1

Let X be a set and let A be a unital C*-subalgebra of l1 (X ) whi h

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Chapter 5: C*-algebras

separates points of X. Then there is a ompa t Hausdor spa e Y su h that X is a dense subset of Y and A = ff jX : f 2 C(Y )g.

PROOF Let Y = sp(A). Given ! 2 Y and f 2 A, we have j!(f)j  kf k1 for the same reason that this is true when A = C(X); namely, for any  2 C satisfying jj > kf k1 we have (f


1X )

1

=

 2

1 1 1 1 + f+ f  X  

!

+




2 A;

so !(f)  = !(f 
1X ) 6= 0. This shows that Y is ontained in the unit ball of A . We laim that Y is a ompa t Hausdor spa e. Sin e dual unit balls are always weak*

ompa t and Hausdor, we need only show that Y is losed. But if (! ) is a net in Y whi h weak* onverges to ! 2 A , it is easy to see that ! is also a unital -homomorphism and hen e belongs to Y . So Y is

ompa t Hausdor. The embedding x 7! x^ allows us to identify X with a subset of Y . Moreover, any f 2 A extends to a ontinuous fun tion f~ on Y de ned ~ = !(f). (In parti ular, f(^ ~ x) = x^(f) = f(x), so f~ really is an by f(!) ~ extension of f.) The set ff : f 2 Ag is then a unital C*-subalgebra ~ 1 ) = f(! ~ 2 ) for all of C(Y ), and it separates points of Y be ause f(! f 2 A implies !1 (f) = !2 (f) for all f, whi h implies !1 = !2 . Thus the extension map f 7! f~ takes A onto C(Y ), and we on lude that A = ff jX : f 2 C(Y )g. Finally, X is dense in Y be ause any fun tion in C(Y ) whi h vanishes on X restri ts to the zero fun tion on X, and hen e must be zero on Y by the above isomorphism of A with C(Y ). Thus, unital C*-subalgebras of l1 (X) orrespond to ompa ti ations of X, i.e., ompa t Hausdor spa es whi h ontain X as a dense subset (up to homeomorphisms xing X). This is our justi ation for thinking of C*-subalgebras as orresponding to topologies. In a way, measure spa es provide a better setting for this kind of result, be ause we an agree that the extra points that have to be added to X onstitute a set of measure zero, and hen e do not essentially alter X as a measure spa e. Here we must assume - niteness, however. We pro eed to formulate a result of this type. It uses the following notion of equivalen e of measure spa es. DEFINITION 5.2.2

© 2001 by Chapman & Hall/CRC

Let X = (X; ) and Y = (Y; ) be - nite

97 measure spa es. Then a measure equivalen e between X and Y is a bije tion  between the measurable sets in X and the measurable sets in Y , both modulo null sets, su h that (a) (;) = ; and (X ) = Y ; (b) (S ) = (S ) ; S S T T ( ) ( Sn ) = (Sn ) and ( Sn ) = (Sn ); and (d) (S ) =  ((S )) for all measurable S; Sn  X . If a measure equivalen e exists then we say that X and Y are measurably equivalent.

Suppose X and Y are measurably equivalent and let  be a measure equivalen e. If f : X ! C has ountable range, then we an partition X into a ountable family of positive measure sets on ea h of whi h f is onstant. Then it is lear how to de ne a fun tion (f ) : Y ! C with the property that (f 1(S )) = (f ) 1(S ) (up to null sets) for every positive measure subset S of C. Now for any measurable fun tion f : X ! C we an nd a sequen e of measurable fun tions fn : X ! C, ea h with ountable range, su h that kf fn k1 ! 0. Setting (f ) = lim (fn ) then yields a orresponden e between the measurable fun tions on X and the measurable fun tions on Y , both up to null sets, whi h respe ts the measure equivalen e . In parti ular, any measure equivalen e between X and Y implements an isometri isomorphism between L1 (X ) and L1 (Y ). We also need a notion of \separating points" whi h is suitable for measurable sets. DEFINITION 5.2.3

Let X be a  - nite measure spa e and let

A  L1 (X ). Let be the smallest -algebra for whi h every fun tion in A is measurable. Then we say that A measurably separates points if

every measurable subset of X has null symmetri dieren e with a set in .

Note that if A is losed under omplex onjugation then an also be des ribed as the smallest -algebra for whi h every real fun tion in A is measurable. PROPOSITION 5.2.4

Let (X; ) be a  - nite measure spa e, let A be a unital C*-subalgebra of L1 (X ), and suppose A measurably separates points. Then there is a ompa t Hausdor spa e Y , a regular Borel measure  on Y , and a

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Chapter 5: C*-algebras

measure equivalen e between X and Y that takes

PROOF

A to C (Y ).

Let Y be the spe trum of A. As in Proposition 5.2.1, is a ompa t Hausdor spa e and the map f 7! f~, where f~(!) = ( ), is a unital -homomorphism, and hen e a ontra tion, from A into C (Y ). We laim that this map is isometri . To see this, let f 2 A and  > 0 and x a subset S0  X su h that jf j  kf k1  on S0 . Then let  be a maximal family of positive measure subsets of X , ea h

ontained in S0 , with the property that S; T 2  implies S \ T 2 . Dire t  by reverse in lusion. For ea h g 2 A and S 2  the essential range of gjS is ompa t, so the interse tion of these essential ranges is nonempty. Moreover, there annot be more than one interse tion point by maximality of . For ea h g 2 A de ne !(g) to be this interse tion point; then ! is a -homomorphism from A into C su h that j!(f )j  kf k1 , whi h establishes the laim. So A embeds in C (Y ), and sin e its image learly separates points, the embedding is onto. So A  = C (Y ). Note that this shows in parti ular that the real fun tions in A are losed under nite latti e operations. Let 0 be a nite measure on X su h that  and 0 are mutually absolutely ontinuous. We laim that A is dense in L2 (X; 0 ). To see this rst let S  X be a set of the form S = f 1(( 1; a℄) for some real fun tion f 2 A and some a 2 R. Without loss of generality a > 0. Then, using the latti e operations in A, we an nd a real fun tion g = a=(f _ a) 2 A su h that 0  g  1 and S = g 1 (1), and onsequently the powers gn onverge to S in L2 (X; 0 ). Now it is straightforward to show that the family of S  X su h that S is in the L2 losure of A

onstitutes a -algebra, and it follows from the above and the de nition of measurable separation that every measurable S  X belongs to this family. So the L2 losure of A ontains all hara teristi fun tions, and hen e equals L2 (X; 0 ).R Now the map f 7! f d0 is a bounded linear fun tional on A, so it orresponds to a bounded linear fun tional on CR(Y ), and Rthus there exists a regular Borel measure  0 on Y su h that f d0 = f~ d 0 for all f 2 A. Now A is dense in L2 (X; 0 ) by the last paragraph, and it is standard that C (Y ) is dense in L2 (Y;  0 ). Moreover, the map f 7! f~ is an isometry in L2 norm by the de nition of  0 . So this map extends to a surje tive isometri isomorphism U : L2 (X; 0 ) ! L2 (Y;  0 ). It is easy to see that U takes hara teristi fun tions to hara teristi fun tions, and therefore it de nes a measure isomorphism between (X; 0 ) and (Y;  0 ). It is also lear that this measure isomorphism takes A to C (Y ). Finally, for ea h Borel set S  Y de ne  (S ) = (S 0 ) where S 0 is the orresponding set in X . Then  and  0 are mutually absolutely

ontinuous, and the result for 0 and  0 transfers to  and  . Y ! f

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99

5.3

Abelian C*-algebras

We saw in the last se tion that C*-subalgebras A of l1 (X ) orrespond to ompa ti ations Y of X , and by the results of Se tion 5.1 algebrai properties of A are exa tly mirrored in topologi al properties of Y . C*algebras are the Hilbert spa e version of this onstru tion. A ( on rete) C*-algebra is a C*-subalgebra A of B (H), for some Hilbert spa e H. If AB = BA for all A; B 2 A then A is abelian. If I 62 A, then we de ne the unitization of A to be the C*-algebra A+ = fA + aI : A 2 A and a 2 Cg. DEFINITION 5.3.1

We will dis uss abstra t C*-algebras in Se tion 5.6. Just as for C (X ), every unital -homomorphism  : A ! B between C*-algebras is automati ally ontra tive. For if A 2 A is self-adjoint and jj > kAk then A I = (I 1 A) is invertible in A by Lemma 3.2.3. (The series that de nes the inverse onverges in A.) It follows that  (A I ) =  (A)  is invertible in B , so that  62 sp( (A)); as this is true whenever jj > kAk, we have k (A)k  kAk by Proposition 3.2.4. Now for any A 2 A the pre eding implies that k (A)k2 = k (A A)k  kA Ak = kAk2 . So  is a ontra tion. Moreover, if  has null kernel then sp(A) = sp( (A)) for all self-adjoint A 2 A; otherwise, let f 2 C (sp(A)) vanish on sp( (A))  sp(A) but not on all of sp(A), and observe that f (A) 6= 0 but  (f (A)) = f ( (A)) = 0,

ontradi ting inje tivity. It follows that kAk = k (A)k for all self-adjoint A 2 A, and hen e kAk2 = kA Ak = k (A A)k = k (A)k2 for all A 2 A. This shows that a -homomorphism with null kernel must be isometri , i.e., a -isomorphism. Example 5.3.2

Let A 2 B (H) be self-adjoint and let C  (A) be the norm losure of the polynomials in A. This is the unital C*-algebra generated by A. But by the ontinuous fun tional al ulus, the norm losure of the polynomials in A is pre isely the set ff (A) : f 2 C (sp(A))g. Therefore, we have C  (A)  = C (sp(A)), and so Proposition 5.1.7 implies that sp(C  (A)) (in the sense of De nition 5.1.1) is naturally identi ed with sp(A) (in the sense of De nition 3.2.1). Example 5.3.3

Let X be a ompa t Hausdor spa e, let  be a regular Borel measure on X , and let X be a measurable Hilbert bundle over X . Then the set of multipli ation operators fMf : f 2 C (X )g on L2 (X ; X ) is a unital C*-algebra.

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100

Chapter 5: C*-algebras The map

f 7! Mf

is a unital

 nor X

isomorphism if neither

-homomorphism,

and it is a

-

is onstantly 0 on any open set.

Both of the above examples are abelian, and the se ond one is general over separable Hilbert spa es. We now pro eed to prove this. LEMMA 5.3.4

Let fA :  2 J g be a pairwise ommuting set of self-adjoint operators in B (H). Assume kA k  1 for all . Then there is a spe tral measure E on the set X = [ 1; 1℄J whi h on ea h oordinate gives rise to the spe tral measure of A in the manner of Lemma 3.5.2.

PROOF

form

For any indi es 1 ; : : : ; n and any fun tion f on X of the

f (x) =

X m

j =1

aj S1j (x1 )


Snj (xn )

with x = (x ) and S1j ; : : : ; Snj  [0; 1℄, we de ne  (f ) to be the operator

 (f ) =

X m

j =1

aj S1j (A1 )


Snj (An ):

Sin e the A ommute, so do polynomials in the A , and by ontinuity so do bounded Borel fun tions of the A obtained by fun tional al ulus. It follows that  is a -homomorphism. If f has a bounded inverse then its inverse is also of the above form, and this allows us to prove that  is ontra tive by the argument that showed this for -homomorphisms between C*-algebras given at the beginning of this se tion. Then  extends by ontinuity to any ontinuous fun tion on X | every oordinate fun tion is learly in the norm losure of the set of fun tions onsidered above, and by the Stone-Weierstrass theorem this implies that all of C (X ) is in the norm losure | and it extends further to the bounded Borel fun tions on X pre isely as in Theorem 3.3.3. We an then de ne, for any Borel set S  X , a proje tion E (S ) by PE (S ) =  (S ). Veri ation that this is a spe tral measure is routine. By onstru tion, E agrees on ea h oordinate  with the spe tral measure of A . THEOREM 5.3.5

Let A  B (H) be a unital abelian C*-algebra and suppose H is separable. Then there is a probability measure  on sp(A), a measurable Hilbert bundle X over sp(A), and an isometri isomorphism U :

© 2001 by Chapman & Hall/CRC

101

L2 (sp(A); X )  = H su h that A = UC (sp(A))U 2 on L (sp(A); X ) by multipli ation.

1

where

C (sp(A))

a ts

PROOF Let fA :  2 J g be the set of all self-adjoint operators in A of norm at most 1, let E be the spe tral measure on X = [ 1; 1℄J provided by the lemma, and let , X , and U be as in Corollary 3.4.3.

Note that  is regular be ause E is regular, in the same sense. De ne a -homomorphism  : C (X ) ! B (H) by  (f ) = UMf U 1 . This map takes the oordinate fun tions on X to the operators A , so it takes polynomials in the oordinates into A, and by ontinuity its image is ontained in A. Conversely, every operator in A is a linear

ombination of self-adjoint operators of norm at most 1, so we on lude that the image of  equals A. Let I = ker( ). By Proposition 5.1.5 there is a losed subset K of X su h that I = ff : f jK = 0g, and  des ends to a map ~ : C (K )  = C (X )=I ! A whi h is a bije tive -homomorphism, and therefore a -isomorphism. K an be identi ed with sp(A) by Proposition 5.1.7, and it is lear that  is supported on K , so we obtain the desired result.

In the nonseparable ase, we an still say that there is a spe tral measure E on sp(A) whi h gives rise to a -isomorphism from C (sp(A)) onto A via approximation of ontinuous fun tions by simple fun tions, as in the proof of Lemma 5.3.4. The omment about invariants that we made after Theorem 3.5.1 also applies here: sp(A),  (up to measure lass), and (Xn ) (up to null sets)

onstitute a omplete set of invariants for the unital C*-algebra A, up to unitary equivalen e. 5.4

The quantum plane

In this se tion we will de ne a quantum analog of the C*-algebra C0 (R2 ) and determine its stru ture. We view this C*-algebra as endowing the quantum plane with topologi al stru ture. The idea is to develop a fun tional al ulus, so that we an de ne operators of the type f (Q; P ) where f is a ontinuous fun tion on the plane whi h goes to zero at in nity and Q and P are the position and momentum operators of Se tion 4.1. Even though Q and P are unbounded and do not ommute, we an still do this for suÆ iently smooth fun tions. The result is alled the Weyl fun tional al ulus. We will use fun tions in the S hwartz lass S (R2 ) onsisting of all C 1 fun tions whi h, together with their partial derivatives of every order, are o((jxj + jy j) n ) as x; y ! 1. The key fa ts about S (R2 ) are that it is losed under

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102

Chapter 5: C*-algebras

sums and produ ts and that it is taken onto itself by the Fourier transform. Also note that C 1 (R2 )  S (R2 ), so S (R2 ) is dense in C0 (R2 ). After we have de ned the operators f (Q; P ) for f 2 S (R2 ), the desired C*-algebra will simply be their norm losure. It is onvenient to work in the tra ial representation of Se tion 4.2. Let h operator Lt1 ;t2 on L2 (R2 ) by

DEFINITION 5.4.1

 0.

For t1 ; t2

2 R de ne a unitary

1 (Lt1 ;t2 g )(x1 ; x2 ) = ei(x1 t1 +x2 t2 ) g (x1 + h t ; x 2 2 2

1 ht ); 2 1

for f; g 2 S (R2 ) de ne the twisted produ t f h g 2 S (R2 ) by Z Z 1 (f h g )(x1 ; x2 ) = f^(t1 ; t2 )Lt1 ;t2 g dt1 dt2 Z Z 2 1 1 1 = f^(t1 ; t2 )g (x1 + ht2 ; x2 ht )ei(x1 t1 +x2 t2 ) dt1 dt2 ; 2 2 2 1 and for f 2 S (R2 ) de ne the twisted multipli ation operator Lf on L2 (R2 ) by Lf (g ) = f h g . We are using the Fourier transform in two variables de ned by Z Z 1 f^(t1 ; t2 ) = f (x1 ; x2 )e i(x1 t1 +x2 t2 ) dx1 dx2 : 2 One an prove that kf h g k2  kf^k1 kg k2 using Minkowski's inequality for integrals, just as one proves the same inequality for the onvolution of f^ and g . Thus Lf is a bounded operator with norm at most kf^k1 . The operators Lt1 ;t2 generalize the unitaries Us and Vt introdu ed in Se tion 4.2; indeed, Ls;0 = Us and L0;t = Vt . They obey the ommutation relation

Ls1 ;s2 Lt1 ;t2 = eih (s2 t1

s1 t2

)

Lt1 ;t2 Ls1 ;s2 ;

whi h is a general version of the Weyl ommutation relations. The operator Lf plays the role of f (Q; P ) in the tra ial representation. The intuition is that Lt1 ;t2 is ei(t1 Q+t2 P ) , so we obtain f (Q; P ) by Fourier expansion: Z Z 1 f (Q; P ) = f^(t1 ; t2 )ei(t1 Q+t2 P ) dt1 dt2 : 2 Applying the right side of this formula to g yields the formula for Lf g . The following result is a fairly straightforward omputation.

© 2001 by Chapman & Hall/CRC

103 PROPOSITION 5.4.2

2 S (R2 ) and a; b 2 C. Then aL + bL = L + . (b) Let f; g; h 2 S (R2 ). Then (f  g )  h = f  (g  h). (a) Let f; g

f

Lf Lg = Lf h g . = L t1 ; t2 for all t1 ; t2 ( ) L t1 ;t2 where g (x1 ; x2 ) = f ( x1 ; x2 ).

2 S (R

g

h

af

bg

h

h

Thus

2 R. For f 2 S (R2 ) we have L = L f

g

R

 ( 2 ) is the norm losure of the operators C0h

DEFINITION 5.4.3

Lf for f

h

2 ).

It follows from Proposition 5.4.2 that C0h (R2 ) is a C*-algebra. The next result is trivial. PROPOSITION 5.4.4

Suppose h 

= 0.

R

C0 ( 2 ) .

Then Lf

= Mf

for all f

2 S (R2 ),

( and C0h

R2) =

Next we onsider the a tion of translation operators on C0h (R2 ). Re all the unitary operator Tt1 ;t2 on L2 (R2 ) introdu ed following De nition 4.2.1. For A 2 B (L2 (R2 )) de ne t1 ;t2 (A) = Tt1 ;t2 ATt1 ;t1 2 . PROPOSITION 5.4.5

For any t1 ; t2

2 R and f 2 S (R2 ) we have t1 ;t2 (Lf )

= LTt1 ;t2 f :

R R



 ( 2 ) onto itself. The map t1 ;t2 restri ts to a -isomorphism from C0h 2 h  This yields an a tion of by automorphisms of C0 ( 2 ). Moreover, 2 h  t1 ;t2 (A) is ontinuous in norm. for any A C0 ( ) the map (t1 ; t2 )

R

R

2

7!

PROOF The rst equality is a straightforward al ulation, and it implies that t1 ;t2 takes C0h (R2 ) into itself. The inverse map  t1 ; t2 , has the same property, so t1 ;t2 restri ts to a -isomorphism from C0h (R2 ) onto itself. It is lear that the map (t1 ; t2 ) ! t1 ;t2 is a group homomorphism. For A = Lf (f 2 S (R2 )) we have k = kL 1 2 L k = kL 1 2 k: This onverges to 0 as t1 ; t2 ! 0 sin e k(T 1 2 f f )^k1 ! 0. But for any A 2 C0 (R2 ) and any  > 0 we an nd L su h that kL Ak  =3 and Æ > 0 su h that jt1 j; jt2 j  Æ implies k 1 2 (L ) L k  =3. Then k 1

t ;t2

(A)

A

Tt ;t f

Tt ;t f

f

f

t ;t

h

f

t ;t

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f

f

f

104

Chapter 5: C*-algebras

jt1 j; jt2 j  Æ also implies kt1 ;t2 (A) Ak  kt1 ;t2 (A

k + kt1 ;t2 (Lf ) Lf k + kLf Ak  : This shows that the map (t1 ; t2 ) 7! t1 ;t2 (A) is norm ontinuous. Lf )

In parti ular, we have Ts;0 Lf Ts;01

and

= LT

T0;t Lf T0;t1

s;0

1 f = V s=h Lf V s=h

= LT0; t f = Ut=h Lf Ut=h1 :

This is the sense in whi h V s=h and Ut=h are \equivalent" to Ts;0 and T0;t , as we indi ated at the end of Se tion 4.2. The remainder of this se tion is devoted to hara terizing the stru ture of C0h (R2 ). We observe rst that C0h (R2 )  = C0ah (R2 ) for all a > 0, i.e., these algebras are mutually -isomorphi for all nonzero values of  . Indeed, h the unitary operator U de ned on L2 (R2 ) by U f (x1 ; x2 ) = p p af ( ax1 ; ax2 ) satis es U 1 Lt1 ;t2 U g (x1 ; x2 )

p 1 p  at2 ; x2 = ei(x1 t1 +x2 t2 )= a g (x1 + h 2

1 p h at1 ); 2

so U 1 Lt1 ;t2 U is the operator Lt1 =pa;t2 =pa for h0 = ah; and integration against f^ yields that onjugation by U takes Lf (for h) to LU 1 f (for  0 = ah h  ). Thus onjugation by U de nes a -isomorphism from C0h (R2 ) onto C0ah (R2 ). To determine the stru ture of C0h (R2 ) more expli itly we must pass to the L2 (R) representation of Se tion 4.1. In this model the operator ~ f 2 B (L2 (R)) de ned by Lf 2 B (L2 (R2 )) is repla ed by the operator L ~ f g (x) = 1 L 2

Z Z

1

 t2 ) g (x + h  t2 ) dt1 dt2 : f^(t1 ; t2 )eit1 (x+ 2 h

~ f I )W where W is the unitary operator of PropoWe have Lf = W 1 (L sition 4.2.2. This an be proven by a dire t omputation using the de nition of W , or more easily by using the formula Lt1 ;t2 = eih t1 t2 =2 Ut1 Vt2 RR 1 f^(t)Lt g dt. and the fa t that Lf g = 2 h  2 ~ ~ f on L2 (R). Then Let C0 (R ) be the norm losure of the operators L ~ f extends to a -isomorphism from C0h (R2 ) onto the map Lf 7! L ~0h (R2 ). We will prove that C~0h (R2 ) = K (L2 (R)), the C*-algebra of C

ompa t operators on L2 (R). First we must develop some general information about ompa t operators.

© 2001 by Chapman & Hall/CRC

105 An operator A 2 B (H) is ompa t if the image under A of the unit ball of H is pre ompa t in norm, or equivalently, for any bounded sequen e (vn ) in H, the sequen e (Avn ) has a luster point. The set of all ompa t operators is denoted K (H). DEFINITION 5.4.6

For example, a proje tion is ompa t if and only if its range is nite dimensional. For A 2 B (H) let jAj be the operator jAj = (A A)1=2 de ned using

ontinuous fun tional al ulus on the self-adjoint operator A A. The next result is known as polar de omposition. (To see why, apply it to an operator in B (C)  = C.) LEMMA 5.4.7

A 2 B (H). Then there is a unique U 2 B (H) su h that = 0 on ran(jAj)? , and U is an isometry on ran(jAj).

Let U

PROOF

A

= U jAj,

For any v 2 H we have

kj j k2 = hj j j j i = hj j2 i = h  i = h i = k k2 Thus the map : j j ! 7 is well-de ned and isometri on ran(j j), and we an extend it to H by setting = 0 on ran(j j)? . Uniqueness A v

A v; A v U

A v

A

v; v

A Av; v

Av; Av

Av

Av

:

A

U

is lear.

A

PROPOSITION 5.4.8 K

(H) is a C*-ideal of B (H).

PROOF It is straightforward to verify that K (H) is a linear subspa e of B (H), and that A 2 K (H) and B 2 B (H) implies AB; BA 2 K (H). Next we show that K (H) is losed. Let (An ) be a sequen e in K (H) whi h onverges in norm to A 2 1 B (H). We may assume that kAj Ak k  n for j; k  n. We must show that A is ompa t. To see this, let (vk ) be a sequen e of ve tors of norm at most 1. De ne vk0 = vk , and for n > 0 indu tively let (vkn ) be a subsequen e of (vkn 1 ) su h that kAn vjn An vkn k  n1 for all j; k . This is possible be ause An is ompa t. Finally, de ne wk = vkk ; this is a subsequen e of (vk ) su h that k

Awj

Awk

k=k k 3

Awj

A

An wj An

+ An wj

An wk

k(k j k + k k k) + k w

w

+ An wk

A n wj

Awk

An wk

k

=n

for j; k  n. Thus (Awk ) onverges, showing that

© 2001 by Chapman & Hall/CRC

A

is ompa t.

k

106

Chapter 5: C*-algebras

Finally, we must show that A 2 K (H) implies A 2 K (H). Let A 2 K (H) and write A = U jAj as in the lemma. Then jAj is ompa t be ause every polynomial in A A is ompa t and K (H) is losed in norm, and therefore A = jAjU  is ompa t by the ideal property of K (H). Self-adjoint ompa t operators have a parti ularly simple stru ture. PROPOSITION 5.4.9

P

Let

2

H

A B ( ) be self-adjoint. Then A is ompa t if and an Pn where the Pn are orthogonal proje tions with nite ranges and an 0.

!

only if A = dimensional

PROOF The reverse dire tion follows from the fa t that K (H) is

losed. For the onverse, suppose A is ompa t and write A = U Mx U 1 as in Theorem 3.5.1. It will suÆ e to show that for ea h  > 0 the set S = fx 2 sp(A) : jxj  g is nite and the orresponding spe tral proje tion P = S (A) has nite dimensional range; for then we an write A= an Pn where the an enumerate sp(A) f0g and Pn = an (A). Fix , S , and P . Then for any v 2 H we have kAv k  kP v k. So

ompa tness of A implies ompa tness of P , and it follows that P has nite dimensional range, and hen e S is nite, as desired.

P

Next we introdu e another on ept whi h will lead to the key theorem that we need in order to identify the C*-algebra C0h (R2 ). DEFINITION 5.4.10 Let A  B (H) be a C*-algebra. Then A is irredu ible if for every v 2 H the set Av = fAv : A 2 Ag is dense in H.

By onsidering the ompa t operators easily sees that K (H) is irredu ible. THEOREM 5.4.11

v

7! hv; wiw (for w 2 H) one

Let A  B (H) be an irredu ible C*-algebra and suppose Then A = K (H).

PROOF

A  K (H).

It follows from Proposition 5.4.9 and the ontinuous fun tional al ulus that A ontains a proje tion with nite dimensional range. Let P 2 A be su h a proje tion whose range has minimal (nonzero) dimension. We laim this dimension is one. Otherwise, let v; w 2 ran(P ) be orthogonal. For any self-adjoint operator A 2 A, the

© 2001 by Chapman & Hall/CRC

107 operator P AP is self-adjoint and by Proposition 5.4.9 it an be de omposed into proje tions. But ran(P AP )  ran(P ), so minimality of P implies P AP = aP for some a 2 C. Thus

h

i= h i=0 so that is orthogonal to . Sin e any 2 A is a linear ombination of self-adjoint operators in A, we have ? A , ontradi ting irredu ibility. Av; w

i=h

w

P AP v; w

Av

a v; w

;

A

w

v

This proves the laim. Now say the range of P is spanned by the unit ve tor v , so that P u = hu; v iv for all u 2 H, and let w be any other unit ve tor. Find a sequen e (An )  A su h that An v ! w and kAn v k = 1 for all n. Let 0 P be the proje tion onto the span of w . Then for every u 2 H

k



0

An P An u

P u

k = kh = kh  2k  !

i i(

u; An v An v u; An v An v

h

An v

w

kk k

i k

u; w w w

) + hu; An v

i k

w w

u :

0 in norm, and hen e P 0 2 A, so A We on lude that An P An P

ontains the proje tions onto all one-dimensional subspa es of H. By Proposition 5.4.9 and de omposition into real and imaginary parts, it follows that A ontains every ompa t operator.

We now pro eed to prove that

(

h

C0

R2) =

(

R)).

2(

K L

LEMMA 5.4.12

Let

k

= (kmn ) 2 l2 (N  N). Then the operator A on l2 (N) de ned by Av

=

X

h

i

kmn v; em en m;n

satis es kAk  kk k and is ompa t.

PROOF Let 2 (N  N), so

v

= (an ); w = (bn )

2 2(N). l

Then v w = (am bn )

2

l

jh

Av; w

ij =

X kmn am bn



= jhk; v wij  kk kkv kkwk:

m;n

Thus kAk  kk k. If only nitely many kmn are nonzero then A has nite dimensional range, and hen e is ompa t. In general the norm estimate shows that every A is approximated by operators of this type, so ompa tness of A follows from the fa t that K (H) is losed.

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108

Chapter 5: C*-algebras

THEOREM 5.4.13 h  2

If h 6= 0 then C0 (R )  = K (L2 (R)).

Let A  = C~0h (R2 )  B (L2 (R)) be the norm losure of the ~ operators Lf for f 2 S (R2 ). As we noted earlier, A is naturally isomorphi to C0h (R2 ). We will show that A = K (L2 (R)). By Theorem 5.4.11, it will suÆ e to show that A  K (L2 (R)) and A is irredu ible. Sin e K (L2 (R)) is losed, to prove the rst assertion we need only ~ f is in K (L2 (R)). In fa t, sin e kL ~ f k  kf^k1 , passing show that ea h L ~ f is approximated to the Fourier transform pi ture shows that every L ~ f with f of the form f (x1 ; x2 ) = by linear ombinations of operators L f1 (x1 )f2 (x2 ) for f1 ; f2 2 S (R). So it will be enough to show that every operator of this form is ompa t. Fix su h a fun tion f (x1 ; x2 ) = f1 (x1 )f2 (x2 ). Then for g 2 S (R) we have

PROOF

ZZ Z Z

~ f g (x) = 1 L 2 1 =p 2 1 =p 2

it (x+ 21 h  t2 ) f^1 (t1 )f^2 (t2 )e 1 g (x + h  t2 ) dt1 dt2

f1 (x +

1 ht )f^ (t )g (x + ht2 ) dt2 2 2 2 2

k (x; s)g (s) ds

where k (x; s) = h1 f1 ((s + x)=2)f^2 ((s x)=h) and we have made the substitution s = x + ht2 . We have k 2 L2 (R2 ). Write k = kmn hm (s)hn (x) where (hn ) is the Hermite basis of L2 (R) and (kmn ) 2 l2 (N  N). Then

P

Z

k (x; s)hi (s) ds

ZX = X =

kmn hm (s)hn (x)hi (s) ds

m;n

kin hn (x):

n

~ f is an operator of the form given in the By linearity, this shows that L ~ f is ompa t. Thus A  K (L2 (R)). lemma, and therefore L To verify irredu ibility let a 2 R and for  > 0 nd f2 2 S (R) su h that f^2  0, f^2 = 1, and f^2 is supported on the interval [a ; a + ℄. This an be done be ause the inverse Fourier transform takes C 1 (R) into S (R). Now for any f1 ; g 2 S (R), de ning f (x1 ; x2 ) = f1 (x1 )f2 (x2 ) yields

R

~ f g (x) L

=

p1

2

© 2001 by Chapman & Hall/CRC

Z

f1 (x +

1 ht )f^ (t )g (x + h  t2 ) dt2 2 2 2 2

109

! p1

2

f1 (x +

1 ha)g (x + ha) 2

~ f g ! p1 Mf T h a g where f (x) = f1 (x + 1 h as  ! 0. That is, L 2  a). 2 Moreover, the onvergen e is uniform in g , so we on lude that the same limit obtains, and hen e p12 Mf T h a g 2 Ag , for all g 2 L2 (R). Sin e a and f1 were arbitrary, this shows that Ag ontains all L2 fun tions whose support is ontained in a translation of the support of g . Thus Ag = L2 (R) for any g 2 L2 (R), and we on lude that A is irredu ible. Together with Proposition 5.4.4, this determines the stru ture of every R2).

( C0h

5.5

Quantum tori

Quantum tori are related to the quantum plane, but they are te hni ally easier to deal with owing to the fa t that the Fourier transform takes the torus T2 = R2 =2 Z2 to the dis rete spa e Z2 . First we des ribe the untransformed model. The underlying Hilbert spa e is L2 (T2 ). By analogy with the operators Us and Vt of Se tion 4.2, we de ne 1 ix U f (x1 ; x2 ) = e 1 f (x1 ; x2 ) h 2 and

1  ; x2 ) = eix2 f (x1 + h 2

V f (x1 ; x2 )

for f 2 L2 (T2 ). These are unitary operators, and they obey the ommutation relation U V = e ih V U . We regard the operators U and V as \exponential fun tions" on a quantum torus, so the C*-algebra they generate should be the algebra of \ ontinuous fun tions." DEFINITION 5.5.1 of the operators

U mV n

T 2Z

 ( 2) Ch m; n .

Let for

be the losed span in

Equivalently, C h (T2 ) is the C*-algebra generated by U and V . The next result is trivial. PROPOSITION 5.5.2 If

h = 0 then

T = C (T2 )

 ( 2) Ch

© 2001 by Chapman & Hall/CRC

.

T

B (L2 ( 2 ))

110

Chapter 5: C*-algebras

Now we pass to the Fourier transform pi ture. Let F~2 : L2 (T2 ) 7! Z be the Fourier transform given by

l2 ( 2 )

(F~2 f )mn =

1 2

Z Z 2

0

2

0

f (x1 ; x2 )e i(mx1 +nx2 ) dx1 dx2

( f. Example 2.3.6). Then onjugation by F~2 gives rise to operators U^ = F~2 U F~2 1 and V^ = F~2 V F~2 1 on l2 (Z2 ) whi h satisfy ^ m;n = e ih n=2 em+1;n Ue

V^ em;n = eih m=2 em;n+1

and

where fem;n g is the standard orthonormal basis of l2 (Z2 ). More generally, for any k; l 2 Z de ne

Lk;l em;n = eih (ml nk)=2 em+k;n+l = eih kl=2 U^ k V^ l em;n : These are analogous to (a Fourier transform version of) the operators Lt1 ;t2 of the last se tion. Let C^ h (T2 ) be the C*-algebra generated by the operators U^ and V^ . It is -isomorphi to C h (T2 ) via onjugation by F~2 . Any polynomial in U^ and V^ an be written as a nite sum A = ak;l Lk;l . We have the following basi fa ts.

P

P ak;lLk;l be a polynomial in the operators Lk;l. Then X eih ml nk = a e Ae =

PROPOSITION 5.5.3

Let A =

m;n

and

(

) 2

k;l

k;l m+k;n+l

hAem;n ; em+k;n+l i = eih (ml nk)=2 ak;l :

For any A 2 C^ h (T2 ) we also have

hAem;n ; em+k;n+l i = eih (ml nk)=2 hAe0;0 ; ek;l i:

The last part of this proposition holds for polynomials in the Lk;l by dire t omputation, and for any A 2 C^ h (T2 ) by ontinuity. We regard the ak;l as the \Fourier oeÆ ients" of ak;l Lk;l . The pre eding suggests that for any A 2 C^ h (T2 ), not just polynomials, we should de ne the Fourier oeÆ ients of A to be

P

ak;l = hAe0;0 ; ek;l i:

P

We must then ask in what sense A = ak;l Lk;l . This is not true in the sense of norm onvergen e; that already fails in the ase h = 0.

© 2001 by Chapman & Hall/CRC

111 (There exists a ontinuous fun tion on the ir le, and hen e also one on the torus, su h that the partial sums of its Fourier series do not

onverge uniformly.) Rather, we must follow the h  = 0 ase and onsider

onvergen e of Cesaro means. We now pro eed to do this. DEFINITION 5.5.4

For A 2 C^ h (T2 ) and m; n 2 Z de ne

(a) ak;l (A) = hAe0;0 ; ek;l i (the Fourier oeÆ ients of A); (b) sN (A) = jkj;jljN ak;l (A)Lk;l (the partial sums of the Fourier series); and 1 ( ) N (A) = (N +1) 2 jkj;jljN (N +1 jkj)(N +1 jlj)ak;l(A) (the Cesaro means of the Fourier series).

P

P

In order to prove onvergen e in the Cesaro sense of the Fourier series, we need an alternative formula for the Fourier oeÆ ients whi h involves the unitary operators Me i(sm+tn) on B (l2 (Z2 )) (s; t 2 R). In the untransformed pi ture, these are oordinate translations. For A 2 B (l2 (Z)) de ne ^s;t (A) = Me i(sm+tn) AMe 1i(sm+tn) : We have the following result. PROPOSITION 5.5.5

For any s; t 2 R and k; l 2 Z we have ^s;t (Lk;l ) = e i(sk+tl) Lk;l . The map ^s;t restri ts to a -isomorphism from C^ h (T2 ) onto itself. This de nes an a tion of R2 by automorphisms of C^ h (T2 ). Moreover, for any A 2 C^ h (T2 ) the map (s; t) 7! ^s;t (A) is ontinuous in norm. It is lear that ^s;t is a -isomorphism from B (l2 (Z2 )) onto ^ itself. Sin e s;t (Lk;l ) = e i(sk+tl) Lk;l (an easy omputation) it follows that ^s;t takes polynomials in U^ and V^ to polynomials in U^ and V^ , and by ontinuity it takes C^ h (T2 ) into itself. As ^s;t1 = ^ s; t has the same property, it follows that ^s;t maps C^ h (T2 ) onto itself. It is lear that the map (s; t) 7! ^s;t is a group homomorphism. The equations ^s;t1 = ^ s; t and ^s;t Æ ^s0 ;t0 = ^s+s0 ;t+t0 are trivial, so ^ h (T2 ). ^ is an a tion of R2 by automorphisms of C If A is a polynomial in the Lk;l , the above expression for ^s;t (Lk;l ) shows that the map (s; t) 7! ^s;t (A) is ontinuous in norm. For any ^ h (T2 ) and any  > 0, nd a polynomial B in the Lk;l su h that A2C kA B k  =3 and nd Æ > 0 su h that jsj; jtj  Æ implies k^s;t (B ) B k 

PROOF

© 2001 by Chapman & Hall/CRC

112 =3.

Chapter 5: C*-algebras

Then jsj; jtj  Æ implies

k^s;t (A)

k  k^s;t (A

k + k^s;t(B ) B k + kB Ak  : This shows that the map (s; t) 7! ^s;t (A) is norm ontinuous. B)

A

Now we an hara terize the Fourier oeÆ ients of A dierently. PROPOSITION 5.5.6

Let A 2 C^ h (T2 ). Then ak;l (A)Lk;l

Z

1

=

(2 )2

2 0

Z

2 0

ei(sk+tl) ^s;t (A) dsdt:

PROOF The integral is taken in the sense of De nition 3.8.3. Sin e nite linear ombinations of basis ve tors em;n are dense in l2 (Z2 ), it suÆ es to he k equality when both sides are paired against these basis ve tors. But 1

Z

(2 )2 = =

2

Z

0

1

(2 )2

n hAe 0

Z

2

0 2

0

h

i

ei(sk+tl) ^s;t (A)em;n ; em0 ;n0 dsdt

Z

2

0

m;n ; em0 ;n0

h

i

0 0 ei(sk+tl) e i((m m)s+(n n)t) Aem;n ; em0 ;n0 dsdt

if m0 = m + k and n0 = n + l otherwise

i

= hak;l (A)Lk;l em;n ; em ;n i: 0

0

So the desired equality holds. Using this result, the proof that the Cesaro means of A onverge is an easy adaptation of its proof in the lassi al ase. THEOREM 5.5.7

Let A 2 C^ h (T2 ). Then N (A) ! A in norm.

PROOF KN (t)

Let KN be the Fejer kernel, =

jnj  eint =

N  X

n= N

1

Then A

=

N

1

(2 )2

© 2001 by Chapman & Hall/CRC

+1

ZZ









1 sin((N + 1)t=2) N +1 sin(t=2)

AKN (s)KN (t) dsdt

2 :

sin e

R 

113 KN (s)ds N (A)

= 2 . Also

=

1

(2 )2

ZZ







^s;t (A)KN (s)KN (t) dsdt

by Proposition 5.5.6. Therefore A

N (A)

1

Z Z

(A ^s;t (A))KN (s)KN (t) dsdt (2 )2 Z Z 1 = (A ^s;0 (A))KN (s)KN (t) dsdt (2 )2 Z Z 1 ^s;0 (A ^0;t (A))KN (s)KN (t) dsdt + (2 )2 Z 1 (A ^s;0 (A))KN (s) ds = 2 Z  Z 1 ^ ^ s;0 (A 0;t (A))KN (t) dt KN (s) ds: + (2 )2 =

This redu es the problem to showing that both of the last two integrals go to zero as N ! 1. The rst integral is small for large N sin e the map R s 7! (A ^s;0 (A)) 1 is norm ontinuous and isR zero for s = 0, while 2 jKN (s)jds = 1 and for any  > 0 we have jsj jKN (s)jds ! 0 as N ! 1. The same argument shows that the inner part of the se ond integral is small, from whi h it follows thatR the whole se ond integral is small be ause ^s;0 is an isometry and 21 jKN (t)jdt = 1. Unlike the quantum plane, quantum tori annot be s aled onto ea h other for dierent values of h. In fa t, the C*-algebras C h (T2 ) are generally not -isomorphi for dierent values of h. We will not prove the full uniqueness result, but only the fa t that C h (T2 ) 6 = C h (T2 ) if h 0 is a rational multiple of  and h is an irrational multiple of  . We need the following interesting fa t. 0

THEOREM 5.5.8

Let H be a Hilbert spa e, suppose  h is an irrational multiple of  , and ~ V~ = e ih V~ U~ . Then the let U~ ; V~ 2 B (H) be unitaries whi h satisfy U  ~ ~ ~ ~ C*-algebra C (U ; V ) generated by U and V has no proper C*-ideals. In parti ular, this is true of C h (T2 ).

PROOF

For ea h m; n 2 Z and A 2 C  (U~ ; V~ ) de ne ~s;t (A)

© 2001 by Chapman & Hall/CRC

= (U~ m V~ n )A(U~ m V~ n )

1

114

Chapter 5: C*-algebras

where s = hn + 2k , t = hm + 2l, and k and l are arbitrary. Note that sin e h= is irrational, distin t values of k and n give rise to distin t values of s and distin t values of l and m give rise to distin t values of ~ ) = e is U~ and ~s;t (V~ ) = e it V~ t, so ~s;t is well-de ned. We have ~s;t (U for all s; t (so it agrees with the automorphism ^s;t on C^ h (T2 )). The values of s and t for whi h ~s;t is de ned are dense in R. Moreover, it is lear that the map (s; t) 7! ~s;t (A) extends to a norm ontinuous map from R2 into C  (U~ ; V~ ) if A = U~ or V~ . Norm ontinuity of the sum and produ t in B (H) implies that the same is true of any polynomial in ~ and V~ , and it then holds for every A 2 C  (U~ ; V~ ) by an =3 argument. U Thus, ~s;t extends to an a tion of R2 on C  (U~ ; V~ ) by automorphisms, whi h we also denote by ~s;t . Consider the map  : C  (U~ ; V~ ) ! C  (U~ ; V~ ) de ned by  (A)

=

1

Z Z 2

(2 )2

0

0

2

~s;t (A) dsdt:

We have  (I ) = I and  (U~ m V~ n ) = 0 when m and n are not both zero. By ontinuity,  (A) is a omplex multiple of I for every A 2 C  (U~ ; V~ ). Now let I be a nonzero C*-ideal of C  (U~ ; V~ ) and hoose A 2 I , A 6= 0. Let v 2 H satisfy Av 6= 0. Then

h~0;0 (A A)v; vi = hA Av; vi = kAvk2 > 0;

and

h~s;t (A A)v; vi = h~s;t (A) ~s;t (A)v; vi = k~s;t (A)vk2  0 for all s; t. Sin e h~s;t (A A)v; v i is a ontinuous fun tion of s and t, it follows that h (A A)v; v i > 0. In parti ular,  (A A) = 6 0.

But the integral whi h de nes  (A A) is approximated in norm by nite sums of the form 1 ~s;t (A A) N

X

where (s; t) ranges over a dis rete N -element subset of T2 ; as ea h term of su h a sum belongs to I , we must have  (A A) 2 I . We on lude that I ontains a nonzero multiple of the identity operator, and hen e that I = C  (U~ ; V~ ). COROLLARY 5.5.9

Under the same hypotheses as the theorem, C  (U~ ; V~ )  = C h (T2 ).

PROOF For k = 1; 2 suppose U~k ; V~k 2 B (Hk ) are unitaries whi h satisfy U~k V~k = e ih V~k U~k . Then let A  B (H1  H2 ) be the C*-algebra

© 2001 by Chapman & Hall/CRC

115 generated by the unitary operators U~1  U~2 and V~1  V~2 . These operators satisfy the ommutation relations required by the theorem, so we

on lude that A has no proper C*-ideals. Consider the restri tion maps k : A ! C  (U~k ; V~k ). These are unital ~k and V~k , so the kernel of k -homomorphisms, and k (A) ontains U is not all of A. Sin e A has no proper C*-ideals, it follows that k has null kernel, and hen e is a -isomorphism. Therefore

C  (U~1 ; V~1 )  = C  (U~2 ; V~2 ): =A In parti ular, this is true if we take U~1 = U and V~1 = V to be the generators of C h ( 2 ).

T

The theorem and orollary both fail when h is a rational multiple of  . Suppose h = 2p=q with p; q 2 N and onsider the operators U~ and V~ a ting on l2 ((Z=q )2 ) by the some formulas as U^ and V^ , namely ~ m;n = e Ue



ihn=

2

em+1;n

and

V~ em;n = eih m=2 em;n+1 ;

Z

but now with m; n 2 =q . It is easy to he k that U~ V~ = e ih V~ U~ still ~ V~ ) is nite dimensional. holds, so Corollary 5.5.9 learly fails, as C  (U; ~ V~ ) We laim that there is a -homomorphism from C^ h ( 2 ) to C  (U; whi h takes U^ to U~ and V^ to V~ ; this will imply that C^ h ( 2 ) has a proper C*-ideal, and hen e is not -isomorphi to C^ h ( 2 ) for h0 any irrational multiple of  . Sin e U~ and V~ obey the same ommutation relation as U^ and V^ , the map U^ 7! U~ , V^ 7! V~ extends -homomorphi ally to the -algebra generated by U^ and V^ . To prove the laim we must show that this map ~ V~ )k  kf (U; ^ V^ )k for any fun tion f of the is ontra tive, i.e., kf (U; form f (x; y ) = jkj;jljN ak;l xk y l . Now given w = (bm;n ) 2 l2 (( =q )2 ) (0  m; n < q ) and M > 0, de ne wM 2 l2 ( 2 ) by

T T T 0

P

Z

Z

n

bm;n for jk j; jlj  M M wm +kq;n+lq = 0 otherwise. Then

^ V^ )wM k=kwM k ! kf (U; ~ V~ )wk=kwk kf (U;

~ V~ )k  kf (U; ^ V^ )k, as desired. We as M ! 1. This implies that kf (U; have proven: COROLLARY 5.5.10

T

If h is a rational multiple of  then C h ( 2 ) has a proper C*-ideal. If 0 also h0 is an irrational multiple of  then C h ( 2 )  6 C h ( 2 ). =

© 2001 by Chapman & Hall/CRC

T

T

116

Chapter 5: C*-algebras

5.6

The GNS onstru tion

We have seen several instan es of apparently dierent C*-algebras whi h turned out to be -isomorphi . In su h ases it an be helpful to draw a distin tion between the abstra t C*-algebra, viewed as a Bana h spa e equipped with a produ t and an involution, and its realization(s) as a C*-subalgebra of B (H). In keeping with this point of view, we now present a general te hnique for nding -homomorphisms from a given C*-algebra into some B (H). Here the order stru ture in C*-algebras be omes important. For A; B 2 B (H) we de ne A  B if hAv; v i  A is positive if A  0.

DEFINITION 5.6.1

h

Bv; v

i for all 2 H. We say v

A multipli ation operator Mf is positive if and only if f  0 almost everwhere, and it follows from the spe tral theorem that A  kAkI for all self-adjoint A 2 B (H). It is also easy to see that A  B implies   C AC  C BC for all A; B; C 2 B (H). PROPOSITION 5.6.2

Let

A

2 (H). The following are equivalent: B

(a) A is positive; (b) A is self-adjoint and sp(A)  [0; 1); and ( ) A = B  B for some B 2 B (H).

PROOF

(a) ) (b): Suppose hAv; v i  0 for all v 2 H. Then by polarization

h

Av; w

i= =

3 X

( i) k

k=0 3 X

( i)k





( + ik w); (v + ik w)

A v



( + ( i)k v ); (w + ( i)k v )

A w

k=0

= hAw; v i; so A is self-adjoint. By the spe tral theorem we may suppose A is multipli ation by some bounded real-valued fun tion f on some L2 (X ); then positivity easily implies that f  0 almost everywhere, so that sp(A), whi h is the essential range of f (Example 3.2.2), is ontained in [0; 1). (b) ) ( ): Suppose A is self-adjoint and sp(A)  [0; 1). Then without loss of generality A = Mf , and the spe trum ondition implies pf . that 2  f  0 almost everywhere. Thus A = Mg = Mg Mg where g =

© 2001 by Chapman & Hall/CRC

117 ( ) ) (a): Suppose A = B  B. Then for any v 2 H we have

hAv; vi = hB  Bv; vi = hBv; Bvi = kBvk2  0; so A  0. By fun tional al ulus, if A belongs to a unital C*-algebra A and A I is invertible in B(H), then (A I) 1 2 A. Thus, part (b) of Proposition 5.6.2 shows that positivity of an element of an abstra t C*-algebra A is independent of the realization of A in B(H). That is, positivity is well-de ned in abstra t C*-algebras. Next we des ribe the obje ts that are used to onstru t representations. Let A be a C*-algebra. A state on A is a bounded linear fun tional ! 2 A su h that k! k = 1 and A  B implies !(A)  !(B). More generally, a weight on A is a linear map ! from a dense subalgebra of A into C with the property that A  B implies !(A)  !(B). DEFINITION 5.6.3

Observe that by fun tional al ulus any self-adjoint operator A an be written as a dieren e of positive operators, so !(A) must be real for any state or weight !. If A is unital, states an equivalently be de ned as linear fun tionals ! whi h preserve positivity and satisfy !(IA ) = 1. This an be shown as follows. If ! preserves positivity then for any self-adjoint A 2 A we have !(A)  !(kAkIA ) = kAk!(IA ); then for any A 2 A, letting a = !(A) we have

j!(A)j2 = !(aA) = !(Re aA + iIm aA) = !(Re aA)  kRe aAk!(IA )  jaj!(IA )kAk: Thus k! k  !(IA ). The reverse inequality is automati , so we on lude that in the presen e of positivity and a unit we have k! k = !(IA ). Example 5.6.4

Let X be a ompa t Hausdor spa e and let  be a regular Borel measure on X with the property that any x X has a neighborhood su h that ( ) < . Then integration against  de nes a weight on C (X ) whose domain is L1 () C (X ). It is a state if and only if  is a probability measure.

O

O 1

© 2001 by Chapman & Hall/CRC

\

2

118

Chapter 5: C*-algebras Example 5.6.5

A  (H) be a C*-algebra and let 2 H be a unit ve tor. Then 7! h i is a state on A.

Let A

B

v

Av; v

As the se ond example indi ates, the on ept of a state on a C*algebra is related to the physi al notion of a state. The next theorem, whi h des ribes the so- alled GNS onstru tion, is a kind of onverse. Let A be a C*-algebra and ! : A0 ! C a weight. De ne a pseudo inner produ t on A0 by setting hA; B i = ! (B  A). Let H! denote the Hilbert spa e formed by fa toring out null ve tors and ompleting, and write A for the element of H! orresponding to A 2 A0 . THEOREM 5.6.6

Let A be a C*-algebra and let ! : A0 ! C be a weight on A. Then there  = AB is a unique -homomorphism  : A ! B (H! ) su h that  (A)B  for all A; B 2 A0 . If A is unital and ! is a state, then IA is a unit ve tor in H! and we have ! (A) = h (A)IA ; IA i.

Let A 2 A0 . We show rst that the map B 7! AB de nes a bounded operator  (A) on H! of norm at most kAk. This is so be ause B  A AB  kA AkB  B , and hen e PROOF

 k2 : kAB k2 = ! (B  A AB )  kA Ak! (B  B ) = kAk2 kB Thus we have a ontra tion  : A0 ! B (H! ). It is learly linear, and it preserves produ ts and adjoints be ause

 (AB )C = ABC =  (A) (B )C and  C i = ! (C  AB ) = ! ((A C ) B ) = hB;   (A )C i h (A)B; for all A; B; C 2 A0 . Sin e  is ontra tive on A0 it extends to a -homomorphism from A into B (H! ). Uniqueness of  follows from density of A0 in A. If A is unital and ! is a state, the remark made after De nition 5.6.3 implies that kIA k2 = ! (IA ) = 1. The nal assertion is trivial. It follows from this theorem that if ! is a state on a unital C*-algebra

A then there is a representation of A on a Hilbert spa e H (i.e., a homomorphism from A into B (H)) whi h ontains a unit ve tor that

gives rise to the state in the manner of Example 5.6.5.

© 2001 by Chapman & Hall/CRC

119 Next we indi ate how some of the representations we have been using

an be obtained from the GNS onstru tion. Example 5.6.7

(a) Consider the quantum torus algebra C^ h (T2 )  B (l2 (Z2 )). The map ! (A) = hAe0;0 ; e0;0 i is a state on C^ h (T2 ), and the GNS onstru tion gives rise to a representation of C^ h (T2 ) whi h is equivalent to the original representation on l2 (Z2 ). The identi ation of H! with l2 (Z2 ) takes L k;l to ek;l . The state ! an also be de ned on C h (T2 ) in terms of the automorphisms ~ used in the proof of Theorem 5.5.8: we have  (A) = ! (A)I . This de nition is less ir ular than the one above be ause it does not depend on already having the l2 (Z2 ) representation. (b) For the non ommutative planeR algebra C0h (R2 ), we have a weight de ned by ! (Lf ) = 2 f^(0; 0) = f for f 2 S (R2 ). Then the GNS

onstru tion gives rise to our representation of C0h (R2 ) on L2 (R2 ). In parti ular, for f; g 2 S (R2 ) we have ! (Lg Lf ) = ! (Lgh f ) =

f so the orresponden e of L isometri .

2 H!

Z

with f

f g;

2 S (R2) 

R

L2 ( 2 ) is

We now indi ate why we all the L2 (R2 ) model of the quantum plane the \tra ial" representation. For ertain Hilbert spa e operators it is possible to de ne a tra e in a way that generalizes the tra e of nite dimensional matri es (see Se tion 6.3). The operators L~ f 2 B (L2 (R)) R fall into this ategory, and we have tr(L~ f ) = 21h f for f 2 S (R2 ). Thus, 2 h times the tra e on B (L2 (R)) restri ts to a weight on C0h (R2 ) su h that the resulting GNS representation is the L2 (R2 ) model. C*-algebras an be hara terized abstra tly. They are just those Bana h algebras equipped with an antilinear involution  su h that ( ) =   and k k = kk2 for all ; . This result, known as the Gelfand-Neumark theorem, is proven by using the GNS onstru tion to nd a Hilbert spa e representation of su h an abstra tly given algebra. It is presented in nearly every standard exposition of C*-algebras; we will not prove it here, be ause it requires signi ant Bana h algebra preliminaries that we prefer to avoid. A tually, one often does not need to use the Gelfand-Neumark theorem, be ause C*-algebras are usually de ned in terms of their representations, or an at least be given a representation without mu h eort. The most prominent ex eption is the quotient onstru tion, whi h is not evidently represented on any Hilbert spa e but is in fa t always an abstra t C*-algebra. One an prove this using the Gelfand-Neumark

© 2001 by Chapman & Hall/CRC

120

Chapter 5: C*-algebras

theorem, but it is not mu h more diÆ ult to prove it dire tly from Theorem 5.6.6; we do this now. For any C*-ideal I of a C*-algebra A, write

kA + Ik = inf fkA + B k : B 2 Ig for the quotient norm in A=I . LEMMA 5.6.8

Let I be a C*-ideal of a unital C*-algebra A and let A 2 A. kA A + Ik = kA + Ik2 .

PROOF

Then

One inequality is easy: for any B 2 I we have

kA A + Ik  k(A + B  )(A + B )k = kA + B k2 ; and taking the in mum over B 2 I yields kA A + Ik  kA + Ik2 . For the onverse, observe rst that if

A

is self-adjoint then

kA + Ik = inf fkA + B k : B 2 I ; B = B  g be ause for any B we have

kA + Re B k  12 (kA + B k + kA + B  k) = kA + B k: Thus let A 2 A and B 2 I be self-adjoint and let  > 0. For any realvalued f 2 C (sp(B )) su h that f (0) = 0, 0  f (x)  1 for all x, and f (x) = 1 for jxj   we have kA A + B k  k(A A + B )(IA f (B ))k  kA A(IA f (B ))k kB (IA f (B ))k  kA A(IA f (B ))k   k(IA f (B ))A A(IA f (B ))k  = kA(IA f (B ))k2   kA + Ik2 : Finally, taking the in mum over all self-adjoint operators B 2 I yields kA A + Ik  kA + Ik2 , whi h is enough. We need one other, slightly more te hni al lemma. LEMMA 5.6.9

Let I be a C*-ideal of a unital C*-algebra A and let A 2 A be positive. Then

kA + bIA + Ik  kA + Ik + b

© 2001 by Chapman & Hall/CRC

121 for all b 2 R.

Say A  B (H ). First onsider the ase b  kA + Ik. Let 2 I ; as in Lemma 5.6.8, we may assume B = B  . Fix  > 0 and a real-valued fun tion f 2 C (sp(B )) su h that f (0) = 0, 0  f (x)  1 for all x, and f (x) = 1 for jxj  , just as in the proof of Lemma 5.6.8. Let U = f (B ). Find a unit ve tor v0 2 H su h that h(I U )A(I U )v0 ; v0 i  k(I U )A(I U )k  and let w0 = (I U )v0 =k(I U )v0 k. Then hoose g 2 C (sp(B )) su h that g (x) = 0 for jxj  , 0  g (x)  1 for all x, and g (x) = 1 for jxj  2. Let V = g(B ). Then (1 g)(1 f ) = 1 f , so k(A + bI )(I V )k  h(A + bI )(I V )w0 ; (I V )w0 i = h(A + bI )(I U )v0 ; (I U )v0 i=k(I U )v0 k2  k(I U )A(I U )k + b   kA + Ik + b :

PROOF

B

But

kA + bI + B k  k(A + bI + B )(I

k  k(A + bI )(I V )k 2 as in Lemma 5.6.8, so we on lude that kA + bI + B k  kA + Ik + b 3. Taking  ! 0 ompletes the proof in the ase b  kA + Ik. If b < kA + Ik then let B0 2 I satisfy B0 = B0 and kA + B0 k < b. Then A0 = kA + B0 kI (A + B0 ) is positive, so the b  kA + Ik ase V)

implies

kA + bI + B0

B

k = kA0 (kA + B0 k + b)I + B k  k A0 + Ik (kA + B0 k + b)  kA + B0 k + b

for all B 2 I . Thus

kA + bI + Ik  kA + B0 k + b ;



and taking the in mum over B0 yields the desired inequality. THEOREM 5.6.10

Let I be a C*-ideal of a C*-algebra A. Then there is a Hilbert spa e H and a -isomorphism from A=I into B (H).

It is lear that the produ t and -operation on A des end to I . We will nd a Hilbert spa e H and a -homomorphism  : A ! H

PROOF

A=

© 2001 by Chapman & Hall/CRC

122

Chapter 5: C*-algebras

su h that ker( ) = I and k (A)k = kA + Ik for all A 2 A. If A has no unit we an adjoin one without loss of generality, be ause I will still be an ideal of A+ and A=I will be a C*-subalgebra of A+ =I . Thus, we assume A is unital. For any positive element A 2 A, let

E = faA + bIA + B : a; b 2 R; B 2 I ; B = B  g be the self-adjoint part of the span of the ideal and IA . Then de ne !1 : E ! R by

I

and the elements A

!1 (aA + bIA + B ) = a
kA + Ik + b: (This map is not obviously well-de ned if A = b0 IA + B 0 for some b0 2 C and B 0 2 I . One an either prove that it indeed is well-de ned even in this ase using Lemma 5.6.9 with A = 0, or one an observe that this parti ular ase is not a tually needed in the sequel.) Lemma 5.6.9 implies that k!1 k  1, and sin e !1 (IA ) = 1 we have k!1 k = 1. By the Hahn-Bana h theorem we an extend !1 to a real linear fun tional !2 on the self-adjoint part of A with k!2 k = 1. Moreover, if B 2 A is positive then kB kIA B has norm at most kB k, so

kB k !2 (B ) = !2 (kB kIA B )  kB k; and hen e !2 (B )  0. Thus the omplex linear extension ! of !2 to A

is a state. For ea h positive A 2 A let ! be su h a state and let ! : A ! B (H! ) be the -homomorphism given by Theorem 5.6.6; then let

=

M

! : A ! B

M

H!



be the dire t sum of these -homomorphisms. It is learly itself a homomorphism and hen e a ontra tion, and its L kernel ontains I , so it des ends to a ontra tion from A=I into B ( H! ). Conversely, for ea h A 2 A we have

h! (A A)IA ; IA i = !(A A) = kA A + Ik for some ! , so

k(A)k2 = k(A A)k  kA A + Ik = kA + Ik2 (using Lemma 5.6.8). Thus  is isometri for the quotient norm. This

ompletes the proof. Thus, quotients of C*-algebras are themselves C*-algebras.

© 2001 by Chapman & Hall/CRC

123 Sin e separable Hilbert spa es are better-behaved than nonseparable ones, it is worth pointing out that we an take H to be separable in Theorem 5.6.10 provided A is a separable C*-algebra. For it is suÆ ient to ensure that  be isometri on a ountable dense subset of A, so that we only need to use ountably many statesL!, and it is also easy to see that ea h H! is separable if A is. So H = H! will also be separable. 5.7

Notes

Basi material on C*-algebras is overed in many texts. Besides the books ited in earlier hapters, we also mention [40℄ on this topi . For more on the quantum plane see [60℄, and for more on quantum tori see [60℄ and [16℄. The latter are also alled irrational rotation algebras when h = is irrational, the ase of greatest interest. Our treatment of K (H) in Se tion 5.4 follows [3℄. The dis ussion of Fourier series on non ommutative tori in Se tion 5.5 follows [71℄.

© 2001 by Chapman & Hall/CRC

© 2001 by Chapman & Hall/CRC

Chapter 6

Von Neumann algebras 6.1

The algebras

l1(X )

In this hapter we dis uss quantum measure theory. This is formally rather similar to quantum topology; in both ases we treat the given stru ture (topologi al or measure theoreti ) in terms of the algebra of

omplex-valued fun tions whi h are ompatible with that stru ture. The algebras whi h arise in the latter ase are a tually a spe ial ase of the former: every von Neumann algebra is a C*-algebra (although this is not always the best way to think about them). We begin with the fun tional analyti obje ts whi h re e t pure set theoreti stru ture, the algebras l1 (X ). The theory of these algebras

losely parallels that of the algebras C (X ) presented in Se tion 5.1. The key dieren e is that l1 (X )  = l1 (X ) is always a dual spa e, and the weak* topology is favored over the norm topology. Two observations about the weak* topology on l1 (X ) will be used repeatedly. First, if (f ) is a bounded net in l1 (X ) then f ! f weak* if and only if f ! f pointwise. The forward dire tion follows by pairing with the hara teristi fun tions x 2 l1 (X ): we have f (x)

=

X

f x

!

X

f x

= f (x)

P

P

for all x 2 X . Conversely, if f ! f pointwise then f g ! f g for every nitely supported fun tion g 2 l1 (X ); sin e (f ) is bounded, an =3 argument then shows that f g ! f g for all g 2 l1 (X ). Se ond, we require the Krein-Smulian theorem, whi h states that a linear subspa e E of a dual Bana h spa e V is weak* losed if and only if the interse tion of E with the losed unit ball of V is weak* losed. That is, E is weak* losed if it is stable under weak* onvergen e of bounded nets. This means that we usually only need to onsider weak* topologies on bounded sets, where they are typi ally easier to des ribe, as we just saw for l1 (X ).

P

P

125 © 2001 by Chapman & Hall/CRC

126

Chapter 6: Von Neumann algebras

The following are the basi on epts surrounding l1 spa es. As in De nition 5.1.1, we immediately formulate the fundamental de nitions at a level of generality that en ompasses the non ommutative (quantum) setting. Let M be a dual Bana h spa e equipped with a produ t and an involution. A W*-subalgebra of M is a weak* losed C*-subalgebra of M. A W*-ideal of M is a weak* losed C*-ideal of M. If M is unital, the weak* spe trum of M is the set sp (M) of all weak* ontinuous, unital -homomorphisms from M to .

DEFINITION 6.1.1

C

The main tool we need in this se tion is a version of the StoneWeierstrass theorem for l1 (X ). It follows easily from the next lemma. LEMMA 6.1.2

Let M be a unital W*-subalgebra of l1 (X ), let and let S  . Then f 1 (S ) 2 M.

R

f

2 M be real-valued,

PROOF Let a = kf k1 and let (g)  C [ a; a℄ be a bounded net of

ontinuous fun tions whi h onverges pointwise to S on [ a; a℄. By the Stone-Weierstrass theorem, we may assume every g is a polynomial. Then the net (g Æ f ) is bounded in M and onverges pointwise to S Æ f = f 1 (S) , so this limit also belongs to M by weak* losure of M in l1 (X ). PROPOSITION 6.1.3

Let X be a set and let M be a unital W*-subalgebra of l1 (X ) whi h separates points. Then M = l1 (X ).

PROOF Let g 2 l1(X ) and let x; y 2 X be distin t. Then there exists f 2 M su h that f (x) 6= f (y ), and by the lemma the hara teristi fun tion of some set whi h ontains x but not y will belong to M. By taking produ ts of su h fun tions, for any nite set S  X not ontaining x we an nd in M a hara teristi fun tion whi h vanishes on S and takes the value 1 on x. But x is a weak* limit of su h fun tions, so x 2 M. As this is true for every x 2 X , we on lude that M ontains every nitely supported fun tion, and hen e, by weak* losure, it equals l1 (X ). Now we pro eed to the relation between algebrai stru ture in l1 (X )

© 2001 by Chapman & Hall/CRC

127 and set theoreti stru ture in in the

l 1 (X )

X.

Quotients and subalgebras are related

setting in the same way they are related in the

C (X )

setting. Example 6.1.4



Let

:

X !Y

be a surje tion.

weak* ontinuously and and

Then the map

-isomorphi ally

embeds

C : f 7! f Æ  l1 (Y ) in l1 (X ),

C (l1 (Y )) is a unital W*-subalgebra of l1 (X ).

C (l1 (Y )) is weak* losed in l1 (X ) is a

onsequen e of the Krein-Smulian theorem. If (C f ) is a bounded net in C (l 1 (Y )) and C f ! g pointwise, then (f ) is bounded in l 1 (Y ) so we an pass to a subnet whi h onverges pointwise to f 2 l 1 (Y ), and then weak* ontinuity implies that C f = g . Thus C (l 1 (Y )) is weak* In this example the fa t that

losed.

PROPOSITION 6.1.5

Let X be a set and let M be a unital W*-subalgebra of l1 (X ). Then there is a set Y and a surje tion  : X ! Y su h that M = C (l1 (Y )). PROOF For x; y 2 X let x  y if f (x) = f (y ) for all f 2 M. Take Y = X=  and let  : X ! Y be the proje tion map. Then the map C : f 7! f Æ  takes l1 (Y ) -isomorphi ally into l1 (X ), and as every

fun tion in

M

respe ts



it follows that

M

is ontained in the image

l1 (Y ) under C . Also, C 1 takes M onto a unital W*-subalgebra of l 1 (Y ) whi h separates points, and hen e onto l 1 (Y ) by Proposition 6.1.3. Thus M = C (l 1 (Y )). of

Next, we turn to ideals and subsets. Example 6.1.6

Let

X

be a set and let

1 a W*-ideal of l (X ).

K  X.

Then

I

=

ff 2

l1 (X ) : f jK

= 0g is

PROPOSITION 6.1.7

Let X be a set and let I be a W*-ideal of l1 (X ). Then there is a subset K  X su h that I = ff 2 l1 (X ) : f jK = 0g. PROOF

satisfy I

+

g jK

K = fx 2 X : f (x) = 0 for all x 2 I g and let g 2 l1 (X ) g 2 I . Let I + = I + C
1X ; then W*-subalgebra of l 1 (X ). Observe that if x; y 62 K then

Let

= 0; we must show that

is a unital

© 2001 by Chapman & Hall/CRC

128

Chapter 6: Von Neumann algebras

x and y are not identi ed by the equivalen e relation in the proof of Proposition 6.1.5. Thus, that result implies g 2 I + . If K 6= ; then this immediately yields g 2 I . However, if K = ; then the ideal property implies that for every x 2 X the hara teristi fun tion x belongs to I . Thus every nitely supported fun tion on X belongs to I , and weak* losure then implies I = l1 (X ).

The following three orollaries are all routine; their proofs are similar to, but easier than, those of the orresponding results in Se tion 5.1. COROLLARY 6.1.8

If I is a W*-ideal of l1 (X ) then l1 (X )=I  = l1 (K ), where K  X is the set of points on whi h every fun tion in I vanishes.

For ea h x 2 X let x^ : f 7! f (x) be the orresponding evaluation fun tional on l1 (X ). COROLLARY 6.1.9

The orresponden e x $ x ^ is a bije tion between X and sp (l1 (X )). COROLLARY 6.1.10

The weak* ontinuous, unital -homomorphisms from l1 (X ) to l1 (Y ) are in a natural bije tion with the fun tions from Y to X . 6.2

The algebras

L1(X )

There are results for L1 (X ) (for X a  - nite measure spa e) whi h are analogous to some of the results of the last se tion for l1 (X ). The main dieren e is that the role of a \quotient" of X is played here by a oarser  -algebra on X . If X is  - nite then L1 (X )  = L1 (X ) , and forRa bounded R net (f ) in L1 (X ) we have f ! f weak* if and only if S f ! S f for all nite measure subsets S of X . The forward impli ation Ris simply a R

onsequen e of the fa t that S 2 L1 (X ), be ause S f = fS is the pairingRof f 2 RL1 (X ) with S . For the reverse impli ation, observe R R that if S f ! S f for all S  X of nite measure, then f g ! fg for all simple fun tions g 2 L1 (X ), by linearity; then density of the simple fun tions in L1 (X ) and the fa t that (f ) is bounded imply that R R f g ! fg for all g 2 L1 (X ), i.e., f ! f weak*. Again, before pro eeding we need a Stone-Weierstrass theorem for L1 (X ).

© 2001 by Chapman & Hall/CRC

129 LEMMA 6.2.1

Let X be a  - nite measure spa e, let M be a unital W*-subalgebra of L1 (X ), let f 2 M be real-valued, and let S  R be Borel measurable. Then f 1 (S ) 2 M.

Let a = kf k1. The map  7! (S ) is a bounded linear fun tional of norm one on M [ a; a℄  = C [ a; a℄ , so there is a bounded net R (g )  C [ a; a℄ whi h onverges to it weak*. That is, we have g d ! (S ) for all  2 M [ a; a℄. By the Stone-Weierstrass theorem we may assume ea h g is a polynomial.R Let h 2 L1 (X ). Then the map g 7! X (g Æ f )h is a bounded linear fun tional on CR [ a; a℄, so there is a measure R 2 M [ a; a℄ su h that R X (g Æ f )h = gd for all g . Then (K ) = f 1 (K ) h holds for any

losed set K  [ a; a℄ and hen e for every Borel set. Therefore

PROOF

Z X

(g Æ f )h =

Z

This shows that g Æ f ! f

g d ! (S ) = 1 (S )

Z X

f

weak*, and so f

1 (S )

1 (S )

h:

2 M.

PROPOSITION 6.2.2

Let X be a  - nite measure spa e and suppose M is a unital W*subalgebra of L1 (X ) whi h measurably separates points. Then M = L1 (X ).

Let be the smallest -algebra with respe t to whi h every fun tion in M is measurable and let 0 be the olle tion of sets S  X su h that S 2 M. The lemma implies that 0 generates as a algebra. But 0 is a -algebra itself, so 0 = . By the de nition of measurable separation of points (De nition 5.2.3), it follows that M

ontains the hara teristi fun tion of every measurable set, and hen e M = L1 (X ).

PROOF

Now we show how W*-subalgebras orrespond to sub -algebras. Example 6.2.3

Let X and Y be  - nite measure spa es and let  : X ! Y be a measurable surje tion su h that  1 (S ) is null in X if and only if S is null in Y . De ne M = ff Æ  : f 2 L1 (Y )g; this is a unital W*-subalgebra of L1 (X ), and the map C : f 7! f Æ  is a weak*

ontinuous -isomorphism from L1 (Y ) onto M.

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130

Chapter 6: Von Neumann algebras

In this example the map C is well-de ned be ause the inverse image under  of a null set is null, and it is an isometry be ause the inverse image of a positive measure set has positive measure. To see that M is weak* losed in the example, let be the -algebra

= f 1 (S ) : S  Y is measurableg: It is ontained, perhaps properly, in the -algebra of measurable subsets of X . Let  and  be the original measures on X and Y , and let 0 be the inverse image of  , de ned on . Possibly repla ing  and  with equivalent measures, we may assume that  is nite and  (S ) = ( 1 (S )) for all measurable S  Y , so that 0 = j . Then M = L1 (X; 0 ). (It is lear that every fun tion in M is 0 -measurable, and

onversely M ontains the hara teristi fun tion of every 0 -measurable set, so it ontains L1 (X; 0 ).) Now let (f ) be a bounded net in M and suppose f ! f weak* in L1 (X ). Sin e  is assumed to be nite, we have L2 (X; )  L1 (X; ). Then hf; gi = 0 for every g 2 L2 (X; ) whi h is orthogonal to M be ause hf ; gi = 0 for every su h g, for all . Thus f belongs to the losure of M in L2 (X; ), whi h is learly L2 (X; 0 ), and sin e f is essentially bounded this implies f 2 M. Thus, M is weak* losed. PROPOSITION 6.2.4

Let X be a  - nite measure spa e and let M be a W*-subalgebra of L1 (X ). Then there is a  - nite measure spa e Y and a measurable surje tion  : X ! Y su h that  1 (S ) is null in X if and only if S is null in Y , and M = C (L1 (Y )). PROOF Let  be the given measure on X ; possibly repla ing it with an equivalent measure, we may assume it is nite. Let be the  -algebra generated by the sets f 1 (S ) for S  R Borel measurable and f 2 M; let  = j ; and let Y be the set X equipped with the measure  . The identity map  : X ! Y is then a measurable surje tion for whi h  1 (S ) is null in X if and only if S is null in Y . It is lear that M  L1 (Y ), and the reverse ontainment follows from Proposition 6.2.2.

Next we hara terize W*-ideals. Example 6.2.5

Let X be a  - nite measure spa e and let K subset. Then

© 2001 by Chapman & Hall/CRC

 X be a measurable

131 I = ff 2 L1 (X ) : f jK = 0g 1 is a W*-ideal of L (X ). PROPOSITION 6.2.6

Let I be a W*-ideal of L (X ). Then there is a measurable subset K of X su h that I = ff 2 L (X ) : f jK = 0g. 1

1

PROOF Let  be the olle tion of subsets S  X with the property that S 2 I . Ordered by in lusion, it is easy to see that  is dire ted upwards. Then the net (S ) for S 2  onverges weak* be ause every fun tion in L1 (X ) an beR written as a linear ombination of positive fun tions in L1(X ), and S h onverges for every positive h 2 L1 (X ). So there must exist f 2 I su h that S ! f weak*. Now f is the hara teristi fun tion of a set K  X with the property that every S 2  is almost everywhere ontained in K . Letting K = X K , the fa t that K 2 I and the ideal property of I imply that every fun tion whi h vanishes almost everywhere on K is in I . Conversely, if f 2 I and f does not vanish almost everywhere on K then (again using the ideal property of I ) it follows that I ontains the hara teristi fun tion of some positive measure subset of K . But this ontradi ts the de nition of K , so every fun tion in I must vanish on K . 0

0

0

0

PROPOSITION 6.2.7

Let I = ff 2 L1 (X ) : f jK Then L1 (X )=I  = L1 (K ).

= 0g be

a weak* losed ideal of

L (X ). 1

The last proposition is trivial. 6.3

Tra e lass operators

Like l (X ) and L (X ), B (H) is a dual spa e. The goal of the present se tion is to prove this result. Then we will dis uss its W*-subalgebras and ideals in the following se tions. It is not too hard to prove that B (H) is a dual spa e using abstra t methods. However, we prefer to spend a little time des ribing its predual

on retely. The predual of B (H) is naturally realized as an ideal (but not weak* losed, or even norm losed) in B (H), mu h as l1 (X ) is an ideal in l (X ). It is alled the \ideal of tra e lass operators on H." We develop the properties of tra e lass operators through a series of lemmas. 1

1

1

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132

Chapter 6: Von Neumann algebras

P

LEMMA 6.3.1

Let A 2 B (H) be positive. If the sum hAe ; e i is nite for some orthonormal basis (e ) of H then it is nite, and onverges to the same value, for every orthonormal basis. If this happens then A is ompa t.

Let (e ) and (~e~ ) be orthonormal bases of H. Then

PROOF

Xh 

P h

A1=2 ~~ ; e~~ ~ Ae

where

Ae ; e

i=

Xk 

A

1=2

e

k2 =

Xh ;~

i

1 =2 A e ; e ~~ 2

i = P;~ hA1=2 e~~ ; ei2 .

is de ned by fun tional al ulus, and similarly we have Thus the two sums are equal; in parti ular, if one is nite so is the other. Now suppose A is not ompa t. By Proposition 5.4.9 it follows that there exists  > 0 su h that E (S ) is in nite dimensional, where E is the spe tral measure of A and S = fx 2 sp(A) : jxj  g. Letting (e ) be an orthonormal basis of H whi h ontains an orthonormal basis of E (S ), we then have hAe ; e i   for every e 2 E (S ). This shows that hAe; e i does not onverge.

P

P

For any A 2 B (H) we de ne tr(A) to be the sum hAe ; e i, provided that it onverges absolutely, and to the same value, for every orthonormal basis (e ) of H. Otherwise we say that tr(A) is not wellde ned. The rank of an operator is the dimension of its range, and an operator has nite rank if its range is nite dimensional. Equivalently, F 2 B (H) has nite rank if its kernel has nite odimension. Sin e ran(F )? = ker(F  ), it follows that F  has nite rank if F does. LEMMA 6.3.2

Suppose F

2 B (H) has nite rank. Then tr(F ) is well-de ned.

Let K = ker(F )? _ ran(F ). We an write F as F = F jK  0 on K  K Then F jK 2 B (K) an be de omposed into its real and imaginary parts, ea h of whi h an then be written as a dieren e of two positive operators. So F an be expressed as a linear ombination of four positive nite rank operators. Ea h of these has a well-de ned tra e by Lemma 6.3.1, so the same must be true of F .

PROOF

?.

LEMMA 6.3.3

Let A; F 2 B (H) and suppose F has nite rank. Then tr(AF ) = tr(F A) and jtr(AF )j  tr(jAj)kF k.

© 2001 by Chapman & Hall/CRC

133

PROOF It is lear that AF and F A both have nite rank, so tr(AF ) and tr(F A) are well-de ned. For the rst statement it suÆ es to take F self-adjoint. In that ase let (e ) be an orthonormal basis of H onsisting of eigenve tors of F and let  be the orresponding eigenvalues (all but nitely many of whi h are zero). Then X

hAF e ; ei =

X

 hAe ; e i =

X

hF Ae ; e i;

so tr(AF ) = tr(F A) as laimed. For the se ond statement, if tr(jAj) = 1 we are done. Otherwise jAj is ompa t and we an nd an orthonormal basis (~e~ ) of H onsisting of eigenve tors of jAj, with orresponding eigenvalues ~~ . Write A = U jAj as in Lemma 5.4.7; then X

jtr(F A)j =



hF U jAje~~ ; e~~ i 

X

~ ~ kF k = tr(jAj)kF k;

as desired. LEMMA 6.3.4

Let A; F

2 B (H) and suppose F

has nite rank. Then

tr(jAj) = supfjtr(AF )j : F has nite rank and kF k  1g = supfjtr(F A)j : F has nite rank and kF k  1g:

PROOF We will show that tr(jAj)  supfjtr(F A)jg; Lemma 6.3.3 implies the rest. Let (e ) be an orthonormal basis of H, write A = U jAj as in Lemma 5.4.7, and let P be the orthogonal proje tion onto the span of some nite set of basis ve tors. Then, taking F = P U  , we have tr(F A) = tr(P jAj); and as P tends to I the right side tends to tr(jAj), whi h is enough. We are now ready for the following de nition. DEFINITION 6.3.5

Let A 2 B (H). Then A is tra e lass if its tra e norm kAkT C = tr(jAj) is nite. The set of all tra e lass operators is denoted T C (H). LEMMA 6.3.6

Let A 2 T C (H) be self-adjoint. Then we an write A = A+ 0  A+ ; A  jAj and tr(A+ ); tr(A )  tr(jAj).

© 2001 by Chapman & Hall/CRC

A where

134

Chapter 6: Von Neumann algebras

De ne A+ = 12 (jAj + A) and A = 21 (jAj A); regarding A as a multipli ation operator, it is lear that 0  A+ ; A  jAj, and this immediately implies that tr(A+ ); tr(A )  tr(jAj).

PROOF

PROPOSITION 6.3.7

T C (H) is a ve tor spa e and k
kTC is a omplete norm. If A 2 T C (H) and B 2 B (H) then A , AB , and BA all belong to T C (H).

PROOF

Everything but ompleteness is a straightforward onsequen e of Lemma 6.3.4. To prove ompleteness, let (An )  T C (H) and suppose kAn kTC < 1; by Lemma 2.1.7, it will suÆ e to show that An onverges in T C (H). Observe rst that kRe An kTC ; kIm An kTC  kAn kTC . Thus by de omposing into real and imaginary parts we an redu e to the ase where ea h An is self-adjoint. Then by Lemma 6.3.6 we an redu e to the ase where ea h An is positive. Now kAn k  tr(An ) in this ase, so An onverges in B (H ). Let A be its limit. Then we have hAv; v i = hAn v; vi for all v 2 H , and summing over an orthonormal basis yields tr(A) = tr(An ) and

P

P

P P

tr(A So A 2 plete.

TC

(H) and

P

N X

P

1 X

An ) ! 0: n ) = tr( n=1 n=N +1

A

A

n = A, and we on lude that

TC

(H) is om-

LEMMA 6.3.8

For any A 2 T C (H), tr(A) is well-de ned. For any A 2 T C (H) and 2 B (H) we have tr(AB ) = tr(BA) and jtr(AB )j  kAkTC kB k. The nite rank operators are dense in T C (H), and T C (H)  K (H). B

PROOF It follows from Lemmas 6.3.6 and 6.3.7 that every tra e

lass operator an be expressed as a linear ombination of positive tra e

lass operators. This implies the rst assertion; together with the easy fa t that positive tra e lass operators are approximated by nite rank operators, it implies density of the nite rank operators in T C (H); and together with Lemma 6.3.1 it implies that every tra e lass operator is

ompa t. The remainder is proven exa tly as in Lemma 6.3.3. The pre eding results give a fairly omplete pi ture of T C (H), and we an now use them to identify the dual of T C (H) with B (H).

© 2001 by Chapman & Hall/CRC

135 THEOREM 6.3.9

For all B 2 B (H) the map !B : A 7! tr(AB ) de nes a bounded linear fun tional on T C (H). The map B 7! !B isometri ally identi es B (H) with T C (H) . On bounded subsets of B (H) the weak* topology agrees with the weak operator topology.

PROOF Boundedness of !B follows from Lemma 6.3.8; in fa t, this shows k!B k  kB k. Conversely, for any ve tors v; w 2 H let Av;w be the operator Av;w u = hu; wiv ; then kAv;w kT C = kv kkwk and jtr(BAv;w )j = jhBv; wij. Taking the supremum over all unit ve tors v and w yields k!B k  kB k. Next, given ! 2 T C (H) , the map (v; w) 7! ! (Av;w ) is a bounded sesquilinear form, and hen e there exists B 2 B (H) su h that hBv; wi = ! (Av;w ) for all v; w 2 H. Thus !B (Av;w ) = tr(BAv;w ) = ! (Av;w ). But the operators Av;w span the nite rank operators, and by density of the latter in T C (H) we have ! = !B . So every bounded linear fun tional on T C (H) is of the form !B for some B 2 B (H). If (B ) is a bounded net in B (H), then B ! B weak* if and only if tr(B A) ! tr(BA) for all A 2 T C (H), whi h holds if and only if tr(B Av;w ) ! tr(BAv;w ) for all v; w 2 H by density of the nite rank operators in T C (H). That is, B ! B weak* if and only if hB v; wi ! hBv; wi for all v; w 2 H. The weak* topology on B (H) is also alled the ultraweak or  -weak topology. 6.4

The algebras

B (H)

Now that we have a weak* topology on B (H), we an onsider W*subalgebras and ideals. We will dis uss general W*-subalgebras in the next se tion. W*-ideals, however, are trivial: PROPOSITION 6.4.1

Let

I be a W*-ideal of B (H). Then I = 0 or I = B (H).

PROOF Suppose I 6= 0 and let A 2 I be nonzero. Then for a suitable operator B the produ t AB 2 I has rank one. Multiplying AB on the left and right by appropriate rank one operators shows that I

ontains every rank one operator, and hen e I ontains all nite rank operators. Now if A 2 T C (H) and the map B 7! tr(AB ) annihilates every nite rank operator then we must have kAkT C = 0 and hen e A = 0.

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136

Chapter 6: Von Neumann algebras

This shows that the nite rank operators are weak* dense in B (H), and therefore we on lude that I = B (H). Although this result is somewhat disappointing, there is a substitute for the notion of an ideal whi h yields a better analog of Propositions 6.1.7 and 6.2.6. First we introdu e the substitute notion and show that it is the same thing as an ideal in the ommutative ase. Let A be a C*-algebra and let B  A be a B is hereditary if A 2 A, B 2 B, and 0  A  B imply

DEFINITION 6.4.2

C*-subalgebra. A 2 B.

PROPOSITION 6.4.3

Let A be an abelian C*-algebra. Then a C*-subalgebra of A is hereditary if and only if it is an ideal. PROOF By Theorem 5.3.5 (and the omment following it) we may assume that A = C (X ) for some ompa t Hausdor spa e X . Sin e any C*-ideal of C (X ) onsists of all fun tions whi h vanish on some

losed subset K  X , it is easy to see that every C*-ideal is hereditary. Conversely, suppose B  A is a hereditary C*-subalgebra and let f 2 A and g 2 B; we must show that f g 2 B. Let f = 3k=0 ik fk and 3 k g= k=0 i gk be de ompositions of f and g into positive fun tions. By fun tional al ulus ea h gk belongs to B , and it will suÆ e to show that fj gk 2 B for all j; k . But for  = 1=kfj k1 we have 0  fj gk  gk , and hen e fj gk 2 B as desired.

P

P

Now we show that the hereditary W*-subalgebras of B (H) orrespond to losed subspa es of H. Example 6.4.4

Let

P

be a proje tion in

B (H).

Then

PB (H)P = fPAP : A 2 B (H)g is a hereditary W*-subalgebra of

PROPOSITION 6.4.5

B (H).

Let M be a hereditary W*-subalgebra of B (H). Then for some proje tion P 2 M.

© 2001 by Chapman & Hall/CRC

M = P B (H)P

137 Let A 2 M be self-adjoint and de ne jAj1=n by fun tional

al ulus. Then jAj1=n onverges weak operator, and hen e weak*, to R f0g (A). So if A 2 M is self-adjoint, then the proje tion onto its range also belongs to M. If P1 and P2 are proje tions and v ? ran(P1 + P2 ) then PROOF

kP1 v k2 = hP1 v; v i  h(P1 + P2 )v; v i = 0:

This shows that the range of P1 is ontained in the losure of the range of P1 + P2 . Thus, by the rst paragraph of the proof the sequen e (P1 + P2 )1=n onverges to a proje tion in M whi h is larger than P1 , and likewise it is larger than P2 . So the set of proje tions in M is dire ted upwards, and by weak* losure its limit belongs to M. Let P be this limit. Now if A 2 B (H) is positive, then P AP is also positive and 0  P AP  kAkP , so P AP 2 M. This shows that P B (H)P  M. Conversely, if A 2 M is positive then P dominates the proje tion onto the range of A, and hen e A = P AP . By linearity we on lude that M  P B (H)P as well. We on lude this se tion by lassifying the weak* ontinuous automorphisms of B (H). With a little more work the assumption of weak* ontinuity an be dropped, and one an a tually lassify all -homomorphisms from B (H) into B (K), but we will not do this. Our result resembles Corollary 6.1.10 in the spe ial ase that Y = X , when the latter implies that the unital -isomorphisms from l1 (X ) onto itself are in one-toone orresponden e with the automorphisms of X . The following is the

orresponding onstru tion for B (H). Example 6.4.6

Let

U

2

B (H) be unitary.

ontinuous unital

Then the map

-isomorphism

from

A 7! UAU 

B (H) onto itself.

is a weak*

PROPOSITION 6.4.7

Let  : B (H)  = B (H) be a weak* ontinuous unital -isomorphism. Then  (A) = UAU  for some unitary U 2 B (H). PROOF Fix a a rank one proje tion P 2 B (H) and unit ve tors v 2 ran(P ) and w 2 ran( (P )). Observe that for any A 2 B (H) we have P AP = f (A)P for some omplex number f (A). Thus

k (A)wk2 = k (AP )wk2 = h (P A AP )w; wi = f (A A)h (P )w; wi = f (A A);

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138

Chapter 6:

Von Neumann algebras

and kAv k2 = f (A A) by a similar omputation. So if A; B 2 B (H) satisfy Av = Bv then k (A B )v k2 = k(A B )v k2 = 0, and this shows that the map U : Av 7!  (A)w is well-de ned. The pre eding also shows that U is an isometry. For any A; B 2 B (H) we have

UAU  ( (B )w) = UABv =  (AB )w =  (A)( (B )w): Thus  (A) = UAU  . Finally, if U is not surje tive then  (A)v = 0 for any v orthogonal to the range of U , ontradi ting surje tivity of  . Thus U is surje tive, and hen e unitary. 6.5

Von Neumann algebras

A von Neumann algebra (or W*-algebra) is a W*-subalgebra of some B (H). As with C*-algebras, sometimes we will onsider them as abstra t spa es. In fa t, they have a simple abstra t hara terization, the easy dire tion of whi h is the following. PROPOSITION 6.5.1 Every von Neumann algebra is a dual spa e.

This proposition follows from standard Bana h spa e fa ts. If B (H) is a von Neumann algebra then M  = (T C (H)=E ) where

M

E = fA 2 T C (H) : tr(AB ) = 0 for all B 2 Mg is the preannihilator of M. The above property a tually hara terizes abstra t von Neumann algebras: an abstra t C*-algebra is an abstra t von Neumann algebra if and only if it is a dual spa e. This result is known as Sakai's theorem; we will not prove it here. Note, however, that the slightly stronger assumption that the C*-algebra A has an order predual (i.e., an ordered Bana h spa e whose dual is isometri ally isomorphi to A as an ordered Bana h spa e) easily implies that A is a von Neumann algebra. For in this ase there trivially exist a family of weak* ontinuous states on A suÆ ient, using the GNS onstru tion, to weak* ontinuously embed A as a W*-subalgebra of some B (H). Next we determine the stru ture of abelian von Neumann algebras. Example 6.5.2

Let X be a  - nite measure spa e, let X be a measurable Hilbert bundle over X , and let H = L2 (X ; X ). Then M = fMf : f 2 L1 (X )g is a von Neumann algebra, and it is -isomorphi to L1 (X ).

© 2001 by Chapman & Hall/CRC

139 THEOREM 6.5.3

Suppose H is separable and let M  B (H) be an abelian von Neumann algebra. Then there is a probability measure  on sp(M), a measurable Hilbert bundle X over sp(M), and an isometri -isomorphism  H su h that M = UL1 (sp(M))U 1 . U : L2(sp(M); X ) = PROOF Let be the -algebra on sp(M) generated by the ontinuous fun tions, i.e., the Baire -algebra. Apply Theorem 5.3.5 to M; this provides a spe tral measure on the Borel sets of sp(M) whi h restri ts to a spe tral measure E on . Then by Corollary 3.4.3 there is a probability measure  on , a measurable Hilbert bundle X , and a surje tive isometry U : L2 (sp(M); X ) ! H su h that US U 1 = E (S ) for every Baire set S . As every ontinuous fun tion on sp(M) an be approximated by Baire measurable simple fun tions, this implies that M = UC (sp(M))U 1 . But C (sp(M))  = M is a W*-subalgebra of L1 (sp(M)) whi h measurably separates points, so we have C (sp(M)) = L1 (sp(M)) by Proposition 6.2.2.

In the nonseparable ase, as one might expe t, M is still -isomorphi to some L1 (Y ). The usual remark about invariants and unitary equivalen e also applies here; see the omments following Theorems 3.5.1 and 5.3.5. Now we turn to W*-ideals and hereditary W*-subalgebras. These have parti ularly simple stru tures. Example 6.5.4

Let M be a von Neumann algebra and let P 2 M be a proje tion. Then P MP is a hereditary W*-subalgebra. If P belongs to the

enter of M, i.e., PA = AP for all A 2 M, then P MP = P M is a W*-ideal of M. PROPOSITION 6.5.5

Let M be a von Neumann algebra. If N is a hereditary W*-subalgebra of M then there is a proje tion P 2 N su h that N = P MP . If N is a W*-ideal then P is in the enter of M.

The proof that every hereditary W*-subalgebra has the form P MP exa tly follows the proof of the same result for B (H) (Proposition 6.4.5). The same argument also applies to any W*-ideal N to show that there is a maximal proje tion P 2 N and N  P MP . Conversely, P MP  N follows immediately from the ideal property. PROOF

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140

Chapter 6: Von Neumann algebras

To show that P is entral, let A 2 M and let E be the range of . Sin e AP 2 N we must have AP = P BP for some B 2 M, and therefore A(E )  E . We have A (E )  E similarly, whi h implies that ? ? A(E )  E . Thus A and P ommute. P

In parti ular, sin e every von Neumann algebra is a W*-ideal of itself, there always exists a proje tion P 2 M whi h satis es AP = A for all A 2 M. That is, every von Neumann algebra has a unit (although it need not be the identity operator in B (H)). Next, we present von Neumann's elebrated \double ommutant" theorem, whi h provides a fundamental algebrai hara terization of von Neumann algebras. DEFINITION 6.5.6 Let M be a subset of B (H). Its ommutant M is the set of all operators in B (H) whi h ommute with every operator in M.

THEOREM 6.5.7

Let M  B (H) be a unital C*-algebra. Then M is a von Neumann algebra if and only if M = M

.

PROOF Suppose (A ) is a bounded net in B (H) and A ! A weak*. If B 2 B (H) ommutes with ea h A then B ommutes with A, be ause hABv; wi = lim hA Bv; wi = lim hA v; B  wi = hBAv; wi

for all v; w 2 H. This shows that the ommutant of any set is weak*

losed. Also, if M is a -algebra it is easy to he k that M is also a -algebra. So if M = M

then M is a weak* losed -algebra, i.e., a von Neumann algebra. For the onverse, we rst prove a seemingly modest density result. Let A0 2 M

and v 2 H, and let P be the proje tion onto Mv . If A; B 2 M then AP (Bv ) = ABv = P A(Bv ); so AP and

PA

agree on ve tors in Mv . But also, if

w

? Mv then

hAw; Bv i = hw; A Bv i = 0; ?

so that Aw ? Mv; thus AP and P A are both zero on Mv . This shows that P 2 M , so we must have P A0 = A0 P . Applying this equation to v we get P A0 v = A0 P v = A0 v , whi h means that A0 v 2 Mv . Thus, for

© 2001 by Chapman & Hall/CRC

141 any  > 0 we an nd A 2 M su h that k(A A0 )v k  . Now it is lear that M  M

; to prove equality we will show that M is weak* dense in M

. Thus let A0 2 M

, B 2 T C (H), and  > 0; we must nd A 2 M su h that jtr(AB ) tr(A0 B )j  . We may assume B > 0. Let (e ) be a basis of H onsisting of eigenve tors of B say pand Be =  e . Sin e  < 1, it follows that v =  e belongs to the dire t sum H1 of in nitely many opies of H. (This ould even be an un ountable sum.) Let M1  = M be the set of operators on H1 of the form A1 = A  A 


for A 2 M, and observe that the orresponding operator A1 0 belongs to the double ommutant of M1 . (The ommutant of M1 onsists of operators C on H1 su h that P CP~ 2 M for all ; ~ where P is the proje tion of H1 onto the th opy of H.) By the last paragraph there exists A 2 M su h that k(A1 A1 0 )v k  =kv k, and hen e

P

jtr(AB )

L

tr(A0 B )j = h(A1

1 )v; vi  ;

A0

for any  > 0. This shows that M is weak* dense in

M

.

We on lude this se tion with the fa tor de omposition of an arbitrary von Neumann algebra. A von Neumann algebra is a fa tor if it has no proper W*-ideals, or equivalently (Proposition 6.5.5) no proper entral proje tions. The idea of fa tor de omposition is to express any von Neumann algebra as a \measurable dire t sum" of fa tors. Given a measurable Hilbert bundle X = (Xn Hn ) over X , a eld of operators on X is a fun tion A~ on X su h that A~(x) 2 B (Hx ) for almost every x, where Hx is the Hilbert spa e lying over x. It is weakly measurable if for any v; w 2 Hn the fun tion x 7! hA~(x)v; wi is measurable on Xn . We require the following fa t: if M  B (H) is a von Neumann algebra and H is separable then M is both separable and rst ountable for the weak* topology. This follows from the fa t that the predual of M is a quotient of T C (H) and hen e is separable; weak* separability and rst

ountability of M are then general Bana h spa e fa ts.

S

THEOREM 6.5.8

Let M  B (H) be a von Neumann algebra and suppose H is separable. Then there is a  - nite measure spa e X , a measurable Hilbert bundle X over X , and a family of fa tor von Neumann algebras Mx  B (Hx ) su h that H an be identi ed with L2 (X ; X ) and M with the bounded weakly measurable elds of operators whi h lie in Mx almost everywhere.

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142

Chapter 6: Von Neumann algebras

The enter of M,

PROOF

M) = fA 2 M : AB = BA for all B 2 Mg; S is an abelian W*-subalgebra of M. Find X , X = (X  H ), and U as in Theorem 6.5.3 and identify H with L2 (X; X ) and Z (M) with L1 (X )  B (L2 (X ; X )) a ting as multipli ation operators. Let fA g be a ountable weak* dense subset of M; we may supZ(

n

n

k

pose this set is losed under sums, produ ts, adjoints, and multipli ation by s alars in Q + iQ. For ea h k , n, and 1  i  n write k fi;j ej ; then de ne a weakly measurable eld of operaAk (1Xn ei ) = k k ~ ~ tors Ak by Ak (x) : ei 7! j fi;j (x)ej . As ea h fi;j is well-de ned almost everwhere, so is A~k . Ea h A~k is essentially bounded be ause for almost every x and all v 2 Hx we have A~k (x)v = (Ak v )(x). Let Nx be the weak* losure in B (Hx ) of the set fA~k (x)g. By the hoi e of the family fAk g, it is lear that almost every Nx is a von Neumann algebra. Let N be the set of bounded measurable elds of operators B~ su h that B~ (x) 2 Nx for almost every x, and let  : N ! B (L2 (X ; X )) be the natural a tion of N on L2 (X ; X ). It is routine to verify that  is a -isomorphism and that  (N ) is a von Neumann algebra. Next we laim that  (A~k ) = Ak for all k . By the de nition of A~k , equality holds when both sides are applied to ve tors of the form 1Xn ei ; sin e both sides ommute with L1 (X )  = Z (M), it follows that

P

~k )(f ei )  (A

P

= Mf  (A~k )(1Xn ei ) = Mf Ak (1Xn ei ) = Ak (f ei )

for all f 2 L1 (X ), and taking linear ombinations yields that  (A~k ) = on a dense set of ve tors in L2 (X ; X ). So the laim is proven. It follows that M   (N ). Conversely, if B 2 M then we an de ne ~ ~k , and BAk = Ak B implies that B in the same way we de ned the A ~ (x) and A~k (x) ommute almost everywhere. Thus B =  (B~ ) 2  (N ) , B and by Theorem 6.5.7 we on lude that M =  (N ). Finally, we must show that almost every Nx is a fa tor. The idea is that if Z (Nx ) were nontrivial on a set of positive measure then there would be a nontrivial eld of operators in Z (Nx ) and this would ontradi t the fa t that Z (N ) = Z (M) = L1 (X ). To prove this rigorously we must ensure that it is possible to nd a nontrivial measurable eld of operators. This an be done as follows. Suppose Z (Nx ) is nontrivial, i.e., it ontains an operator whi h is not a s alar multiple of the identity, on a set S of positive measure. Without loss of generality we an assume S  Xn for some n. Let D be a ountable dense subset of Hn . Then for ea h x 2 S there exists Bx 2 Z (Nx ), v; w 2 D, and  > 0 su h that kBx k  1, kBx v wk < , Ak

© 2001 by Chapman & Hall/CRC

143 and kav wk  2 for every a 2 C. Sin e D is ountable and  an be assumed rational, by restri ting S we an ensure that there exists a single triple (v; w; ) su h that for every x 2 S , some Bx 2 Z (Nx ) satis es the pre eding. Thus, writing D = fvi : i 2 Ng, for ea h x and ea h m 2 N there is a nite linear ombination B~m (x) = ak A~k (x) with oeÆ ients ak 2 Q + iQ su h that kB~m (x)k  1, kB~m (x)v wk < , and jh(B~m (x)A~k (x) A~k (x)B~m (x))vi ; vj ij < 1 kvi kkvj k

P

m

for all i; j; k  m. Also, for ea h su h oeÆ ient sequen e (ak )  Q + iQ (with only nitely many terms nonzero) the set of x su h that ak A~k (x) has the above property is measurable; we an therefore nd, for ea h m, ~k = B~m has the desired a sequen e fkm 2 L1 (S ) su h that fkm A property almost everywhere. Then Bm = fkm Ak belongs to M, and kBmk  1 for all m, so there is a weak* luster point B 2 M of the sequen e (Bm ), and  1 (B ) = B~ evidently satis es B~ (x) 2 Z (Nx ) but ~ (x)v 6= av , for almost every x and all a 2 C. Thus B 2 Z (M) but B B 62 L1 (X ), a ontradi tion. So almost every Nx must be a fa tor.

P

P

P

Measure theoreti ompli ations make it diÆ ult to formulate a meaningful version of this theorem in the nonseparable ase, but it seems morally true in general. 6.6

The quantum plane and tori

We return to the quantum plane and tori. The operator analog of the set of bounded measurable fun tions is the weak* losure of the operator analog of the set of ontinuous fun tions (whi h vanish at in nity). This motivates the following de nition. 2 1 (T2 ) respe tively be the Let Lh1  (R ) and Lh  h  2 weak* losures of the C*-algebras C0 (R )  B (L2 (R2 )) and C h (T2 )  2 ^ h1 B (L2 (T2 )) de ned in Se tions 5.4 and 5.5. Also let L  (T ) be the h  2 2 2 ^ weak* losure of C (T )  B (l (Z )). DEFINITION 6.6.1

2 1(T2 ) are unitarily equivalent via the Fourier ^ h1 Thus L  (T ) and Lh  transform on the torus. Our rst result follows from Propositions 5.4.4 and 5.5.2, together with Proposition 6.2.2.

PROPOSITION 6.6.2

2 1 (T2 )  2  1 If h = 0 then Lh1 = L1 (T2 ).   (R ) = L (R ) and Lh

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144

Chapter 6: Von Neumann algebras

Re all the automorphisms s;t of B (L2 (R2 )) and ^s;t of B (l2 (Z2 )) introdu ed in Se tions 5.4 and 5.5. These were shown to restri t to automorphisms of C0h (R2 ) and C^ h (T2 ) in Propositions 5.4.5 and 5.5.5, 2 respe tively. We now note the orresponding statement for Lh1  (R ) and 1 2 ^ h (T ). L

2

PROPOSITION 6.6.3

For ea h s; t R the map s;t restri ts to a weak* ontinuous auto2 ^ morphism of Lh1  (R ) and the map s;t restri ts to a weak* ontinu1 2 ^ h (T ). This de nes a tions of R2 by automorous automorphism of L 2 2 2 ^1 phisms of Lh1 Lh1  (R ) and Lh  (T ). Moreover, for every A  (R ) and 1 2 ^ ^ B Lh (T ) the maps (s; t) s;t (A) and (s; t) s;t (B ) are weak*

ontinuous.

2

7!

7!

2

Most of the proof of this proposition resembles the proofs of Propositions 5.4.5 and 5.5.5. Ea h s;t and ^s;t is weak* ontinuous be ause it is given by onjugation with a unitary. Weak* ontinuity of the maps (s; t) s;t (A) and (s; t) ^s;t (B ) follows from the fa t that the asso iated unitaries Us;t onverge \strongly" to I as s; t 0 in the sense that Us;t v v for all v . 2 ^ h1 We will now show how C^ h (T2 ) is distinguished inside of L  (T ). Un2 h  fortunately, no su h result is available for C0 (R ), but we will indi ate a substitute. ^ 1 (T2 ) de ne Fourier oeÆ ients ak;l (A), partial sums For A L sN (A), and Ces aro means just as in De nition 5.5.4. We have the fol2 ^ h1 lowing hara terization of L  (T ) ( f. Proposition 5.5.3).

7!

!7 2H

!

!

2

2 B(l (Z )) belongs to L^ 1(T ) if and only if hAe ; e i=e hAe ; e i for all k; l; m; n 2 Z. The partial Fourier sums s (A) onverge weak ^ 1 (T ). operator to A for all A 2 L PROPOSITION 6.6.4

2

An operator A

m;n

2

m



h

(

ih ml

+k;n+l



h

2

) 2

nk =

0;0

k;l

N

2

PROOF

It is easy to verify that the stated equality is satis ed for A = U^ m V^ n and hen e for any polynomial in U^ and V^ . Sin e these 2 ^ h1 polynomials are weak* dense in L  (T ), the forward impli ation follows. 2 2 Conversely, let A B (l (Z )) and suppose A satis es the stated equality. Then sN (A) A weak operator be ause

!

2

hAv; wi =

Xh

k;l;m;n

© 2001 by Chapman & Hall/CRC

ih

ih

v; ek;l Aek;l ; em;n em;n ; w

i

145

X

and

hs

N

(A)v; wi =

j

k

jj

m ; l

j

n

hv; e ihAe k;l

k;l

ih

; em;n em;n ; w

i

N

2 ^ h1 for any v; w 2 l2 (Z2 ). Sin e sN (A) belongs to L  (T ) for all N , this 2 ^ h1 implies that A belongs to the weak operator losure of L  (T ). But 2 ^ h1 the double ommutant theorem (Theorem 6.5.7) implies that L  (T ) is 1 2 ^ weak operator losed. So A 2 Lh (T ). PROPOSITION 6.6.5 Let

2 ^ A1 = fA 2 L^ 1  (T ) : (s; t) 7!  (A) 1 2 ^ A2 = fA 2 L (T ) : kA  (A)k ! 0g: ^  (T2 ) = A1 = A2 C s;t

h

h

Then

h

g

is norm ontinuous

N

.

It is straightforward to he k that A1 is a C*-algebra whi h

ontains U^ and V^ , and therefore C^ h (T2 )  A1 . The fa t that A2  ^ h (T2 ) is trivial be ause C^ h (T2 ) is norm losed and every N (A) 2 C ^ h (T2 ). It remains to show that A1  A2 . But the only property of C ^ C h (T2 ) used in the proof of Theorem 5.5.7 was the fa t that ^s;t (A) is norm ontinuous for all A 2 C^ h (T2 ). Thus that argument a tually shows A1  A2 .

PROOF

The orresponding statement for C0h (R2 ) and s;t already fails in the h = 0 ase: the fun tions in L1 (R2 ) for whi h translations are norm

ontinuous are pre isely the bounded uniformly ontinuous fun tions, not the ontinuous fun tions vanishing at in nity. Following this model, we an de ne the orresponding algebra Cbh (R2 ) for h  6= 0 to be the 2 ) for whi h (s; t) 7!  (A) is norm ontinuous. It is set of A 2 Lh1 ( R s;t  straightforward to he k that this is a C*-algebra, and in some respe ts it is ni er than C0h (R2 ). Its prin ipal drawba k is that it is not separable. 2 1 (T2 ) Finally, we examine the stru ture of the algebras Lh1  (R ) and Lh  when h  6= 0. The former is quite simple: PROPOSITION 6.6.6 If

2 2  h 6= 0 then Lh1  (R ) = B (L (R)).

PROOF We showed in Theorem 5.4.13 that C~0h (R2 ), realized as a subalgebra of B (L2 (R)), is pre isely the ideal K (L2 (R)) of ompa t

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146

Chapter 6:

Von Neumann algebras

operators. Thus its weak* losure equals B (L2 (R)). Conjugation by 2  2  the unitary W in Proposition 4.2.2 yields L1  (R ) = B (L (R)) I = 2 B (L (R)). h

2 The algebras L1  (T ) are mutually -isomorphi for all positive values of h , despite the fa t that the orresponding algebras C  (T2 ) are not. 2 We will not prove this result, but we will show that L1  (T ) is not isomorphi to any B (H). Interestingly, however, like B (H) it is a fa tor: any operator A 2 B (l2 (Z2 )) that ommutes with U^ and V^ must satisfy h

h

h

hAem;n ; em+k;n+l i = e

(

ih ml

) 2

nk =

hAe0;0 ; ek;l i

for all k; l; m; n 2 Z, and it follows from Proposition 6.6.4 that the enter 2 1 2 of L^ 1  (T ) is trivial. So the same must be true of L (T ). h

h

PROPOSITION 6.6.7

1 Lh 

(T2 ) is not -isomorphi to any B (H) for any value of h.

PROOF Sin e L1 (T2 ) = L^ 1 (T2 ) we an work with the latter. Con2 sider the weak* ontinuous linear fun tional  : L^ 1  (T ) ! C de ned h

h

h

by

( ) = hAe0 0 ; e0 0i: If A and B are polynomials in U^ and V^ then  (AB ) =  (BA), so by 2

ontinuity this equality holds for all A; B 2 L^ 1  (T ). Also  (I ) = 1. If H is in nite dimensional there is no weak* ontinuous linear fun tional on B (H) with the above properties. To see this let P and Q be rank one proje tions in B (H). Then there is a unitary W su h that W  P W = Q. So if  2 B (H) satis es  (AB ) =  (BA) for all A; B 2 B (H) then  (P ) =  (Q) for all rank one proje tions P and Q. Let  be an orthonormal basis of H and for any nite subset S   let P be the proje tion onto its span. Then (P ) is a net of nite rank proje tions whi h onverges weak* to the identity operator. But  (P ) = an where n is the ardinality of S and a is the value of  on any rank one proje tion. So weak* ontinuity is in onsistent with the

ondition  (I ) = 1.  A

;

;

h

S

S

S

6.7

Notes

[19℄ and [66℄ are good referen es on von Neumann algebras. Most of the material in Se tion 6.6 was taken from [71℄.

© 2001 by Chapman & Hall/CRC

Chapter 7

Quantum Field Theory 7.1

Fo k spa e

The subje t matter of this hapter lies outside the main line of the book. Elsewhere we use physi s only as motivation for our treatment of mathemati al topi s, but here we take the reverse attitude. In this

hapter we will apply the ideas of previous hapters to the problem of modelling free quantum elds. We hoose this topi be ause it is here that the best argument an be made for the relevan e of C*-algebras to physi s. As we will explain in Se tion 7.5, C*-algebras are needed to onstru t relativisti ally invariant models of free quantum elds in

urved spa etime. The onstru tion of su

essful models of quantum elds is predi ated on a fundamental understanding of eld dynami s. Unfortunately, relativisti dynami al problems are extremely diÆ ult even in the simplest eld systems, and despite half a entury of heroi mathemati al eorts, this area still seems not to be fully understood at a basi level.1 For this reason we will dis uss only systems with trivial dynami s, i.e., the ase of free (nonintera ting) elds. We begin with nonrelativisti elds. Consider a lassi al real s alar eld on R3 : su h a eld is des ribed by a real value at ea h point of spa e. By ontrast, for example, the ele tromagneti eld is a ve tor eld and is des ribed at ea h point by a ve tor with six real dimensions, three ele tri and three magneti . Fields with more than one omponent an be built up from the s alar ase without any fundamental obstru tion, so we will onsider only this simplest ase. It is easier to on eptualize the transition to quantum me hani s if we temporarily repla e R3 with a dis rete set X . We an model a

lassi al eld on X by a olle tion of one-dimensional parti les indexed by the points of X , where the \position" of the nth parti le is the eld 1 In the words of [48℄ it is a \deep gloomy mess."

147 © 2001 by Chapman & Hall/CRC

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Chapter 7: Quantum Field Theory

strength at the nth point. Thus, the quantum me hani al version of the eld is modelled at ea h point of X by the Hilbert spa e L2 ( ) of a one-dimensional quantum me hani al parti le (see Se tion 4.1), and des ribed as a whole by the tensor produ t of a set of opies of L2 ( ) indexed by the set X . If X is in nite, taking this tensor produ t requires the sele tion of a distinguished unit ve tor in ea h fa tor (De nition 2.5.1). This suggests that at ea h point we use the l2 ( ) model of a one-dimensional parti le dis ussed in Se tion 4.3. We an then take the tensor power of l2 ( ) as in Example 2.5.7, and the result is the symmetri Fo k spa e over 2 l (X ). Thus, the Hilbert spa e of a quantum real s alar eld over the dis rete set X is Fs l2 (X ). When the underlying spa e is not dis rete (e.g., X = 3 ), we an, as dis ussed in Se tion 2.5, retain the intuition of a \measurable tensor produ t over X " by taking the Hilbert spa e to be Fs L2 (X ). This is the Hilbert spa e of a free s alar Boson eld over X . Similarly, the antisymmetri Fo k spa e Fa L2 (X ) is the Hilbert spa e of a free s alar Fermion eld; here the lassi al version involves a two-state system at ea h point in spa e, so that the Hilbert spa e of the quantum system is intuitively a measurable tensor produ t of two-dimensional Hilbert spa es. For the sake of simpli ity we will dis uss only Boson elds in this hapter; mu h of what we do an be transferred to the Fermioni

ase by a simple substitution of antisymmetri Fo k spa es for symmetri Fo k spa es. Fo k spa es are onstru ted as dire t sums of symmetri and antisymmetri tensor powers of a Hilbert spa e, in our ase L2 ( 3 ) (De nition 2.5.8). This leads to a parti le interpretation of quantum elds whi h has no lassi al analog. Namely, we interpret an element of 2 3 ) n  F L2 ( 3 ) as des ribing a system of n identi al parti les. L ( s s 2 3 n L ( )

s is alled the n-parti le spa e. To omplete the basi model of a free nonrelativisti eld, we must des ribe its dynami s. This an be done by going ba k to the dis rete model and letting X be a latti e in 3 . Then we an treat the lassi al eld as a dis rete family of one-dimensional parti les, write an expression for their total energy, follow the standard pres ription from physi s for obtaining the quantum Hamiltonian (i.e., ih  times the generator of time evolution) from the lassi al energy, and nally pass to a ontinuous limit. Of ourse some are must be taken to ensure that the limit exists; we will not go into the details. The result an be expressed in terms of the unitary operator Ut 2 B (L2 ( 3 )) de ned by

R

R

N

N

R

R

R R

R

R

R

(Ut f )^(p) = e

i h 2 jpj2 t

^( )

f p ;

where f^ is the three-dimensional Fourier transform; namely, the

© 2001 by Chapman & Hall/CRC

-

n

149 parti le subspa e L2 (R3 ) s


s L2 (R3 )  Fs l2 (R3 ) evolves a

ording to Ut


Ut . Here  is a onstant whi h arises from the strength of the oupling between adja ent parti les in the dis rete approximation and is interpreted as wave velo ity in the ontinuous limit. By omparison, a single free nonrelativisti parti le in R3 evolves a

ording to 2 (Ut f )^(p) = e ih jpj t=2m f^(p); where m is its mass. Thus the free nonrelativisti eld behaves dynami ally like an ensemble of free parti les, with the wave velo ity  playing the role of (2m) 1=2 . This ompletes our initial dis ussion of free nonrelativisti quantum elds. Now we want to introdu e a dierent model of symmetri Fo k spa e whi h will be useful in the sequel. First we re ord some basi fa ts about Fs H (similar statements hold for Fa H). PROPOSITION 7.1.1

Let

H be a Hilbert spa e.

(a) If E is a losed subspa e of H then there is a natural isometri isomorphism between Fs H and Fs E Fs E ? , and Fs E naturally embeds in Fs H by the map v 7! v 1. More generally, if H = E then Fs H  = Fs E .

N

L

Fs H with the ompletion of the dire ted union S(b)FsWeE of anthe identify symmetri Fo k spa es over all nite dimensional subspa es E of

H.

The in nite tensor produ t in part (a) is taken with respe t to the unit ve tors 1 2 C  Fs E (see De nition 2.5.1). Proposition 7.1.1 is a routine onsequen e of Proposition 2.5.3 and De nition 2.5.8. Noti e that part (a) implies Fs L2 (X )  = Fs L2 (S ) Fs L2 (X S ) for any S  X , in keeping with the idea that Fs L2 (X ) is a measurable tensor produ t of Hilbert spa es indexed by X . Now Fs C an be identi ed with l2 (N) and subsequently with L2 (R) (Proposition 4.3.1). By Proposition 7.1.1 (a), taking nite tensor powers then yields a natural isometri isomorphism of Fs Cn with L2 (Rn ). Our next goal is generalize this to in nite dimensions and model any symmetri Fo k spa e as a kind of spa e of L2 fun tions. In general, let HR be a real Hilbert spa e and let H = HR  iHR be its omplexi ation. We want to say Fs H  = L2 (HR ). Of ourse L2 (HR ) does not immediately make sense sin e HR does not arry a natural measure, but it

an be de ned as a limit over nite dimensional subspa es of HR . This requires us to oordinate the L2 fun tions on dierent nite dimensional subspa es.

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Chapter 7: Quantum Field Theory

Let h be a xed positive real number. DEFINITION 7.1.2 Let HR be a real Hilbert spa e. On ea h nite 2 dimensional subspa e ER of HR de ne the fun tion hER (v ) = e kvk =2h and let  = ER be (h ) n=2 times Lebesgue measure, where n is the 1  E2 dimension of ER . Then khER k = 1 in L2 (ER ; ), and if ER = ER R then hER = hER1 hER2 . Thus the map f 7! f hER2 isometri ally em1 ; ) in L2 (E ; ) and these embedding maps are onsistent. beds L2 (ER R 2 (H ) to be the ompletion of the dire ted union We therefore de ne L R S L2(E ) over all nite dimensional subspa es ER of HR . R

Observe that hER an a tually be de ned on any subspa e of HR . Thus, for any f 2 L2 (ER ; ) (ER nite dimensional) let f~ : HR ! C be the fun tion f~ = f hER? . The map f 7! f~ is onsistent with the embeddings in De nition 7.1.2, so it is possible to regard L2 (HR ) as the

ompletion of the set of fun tions of the form f~ for f 2 L2 (ER ; ) with ER a nite dimensional subspa e of HR . Thus, although L2 (HR ) is not a tually a spa e of fun tions of HR , it has a dense subspa e whi h is. Also note that ea h L2 (ER ; ) is naturally isometri to L2 (ER ; m), where m is Lebesgue measure, by the map f 7! (h ) n=4 f . By the

omment made after Proposition 7.1.1, we see that there is a natural isomorphism Fs E  = L2 (ER ; ) for any nite dimensional real Hilbert spa e ER , where E = ER  iER . Taking dire t limits and applying Proposition 7.1.1 (b) yields the following. PROPOSITION 7.1.3

Let HR be a real Hilbert spa e and let H = HR  iHR . Then there is a natural isomorphism between Fs H and L2 (HR ). 7.2

CCR algebras

The HR ) model of Fo k spa e allows us to de ne eld observables whi h are analogous to the position and momentum operators of the one-dimensional parti le. Namely, for ea h v 2 HR de ne operators Qv and Pv on L2 (HR ) by

L2 (

Qf (w) = hw; vif (w)

P f (w) =

ih

f (w); v

where f 2 L2 (HR ) is a fun tion on HR of the type des ribed in the remark following De nition 7.1.2. Equivalently, given a omplex Hilbert spa e H and v 2 H, let E be the omplex span of v and write Fs (H)  = Fs (E ) Fs (E ? ); then Qv = Q I and Pv = P I , after the usual

© 2001 by Chapman & Hall/CRC

151 identi ation of Fs (E )  = Fs (C) with L2 (R). This se ond de nition makes sense for any v in H, not just HR , and is learly independent of the de omposition H = HR  iHR . However, the rst de nition is probably easier to visualize. We think of Qv as measuring the strength of the v omponent of the eld and we think of Pv as measuring its rate of hange. For the one-dimensional parti le we went beyond Q and P and onstru ted C*- and von Neumann algebras of observables (Se tions 5.4 and 6.6). (This is a slight abuse of language. Only the self-adjoint elements of the algebras, at most, should be regarded as genuine observables.) Now we would like to do the same thing for a s alar eld, but we en ounter the following diÆ ulty. In the ase of a one-dimensional parti le the lassi al phase spa e is R2 , and lassi al observables are fun tions on the plane. For the orresponding quantum me hani al system we have a C*-algebra C0h (R2 ) of observables whi h redu es, in the ase  = 0, to the ontinuous fun tions on R2 whi h vanish at in nity. But h for elds, the lassi al phase spa e is in nite dimensional and hen e not lo ally ompa t. So there is no sensible notion of \ ontinuous fun tions vanishing at in nity." Thus, we need to nd a substitute for the lassi al algebra C0 (X ). In order to do this we must take into a

ount the stru ture of the phase spa e of a free lassi al eld. Namely, it is always a symple ti spa e. That is, it is a real ve tor spa e V equipped with a symple ti form, an antisymmetri bilinear map f
;
g : V  V ! R. We topologize V with the weakest topology that makes the map w 7! fv; wg ontinuous for all v 2 V. Generally, in the nonrelativisti ase we an take V to be a omplex Hilbert spa e, with symple ti form fv; wg = Imhv; wi. For the free s alar eld V would just be L2 (R3 ), where we identify the real fun tions in L2 (R3 ) with the possible on gurations of the lassi al eld and the purely imaginary fun tions with their time derivatives. This is analogous to identifying the phase spa e of a one-dimensional parti le with the

omplex plane, taking position to be the real axis and momentum to be the imaginary axis. However, this onstru tion is not anoni al, and as we will see in Se tion 7.5, in urved spa etime it breaks down ompletely. At that point V will really have only a symple ti stru ture, but even then we will be able to embed it as a real linear subspa e of a omplex Hilbert spa e, in su h a way that the symple ti form agrees with the imaginary part of the inner produ t. So this is the most general lass of symple ti spa es we will need to onsider. In this situation we always take fv; wg = Imhv; wi. Now on any symple ti spa e there is a ri h supply of well-behaved exponential fun tions "v : w 7! eifv;wg . We an use these fun tions to

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152

Chapter 7: Quantum Field Theory

build a ni e algebra of ontinuous fun tions whi h an then be deformed to yield a C*-algebra of observables for a quantum eld. The fun tions "v do not vanish at in nity, but they are periodi . Moreover, linear ombinations of the "v are still \almost" periodi in the following sense. Let V be a symple ti spa e. For v 2 V de ne : l1 (V ) ! l1 (V ) by (Tv f )(w) = f (w v ). A fun tion f 2 l1 (V ) is almost periodi if the set fTv f : v 2 Vg is pre ompa t in sup norm. The set of ontinuous almost periodi fun tions is denoted AP (V ). DEFINITION 7.2.1

Tv

The point is that when V is in nite dimensional we an use AP (V ) in pla e of (the nonexistant) C0 (V ). Observe that Tu "v = e ifv;ug "v , so the set fTu "v : u 2 Vg is homeomorphi to a ir le, and hen e is ompa t. Thus "v is indeed almost periodi . The fa t that linear ombinations of the "v are also almost periodi follows from the next result. PROPOSITION 7.2.2

Let V be a symple ti spa e. Then l1 (V ).

V ) is a unital C*-subalgebra of

AP (

PROOF First we show that AP (V ) is a -algebra. It is lear that any s alar multiple of an almost periodi fun tion is almost periodi , as is its omplex onjugate. If f and g are almost periodi then the set

f(T f; T v

w

g)

: v; w 2 Vg  l1 (V )  l1 (V )

is pre ompa t, so its images under the sum and produ t maps, whi h

ontain fTv (f + g ) : v 2 Vg and fTv (f g ) : v 2 Vg, are also pre ompa t. We on lude that f + g and f g are also almost periodi , and this shows that AP (V ) is a -algebra. Next we prove norm losure. Let (fn ) be a sequen e of almost periodi fun tions whi h onverges in norm to f 2 l1 (V ). Given  > 0, hoose n su h that kf fn k  =3 and nd v1 ; : : : ; vk 2 V su h that every Tv fn (v 2 V ) is within =3 of Tvi fn for some i. Then we have

kT f v

Tv i f

k  kT (f v

k + kT

fn )

v

fn

Tvi fn

k + kT

vi

(fn

f)

k  :

This shows that fTv f : v 2 Vg is totally bounded and hen e pre ompa t. Thus AP (V ) is losed in norm, so it is a C*-subalgebra of l1 (V ). It is

© 2001 by Chapman & Hall/CRC

153

lear that AP (V ) is unital. In fa t, if H is a omplex Hilbert spa e with the standard symple ti form Imh
;
i, then AP (H) is pre isely the losed linear span of the fun tions "v . One dire tion of this assertion is lear, as every "v belongs to AP (H). For the onverse dire tion one needs to develop a theory of Fourier expansion of almost periodi fun tions. It turns out that any almost periodi fun tion has a ountably supported Fourier transform de ned on H, and an be approximated in norm by fun tions with nitely supported Fourier transforms, i.e., linear ombinations of the "v . Now suppose V is only a real linear subspa e of a omplex Hilbert spa e H with the inherited symple ti form. We next de ne an algebra of quantum observables, analogous to AP (V ), whi h a ts on Fs H, and then we show that these algebras depend only on V , not on H. DEFINITION 7.2.3 Let H be a omplex Hilbert spa e and let V be a real linear subspa e of H. For ea h v 2 V let Wv = eiQv where Qv is the eld operator introdu ed at the beginning of this se tion. Equivalently, let E be the omplex span of v , let ER be the real span of v , and identify Fs E with L2 (ER ); then Wv is de ned on Fs H  = Fs E Fs E ? it k v k by Wv = Mf I where f (tv ) = e and Mf is the multipli ation operator a ting on L2 (ER ). These are alled Weyl operators. The CCR algebra is the C*-algebra CCR(V ) generated by the operators Wv for v 2 V.

The pre eding onstru tion is more subtle than it rst appears. Suppose H = C is one-dimensional and let fe1 ; e2 g be its anoni al real basis, so that e2 = ie1 . Then We1 and We2 an both be represented as the multipli ation operator Meix , but this is by way of dierent identi ations of Fs C with L2 (R). The Hermite fun tion hn in the e1 pi ture n

orresponds to the ve tor e

1 2 Fs C, whereas hn in the e2 pi ture or

n

n n responds to e2 = i e1 . So the two identi ations are related by the unitary map U : L2 (R) ! L2 (R) whi h takes the nth Hermite fun tion hn to in hn . By onsidering the a tions of Q and P on the Hermite fun tions (whi h

an be dedu ed from the expressions for (Q + iP )hn and (Q iP )hn given after Proposition 4.3.1), it is easily seen that U 1 QU = P . Thus in the e1 pi ture we have We1 = eiQ and We2 = U 1 eiQ U = eiP . This analysis allows us to dedu e ommutation relations for the Weyl operators. If v and w are ( omplex) orthogonal, or if w is a real multiple of v , it is easy to see that Wv and Ww ommute. On the other hand, if kvk = 1 then the pre eding omment (together with Theorem 4.1.3 ( )) shows that Wsv Witv = e ih st Witv Wsv . By linearity this generalizes to

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154

Chapter 7: Quantum Field Theory

the following result: PROPOSITION 7.2.4

Let

V

be as in De nition 7.2.3. For any v; w

eih fv;wg Ww Wv .

2 V

we have Wv Ww =

This is a generalization of the Weyl form of the anoni al ommutation relations presented in Theorem 4.1.3 ( ). Next we show that the algebra CCR(V ) is uniquely determined by the Weyl relations. This fa t requires that V be a real linear subspa e of a omplex Hilbert spa e and that V be nondegenerate in the sense that for every nonzero v 2 V there exists w 2 V su h that fv; wg 6= 0. THEOREM 7.2.5

Let V be a nondegenerate real linear subspa e of a omplex Hilbert spa e ~ v (v 2 V ) be unitary operators whi h a t on some other Hilbert and let W ~ v . Then the C*-algebra the W ~v ~ w = eih fv;wg W ~ wW ~ vW spa e and satisfy W generate is -isomorphi to CCR(V ).

PROOF Let A be the C*-algebra generated by the W~ v . Our proof of uniqueness uses the notion of A-valued almost periodi fun tions on V . These an be de ned as fun tions from V to A whose omposition with any ! 2 A is almost periodi , or one an dire tly mimi De nition 7.2.1, repla ing l1 (V ) with the set of bounded A-valued fun tions on V . The key fa t that we need is that every ontinuous almost periodi fun tion f possesses a mean value  (f ) 2 A with the following properties: (1) k (f )k  kf k; (2) f  0 implies  (f )  0; (3)  (af + bg ) = a (f )+ b (g ) for any a; b 2 C; (4)  (Tv f ) =  (f ) for all v ; and (5) if f (v ) = A

onstantly then  (f ) = A. This is a general fa t whi h is true of almost periodi fun tions from any group into a Bana h spa e. Now de ne an a tion  of V on A by letting w be onjugation with ~ w , i.e., w (A) = W ~ w 1 AW ~ w . For ea h v 2 V the map fW~ : w 7! W v ~ v ) is periodi , and an argument like the one in Proposition 7.2.2 w (W then shows that the map fA : w 7! w (A) is almost periodi for all A 2 A. So it has a mean value  (fA ). We have k (fA )k  kAk, so the ~ v we have fA = "h v A, map A 7!  (fA ) is ontinuous. Also, when A = W and properties (3) and (4) of the mean value imply  (fA ) =  (Tw fA ) = eih fw;vg  (fA );

so that nondegenera y of V implies  (fA ) = 0 unless v = 0, when  (fA ) = I by property (5). By ontinuity it follows that  (fA ) 2 C
I for any A 2 A. Also, A  0 implies  (fA )  0. We an now prove that A

© 2001 by Chapman & Hall/CRC

155 has no proper C*-ideals by the reasoning used in the proof of Theorem 5.5.8, and on lude that A  = CCR(V ) just as in Corollary 5.5.9. This uniqueness result shows in parti ular that the onstru tion of V ) in De nition 7.2.3 depends only on V , and not on the ambient Hilbert spa e H. However, it is important to note that embeddings of V in dierent Hilbert spa es will generally give rise, via De nition 7.2.3, to representations of CCR(V ) whi h are not unitarily equivalent. In parti ular, the von Neumann algebra generated by the Weyl operators does depend on H. We an now introdu e a representation of CCR(V ) whi h is similar to the L2 (R2 ) representation given in Se tion 4.2. For v 2 V  H de ne a unitary operator Lv on L2 (H) (in the sense of De nition 7.1.2, treating H as a real Hilbert spa e) by CCR(

Lv f (w)

= eiImhv;wi=2 f (w

hv ):

These are analogous to the operators Lt1 ;t2 , although our onventions here are slightly dierent. Sin e they obey the same ommutation relations as the Wv , it follows from Theorem 7.2.5 that the C*-algebra they generate is -isomorphi to CCR(V ). But, as in the ase of the quantum plane, this representation allows a de nition of CCR(V ) when h = 0. (The above formula also de nes a representation of CCR(V ) on the non-separable Hilbert spa e l2 (H); this is the GNS representation onstru tion analogous to the one in Example 5.6.7 (a).) If we set h = 0 then Lv is just multipli ation by "v=2 . So the next result is a onsequen e of the omment following Proposition 7.2.2. PROPOSITION 7.2.6 Let

H

be a omplex Hilbert spa e. If

h = 0 then CCR(H)  = AP (H).

Thus, for h > 0 the C*-algebra CCR(H) is a \deformation" of AP (H) in the same way that C0h (R2 ) and C h (T2 ) are deformations of C0 (R2 ) and C (T2 ). 7.3

Relativisti parti les

In this se tion we address the question of how free parti les behave in Minkowski spa etime. The orresponding eld theory will be des ribed in Se tion 7.4. In spe ial relativity, ea h inertial observer sees a three-dimensional sli e of spa etime at ea h instant of time. Thus the state of a spinless parti le at any moment is des ribed by a normalized fun tion in L2 (R3 ). (Spin is a

omodated by instead using a dire t sum of some number

© 2001 by Chapman & Hall/CRC

156

Chapter 7: Quantum Field Theory

of opies of L2 (R3 ); for simpli ity we will sti k to the spinless ase.) Furthermore, omplete knowledge of the parti le's state at any given moment determines its entire past and future. Thus L2 (R3 ) models the phase spa e of the system, just as in nonrelativisti quantum me hani s. We must now des ribe how a state given at time t = 0 evolves as t

hanges, and also how it will appear to other inertial observers. The time evolution of a free relativisti parti le is similar to the nonrelativisti ase (Se tion 7.1). As there, the evolution operators Ut are diagonal (i.e., appear as multipli ation operators) in the Fourier transform pi ture; the appropriate relativisti formula is (Ut f )^(p) = e

i( h2 2 jpj2 +m2 4 )1=2 t=h 

^( ) = e

f p

ip0 t

^( )

f p

where is the speed of light, m is the mass of the parti le (assumed throughout to be stri tly positive), and p0 = ( 2 jpj2 + m2 4 h 2 )1=2 is  1 times the relativisti energy of a quantum me hani al parti le with h mass m and momentum h p = h  (p1 ; p2 ; p3 ). Now we must say how states transform under Lorentz transformations. That is, if L is a 4  4 matrix whi h leaves the relativisti length 2 x20 2 2 2 x1 x2 x3 invariant (here x0 = t is the time oordinate), then we must determine how the state of a parti le given on the t = 0 sli e in the old frame appears on the t = 0 sli e in the new frame, i.e., the set of points fL 1 (0; x) : x = (x1 ; x2 ; x3 ) 2 R3 g. We assume throughout that L preserves the dire tion of time and the orientation of spa e. The way to pro eed is partially lari ed by onsidering \momentum eigenstates" of the form eip
x where x = (x1 ; x2 ; x3 ) is the spa e variable and p = (p1 ; p2 ; p3 ) is xed. Of ourse, these fun tions do not belong to L2 (R3 ), so they are not really states at all, but we an imagine them as being approximated by L2 fun tions if we wish. In the Fourier transform pi ture these momentum eigenstates appear as delta fun tions

on entrated at p. Now the point is that under time evolution they remain delta fun tions, and we expe t that in any frame, at any time they will appear as momentum eigenstates. In parti ular, in a new frame related to the original one by a Lorentz transformation L, we expe t an eigenstate of momentum h  p to transform into one of momentum h  p0 su h ~ (p0 ; p), where L ~ = (L 1 )T is the inverse transpose of L. that (p00 ; p0 ) = L (The transpose arises be ause we have taken the Fourier transform.) It may seem that the pre eding ompletely determines how states should transform: given a state f , one might think, we simply have to take the Fourier transform of f , transform ea h momentum value in the above manner, and nally apply the inverse Fourier transform. However, there is an ambiguity due to the fa t that momentum eigenstates are not L2 fun tions. If they were, we would have to normalize them before

arrying out the above pres ription. Thus, we a tually expe t only to

© 2001 by Chapman & Hall/CRC

157 transform N (p)eip
x to N (p0 )eip
x where N (p) is some \normalizing" fa tor. The value of N (p) is determined by the requirement that transformation of states must be unitary. It is helpful here to onsider the mass shell Xm = (p0 ; p) : p0 = ( 2 p 2 + m2 4 h 2 )1=2 ; 0

f

jj

g

whi h is an orbit of the group of Lorentz transformations a ting in momentum spa e in the inverse transpose manner des ribed above. Proje tion onto the p oordinates de nes a homeomorphism between Xm and R3 , so Lebesgue measure dp on R3 an be transferred to Xm . However, this measure is not Lorentz invariant; the measure dp=p0 is. Thus, ~ de nes a unitary map on L2 (Xm ; dp=p0 ).

omposition with any L (The sense of the notation dp=p0 is the following. If a fun tion f (p0 ; p) is R de ned on Xm then its integral with respe t to the measure dp=p0 is R3 f (p0 ; p) dp=p0.) Therefore, we de ne a unitary map T : L2 (R3 ) L2 (Xm ; dp=p0 ) by

p

!

!

(T f )(p0 ; p) = p0 f^(p);

we let VL : L2 (Xm ) L2 (Xm ) be the operator of omposition with L~ 1 , and we de ne the a tion of L on the original L2 (R3 ) by UL = T 1 VL T . That is, UL f is de ned by passing to L2 (Xm ) via T , omposing with ~ 1 , and then passing ba k to L2 (R3 ). This is manifestly a unitary L transformation, and it permutes momentum eigenstates in the desired manner. We see that the s alar N (p) equals p0 1=2 . This ompletes our des ription of relativisti parti les. But the time evolution operators Ut de ned near the beginning of this se tion have a pe uliar property whi h is worth dis ussing. Suppose f L2 (R3 ) has unit norm and is supported on a ompa t set. Then f^ L2 (R3 ) is real analyti , meaning that it is the restri tion to R3 of the fun tion

2

f^(z ) =

1 (2 )3=2

Z

f (x)e

2

ix
z dx

on C3 , whi h is separately analyti in z1 , z2 , and z3 . (This follows from the fa t that f L1 (R3 ) and so we an dierentiate under the integral sign.) Now if this is the ase, then (Ut f )^ annot be real analyti for any nonzero value of t, be ause of the square root in the exponent whi h de nes Ut . Thus the state whi h equals f at time t = 0 will not be supported in any ompa t subset of position spa e for any t = 0. This phenomenon has serious impli ations. Let K1 and K2 be two well-separated ompa t spatial regions and let P1 and P2 be the orthogonal proje tions of L2 (R3 ) onto L2 (K1 ) and L2 (K2 ). Then x t small enough that light annot travel from K1 to K2 in time t, and de ne

2

6

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Chapter 7: Quantum Field Theory

P20 = U 1P2 U .

We laim that P1 and P20 do not ommute in general. To see this, nd a state f whi h is supported on K1 at time 0 and has a nonzero omponent in K2 at time t; if P1 P20 f = P20 P1 f = P20 f then P20 f would be supported on K1 at time 0 and on K2 at time t, violating the

on lusion of the pre eding paragraph. One an show that an agent lo ated in K1 ould use this ee t to send a faster-than-light signal to an agent lo ated in K2 , violating relativisti

ausality. This is be ause, given a parti le whose prior state is known, merely measuring whether the parti le is in K1 at time 0 an alter the probability that it is in K2 at time t in su h a way that the se ond agent

ould infer that a measurement had been made, regardless of the result of the rst measurement. The moral is that one annot make lo al measurements of individual parti les in relativisti quantum me hani s! This point will be ome more lear in the next se tion when we dis uss the quantum eld observables whi h a tually an be measured lo ally. It turns out that these lo al measurements, when applied to a quantum eld in a single-parti le state, always yield a result whi h has a nonzero many-parti le omponent (though this omponent an be made arbitrarily small by making the measurement over a large enough region). The phenomenon is similar to what happens when one measures whether a diagonally polarized photon is horizontally polarized: the result of su h a measurement is either a horizontally or verti ally polarized photon, so the photon annot remain in a diagonally polarized state. The dieren e is that in the photon example there is no lo ality requirement whi h prevents one from dire tly measuring diagonal polarization. Thus, while we do have a single-parti le spe ial relativisti theory, it is in ompatible with lo al observers. (General relativity is more severely in onsistent with single parti les; see Se tion 7.5.) t

t

t

6       r            

 r  



 

x

-



\A parti le whi h travels faster than the speed of light appears twi e in some frame"

Figure 7.1

Other, in orre t inferen es have also been drawn. For instan e, the

© 2001 by Chapman & Hall/CRC

159 argument is sometimes made that a parti le that travels faster than light must appear twi e in some referen e frame, and hen e one annot have a single-parti le relativisti model. A related argument laims to demonstrate that relativity implies the existen e of antiparti les. The idea here is that if a parti le lo alized in region K1 travels faster than light to another region K2 , then in some other frame it would appear as an antiparti le travelling from K2 to K1 . Although it undeniably stimulates the imagination, this argument too is wrong. The model presented in this se tion is an expli it ounterexample to both of these suggestions: although it does exhibit the kind of superluminal travel at issue, it is manifestly a single-parti le model, with no other parti les or antiparti les. Thus, the most one an say is that relativity implies the existen e of phenomena whi h are reminis ent of multiple parti les or antiparti les. But even this seems misleading. 7.4

Flat spa etime

The free s alar eld in at (Minkowski) spa etime is easily des ribed. As in the nonrelativisti ase (Se tion 7.1), its Hilbert spa e is Fs L2 (R3 ). Its dynami s are based on the operators Ut and UL given in Se tion 7.3. These give rise to unitary operators on Fs L2 (R3 ) in the natural way: time evolution and Lorentz transformations a t on the n-parti le spa e 2 3 n by U n and U n . L (R )s t L Also as in the nonrelativisti ase (Se tion 7.2), we have observables Qv and Pv and Weyl operators Wv for v 2 L2 (R3 ). The intuition for Qv and Pv mentioned in Se tion 7.2, that they respe tively measure the strength and rate of hange of the v omponent of the eld, is still valid. However, this does not tell us whi h observables an be measured in whi h regions of spa e, an issue of some urgen y given the paradox dis ussed at the end of Se tion 7.3. To answer this question we will need to orrelate quantum observables with elements of lassi al phase spa e; then a quantum observable will be measurable within some region K if and only if the orresponding lassi al eld state is supported in K . Any state of the lassi al eld an be des ribed at a given time by the eld strength f and its time derivative f . For de niteness let us suppose 1 3 f and f are real-valued fun tions in C (R ). Thus the lassi al phase spa e V is the set of all su h pairs (f; f ). Now given (f; f ) we want to identify a ve tor v = T q (f; f ) 2 L2 (R3 ) su h that Qv and Pv an be interpreted as the strength and rate of

hange of the lassi al mode (f; f ) of the quantum eld. In this way we will orrelate quantum observables with elements of lassi al phase spa e mu h as the operators Q and P orrespond to lassi al position and momentum. There are three natural onsisten y onditions that an be pla ed

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Chapter 7: Quantum Field Theory

on this lassi al labelling T q : V ! L2 (R3 ) of quantum observables: it should respe t time evolution, the a tion of the Lorentz group, and symple ti stru ture. It turns out that these requirements ompletely determine T q . The dynami s of the lassi al eld are des ribed by the Klein-Gordon equation  m 2 1 2f 2f 2f 2f f: = 2 2 2 2 2

 x0  x1  x2  x3  h It follows that in the Fourier transform pi ture the time derivative of f^ is f^ and the time derivative of f^ is p20 f^. Lorentz transformations a t in both the lassi al and quantum pi tures by omposition. Finally, the symple ti stru ture on V is given by

f(

f; f

); (g; g )g =

Z

R3



f g



fg

and on L2 (R3 ) it is the imaginary part of the inner produ t. The unique (up to multipli ation by a s alar of modulus one) quantization map T q whi h satis es the three onsisten y onditions is T

1=2 1=2 )^ = p0 f^ + ip0 f^ :

q (f; f

Consisten y with time evolution follows from the omputations T

q

(ft ; (f )t )^ = p10=2 f^t + ip0 1=2 (f^ )t =

and

(T q (f; f )^)t =

(

1=2

ip0 p0

3=2

ip0

^ + p10=2 f^

f

^ + ip0 1=2 f^ ):

f

Invarian e under Lorentz transformations follows from the omputation T

q

(f Æ L

1

 Æ L 1)^ = T q (f; f )^ Æ LT = T q (f; f )^ Æ L~

;f

1

:

Finally, preservation of symple ti stru ture follows from the omputations Z Z f(f; f ); (g; g)g = f g f g = f^ g^ f^g^ and ImhT q (f; f ); T q (g; g )i = Im = Im

Z Z

(p10=2 f^ + ip0 1=2 f^ )(p10=2 g^ + ip0 1=2 g^ ) (p0 f^g^ + if^ g^

^^ + p0 1 f^ g^ ):

if g

Thus T q has the desired properties and this enables us to asso iate to any (f; f ) 2 V an element of L2 (R3 ) whi h then gives rise to a Weyl

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161 operator on Fs L2 (R3 ). The important point here is that if (f; f ) and (g; g ) are disjointly supported then f(f; f ); (g; g )g = 0 and so the orresponding Weyl operators ommute. Moreover, sin e lassi al solutions of the Klein-Gordon equation propagate at the speed of light, this remains true for Weyl operators asso iated to lassi al elds supported on spatial regions at dierent times that are suÆ iently separated to prevent light travelling between them. The ommutation of these \lo al" operators implies that one annot use the measurement pro ess to send a faster-than-light signal. 7.5

Curved spa etime

Free quantum eld theory an also be formulated against a urved spa etime ba kground. This means that we repla e Minkowski spa etime with a four-manifold M satisfying the following onditions: (1) M is smooth; (2) M is Lorentzian, meaning that its tangent bundle is equipped with a smooth bilinear form [
;
℄ whi h an be expressed in lo al oordinates as [x; y ℄ = 2 x0 y0 x1 y1 x2 y2 x3 y3 ; and (3) the Lorentz form satis es Einstein's eld equations. We must also require that M be globally hyperboli ; this means that there exists a Cau hy surfa e, i.e., a smooth three-manifold M0  M su h that every point in M lies in the past or future of exa tly one point on M0 . In other words, any point in M an be onne ted to a point in M0 by a urve all of whose tangent ve tors v satisfy [v; v ℄ > 0. Global hyperboli ity is essential to our approa h be ause Cau hy surfa es will play the role that onstant t sli es play in at spa etime. However, it is a strong restri tion: it implies that M is homeomorphi to M0  R and is in fa t foliated by Cau hy surfa es. The restri tion of the negation of the Lorentz form to any Cau hy surfa e M0 makes M0 a Riemannian manifold, so it has a unique volume measure and there is a anoni al L2 (M0 ). Following the onstru tion of free quantum elds in at spa etime, it seems natural to take the Hilbert spa e of a free s alar eld on M to be Fs L2 (M0 ), with oneparti le spa e L2 (M0 ). However, here we run into diÆ ulties be ause this onstru tion is not relativisti ally invariant. In parti ular, an element of the one-parti le spa e on a Cau hy surfa e M0 will in general not evolve into an element of the one-parti le spa e on a dierent Cau hy surfa e M00 . In other words, there simply is no relativisti ally invariant model of a single parti le on general globally hyperboli spa etimes (and it seems that this is the ase even in Minkowski spa etime, if one allows noninertial observers). Even worse, the Fo k spa e onstru tions on different Cau hy surfa es are in general not unitarily equivalent, in a sense

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Chapter 7: Quantum Field Theory

that we will explain below. That is, general relativisti dynami s not only fail to take the one-parti le spa e of a given Cau hy surfa e to the one-parti le spa e of another Cau hy surfa e, they fail to take the Fo k spa e of states on one surfa e to the Fo k spa e of states on another! For these reasons, it seems hopeless to try to nd a anoni al Hilbert spa e onstru tion of free elds in urved spa etimes. However, it is still possible to formulate a general model of free elds if one adopts the point of view that the CCR algebras of observables are primary. We now des ribe this approa h. Let M be a globally hyperboli spa etime and for any Cau hy surfa e M0 let V0 be the set of pairs (f; f ) with f and f real-valued fun tions in C 1 (M0 ). The dynami s of lassi al elds are des ribed by the KleinGordon equation  2 ; r2  = m h

where r2 = r
r is the four-dimensional divergen e of the gradient, so that in lo al oordinates whi h diagonalize the Lorentz form at a point we have 2 2 2 2 : r2 = 12 x 2 2 2 x1 x2 x23 0 The general theory of partial dierential equations implies that for any Cau hy surfa e M0 , any pair (f; f ) 2 V0 is the initial data of a unique solution  of the Klein-Gordon equation su h that jM0 = f and the forward normal derivative of  on M0 is f . Moreover,  will be smooth and have spatially ompa t support in the sense that its restri tion to any Cau hy surfa e has ompa t support. Thus, if we let the lassi al phase spa e V of a free s alar eld be the set of all smooth solutions of the Klein-Gordon equation with spatially ompa t support, then for any M0 there is a natural bije tion between V and V0 . There is a natural symple ti form on V de ned by Z f(f; f); (g; g )g = f g fg: M0

It is independent of M0 . To see this let M00 be another Cau hy surfa e that lies in the future of M0 and let M1 be the four-dimensional region bounded by M0 and M00 . Let  and be solutions of the Klein-Gordon equation with respe tive data (f; f ) and (g; g ) on M0 and (f 0 ; f0 ) and (g 0 ; g0 ) on M00 . Then Z Z Z 0 0 0 0 ( r r )
n (f g fg ) = (f g f g ) M00

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M0

M1

163 where n is the outward normal ve tor. By the divergen e theorem this equals Z Z r
( r r ) = ( r2  r2 ) = 0 M1

M1

R

sin e rR2 = m2 2 h 2 and r2  = m2 2 h 2 . Thus M0 (f g fg ) = M (f0 g 0 f 0 g0 ). This was shown for a surfa e M00 lying in the 0 future of M0 , but given any two Cau hy surfa es we an nd a third lying in both of their futures, so we on lude that the symple ti form is the same on every Cau hy surfa e. Thus V is equipped with a well-de ned symple ti form. 0

M00 M1 M0 Figure 7.2

M0 M0 ,

0

, and

M1

Now it is possible to embed V in a omplex Hilbert spa e in su h a way that its symple ti form agrees with the imaginary part of the inner produ t. For example, mimi king the orresponding onstru tion in at spa etime, we an embed V  = V0 in L2 (M0 ) by

T q (f; f ) = A1=4 f + iA

1 =4

f

where A = 2 r2M0 + m2 4 h 2 , r2M0 being the three-dimensional Riemannian Lapla ian on M0 . However, the onsisten y onditions whi h provided motivation for this de nition in at spa etime are no longer meaningful, and the resulting CCR(V ) representations (De nition 7.2.3) for dierent Cau hy surfa es are not unitarily equivalent. Nonetheless, this does allow us to de ne CCR(V ) and to know that it has at least one Hilbert spa e representation. Re all that by Theorem 7.2.5 the C*algebra CCR(V ) is independent of M0 . Let us take sto k of the situation. We now have a well-de ned algebra of observables CCR(V ), but we do not have a anoni al embedding of CCR(V ) in a \one-parti le" Hilbert spa e be ause the latter on ept does not have a relativisti ally invariant meaning. But suppose we are given any representation of CCR(V ) on a Hilbert spa e H. Then for any

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Chapter 7: Quantum Field Theory

Cau hy surfa e M0 and any (f; f ) 2 V0 , there is a unique  2 V with initial data (f; f ) and a orresponding Weyl operator W 2 CCR(V ) whi h a ts on H. We an therefore do quantum me hani s in the usual way, and by orrelating the Weyl operators with various Cau hy surfa es we are able to interpret the results of observations made by dierent observers. In parti ular, for any region S  M0 the observables that an be measured in S are pre isely those in the sub-C*-algebra of CCR(V ) generated by the operators W su h that the orresponding f and f on M0 are supported in S . Thus, if we a

ept that there simply is no anoni al Hilbert spa e

onstru tion, we an still be satis ed with interpreting dierent representations of CCR(V ) as dierent realizations of a free eld on M . There is not even any good reason to restri t ourselves to Fo k spa e

onstru tions, although there are various other physi ally and mathemati ally motivated restri tions on whi h representations ought to be allowed. On this view, among the allowed representations of CCR(V ) none is a

orded a fundamental status. The leanest mathemati al expression of this approa h is to take C*-algebrai states in the sense of De nition 5.6.3 as primary, giving rise to Hilbert spa e representations via the GNS onstru tion. The drawba k is that a parti le interpretation be omes diÆ ult in general. But this seems to be a feature of general relativity that one just has to a

ept. Presumably one should still be able to de ne parti les lo ally via a at spa etime approximation. The fa t that representations of in nite dimensional CCR algebras are not unique up to unitary equivalen e has been put forward as an argument for the value of C*-algebrai methods in quantum eld theory. Using the CCR algebra point of view, one is able to re ognize dierent representations of the same algebra as being realizations of the \same" quantum eld; moreover, the language of C*-algebrai states is onvenient for onstru ting representations. Quantum eld theory in urved spa etime provides an even stronger version of this argument be ause here the Hilbert spa e onstru tions of dierent observers are generally inequivalent, and it is only through the C*-algebra CCR(V ) that one is able to relate the experien e of states on dierent Cau hy surfa es. That is, the theory annot be formulated in a relativisti ally invariant manner at the level of Hilbert spa es, only at the level of C*-algebras. 7.6

Notes

A thorough treatment of nonrelativisti elds is given in [36℄. For a Fermioni version of the L2 (H) onstru tion see [6℄. [2℄ is another good mathemati al referen e on quantum eld theory. Standard referen es on CCR and CAR algebras (the Fermioni version of CCR algebras) are [10℄ and [32℄. Almost periodi fun tions are treated

© 2001 by Chapman & Hall/CRC

165 in [14℄, and their relation to CCR algebras is dis ussed in [72℄. Relativisti single-parti le systems, in luding spin, are treated in [68℄. For more on lo alizability and superluminal signalling see [76℄ and [35℄. Spe ial relativisti quantum eld theory is treated in [33℄, with emphasis on the use of C*- and von Neumann algebras to des ribe the observables whi h an be measured lo ally. Free elds in urved spa etime are extensively dis ussed in [30℄ and [69℄. The embedding of T q : L2(M0 ) des ribed in Se tion 7.5 was given in [4℄; the fa t that there exist spa etimes in whi h it has undesirable properties follows from [42℄. Also see [4℄ for further dis ussion of the

on eptual basis of the Cau hy surfa e approa h to quantum me hani s in urved spa etime. The argument that shows the symple ti form is independent of the Cau hy surfa e was shown to me by Renato Feres.

V!

© 2001 by Chapman & Hall/CRC

© 2001 by Chapman & Hall/CRC

Chapter 8

Operator Spa es 8.1

The spa es

V (K )

The topi s in Chapters 2, 3, 5, and 6 onstitute the \elementary" theory of mathemati al quantization. In the remaining hapters we will introdu e some more advan ed topi s. This hapter is about operator spa es, whi h are norm losed linear subspa es of B (H) and are thought of as a quantum version of Bana h spa es. In order to make sense of this interpretation, we need to take a on rete view of Bana h spa es. We will a

omplish this by introdu ing a

lass of obje ts alled \dual unit balls" and de ning, for ea h dual unit ball K , a spa e of fun tions V (K )  l1 (K ), su h that the spa es V (K ) are on rete realizations of Bana h spa es. Then operator spa es will be the orresponding subspa es of B (H). DEFINITION 8.1.1

A subset

K

of a topologi al ve tor spa e (TVS)

balan ed if x 2 K and a 2 C, jaj = 1, implies ax 2 K . A dual unit ball is then a ompa t, onvex, balan ed subset of a lo ally onvex TVS. For any dual unit ball K , let V (K ) be the spa e of ontinuous fun tions f from K into C whi h are linear in the sense that f (ax + by ) = af (x) + bf (y ) whenever x; y; ax + by 2 K . Give V (K ) the supremum norm it inherits from l 1 (K ). is

Note that this de nition of linearity is equivalent to asserting that f extends to a linear fun tion on the span of K . It is easy to see that V (K ) is a norm losed subspa e of l1 (K ), so it is a Bana h spa e. We write (V )1 for the losed unit ball of a Bana h spa e V . Example 8.1.2

Let

V

be a Bana h spa e and let

K = (V  )1 be the unit ball of the 167

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168

Chapter 8:

dual spa e

V .

Then

K

Operator Spa es

is a dual unit ball.

PROPOSITION 8.1.3 Let

K

ball of

be a dual unit ball. Then

V (K ) .

K

is linearly homeomorphi to the unit

Let V = V (K ) and for x 2 K de ne x^ 2 (V  )1 by x^(f ) = f (x). It is easy to he k that the map x 7! x ^ is linear and that it is ontinuous going into the weak* topology on V  . It is also one-toone sin e lo al onvexity implies that the ontinuous linear fun tionals separate points of K . Thus K is linearly homeomorphi to a weak*

ompa t, onvex subset of (V  )1 . To show that the map x 7! x^ is onto, suppose ! 2 V  is not in its image. Then by a standard separation theorem there exists a weak*

ontinuous linear fun tional T on V  su h that

PROOF

Re T (^ x)  1 < Re T (! ) for all x 2 K . It is also standard that there exists T () = (f ) for all  2 V  , so we have

f

2V

su h that

Re f (x)  1 < Re ! (f ) x 2 K . Sin e K is balan ed and f is linear, this implies that kf k  1, and hen e k!k > 1. We on lude that K  = (V  )1 .

for all

The pre eding result shows that every abstra t dual unit ball gives rise to a Bana h spa e of whi h it is the dual unit ball. Conversely, we have the following result. PROPOSITION 8.1.4 Let

V

be a Bana h spa e and let

isomorphi to

V (K ).

K

= (V  )1 .

Then

V

is isometri ally

The weak* ontinuous linear fun tionals on V  are pre isely ^ the maps  : ! 7! ! ( ) for  2 V . Ea h ^ restri ts to a ontinuous linear fun tion on K , and onversely, by the Krein-Smulian theorem every

ontinuous linear fun tion on K extends to a weak* ontinuous linear fun tional on V  . Thus the map  7! ^jK is a linear isomorphism of V onto V (K ). It is isometri by the Hahn-Bana h theorem.

PROOF

Next we observe that maps between Bana h spa es orrespond to maps between their dual unit balls.

© 2001 by Chapman & Hall/CRC

169 Example 8.1.5

!

Let K and L be dual unit balls and let  : K L be a ontinuous linear map (in the same sense as in De nition 8.1.1). Then omposition with  de nes a linear ontra tion from V (L) into V (K ). Example 8.1.6

V

W

V!W

Let and be Bana h spa es and let T : be a linear

ontra tion. Then omposition with T de nes a weak* ontinuous linear map from (  )1 into (  )1 .

W

V

The veri ation of both examples is routine. 8.2

Matrix norms and onvexity

We found in Se tion 8.1 that every Bana h spa e an be realized on retely as a spa e of ontinuous linear fun tions on a dual unit ball, and thus Bana h spa es are the abstra t version of the spa es V (K ). Ea h V (K ) is a norm losed linear subspa e of l1 (K ), and onversely, every norm losed linear subspa e of any l1 (X ) is a Bana h spa e. This motivates the following de nition. DEFINITION 8.2.1

spa e of some

B (H).

An

operator spa e is a norm losed linear sub-

Operator spa es are the quantum version of dual unit balls in the same way that C*-algebras are the quantum version of topologi al spa es and von Neumann algebras are the quantum version of measure spa es. In ea h ase we have a lassi al obje t (topologi al spa e, measure spa e, dual unit ball), a fun tion theoreti parallel (C (X ), L1 (X ), V (K )), and an operator analog (C*-algebra, von Neumann algebra, operator spa e). Now every operator spa e is a Bana h spa e, so at rst it appears that there is no dieren e between operator spa es and Bana h spa es. However, B (H) has an extra level of stru ture whi h we have not dis ussed yet, and this extra stru ture is inherited by operator spa es. Namely, any n  n matrix of bounded operators [Aij ℄, with ea h Aij in B (H), an be viewed as operating on the n-fold dire t sum Hilbert spa e Hn . Thus the matrix [Aij ℄ has a natural norm, and this norm is ompatible with matri es of dierent sizes in a ertain way. This suggests the following de nition. To simplify notation, we will write Mn for Mn (C). We will also write Mm;n for the spa e of all m  n omplex matri es. Note that Mm;n an be identi ed with the set of linear maps from Cn to Cm ; we therefore take the norm of an element in Mm;n to be its operator norm.

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170

Chapter 8: Operator Spa es

For any ve tor spa e V and any n 2 N, let be the ve tor spa e of all n  n matri es with entries in V . We an multiply matri es over V with s alar matri es using the usual formula for matrix produ ts. A matrix norm on V is a sequen e of norms de ned on ea h Mn (V ) with the property that DEFINITION 8.2.2

M (V ) n

kAB k  kAkkkkB k for all A 2 Mm;n , B 2 Mn;m , and  2 Mn (V ). Any linear map T : V ! W indu es a linear map T (n) : Mn (V ) ! Mn(W ) de ned entrywise. If V and W are matrix normed spa es, then a linear map T : V ! W is ompletely bounded if its ompletely bounded (CB) norm kT k b = supn kT (n)k is nite. T is ompletely ontra tive if kT k b  1 and ompletely isometri if ea h T (n) is isometri .

We present two examples of matrix norms. Example 8.2.3

Let V  B (H) be an operator spa e. Give Mn (V ) the norm it inherits from Mn (B (H))  = B (Hn ). This is a matrix norm on V . Example 8.2.4

Let V be a Bana h spa e. Then V  = V (K ) where K = (V  )1 , 1 and V (K )  l (K ). De ne the norm of any matrix F = [fij ℄ 2 Mn (V (K )) by kF k = sup kF (x)k = sup k[fij (x)℄k;

x2K

x2K

where k[fij (x)℄k is the usual operator norm of a s alar matrix. This is a matrix norm on V (K )  = V.

The matrix norm in Example 8.2.4 agrees with the matrix norm in Example 8.2.3 when we embed l1 (K ) in B (l2 (K )) in the usual way as multipli ation operators. Next we give examples of omplete ontra tions. Example 8.2.5

Let A and B be C*-algebras and let  : A ! B be a -homomorphism. Then  is a ontra tion by the omment following De nition 5.3.1. But Mn (A) and Mn (B) are also C*-algebras and  (n) : Mn (A) ! Mn (B) is a -homomorphism for all n, so  is in fa t ompletely

ontra tive.

© 2001 by Chapman & Hall/CRC

171 This example shows why matrix norms have not played a role in earlier hapters. When one is dealing with -homomorphisms, the higher matrix levels are typi ally irrelevant. However, in the setting of general linear maps between operator spa es the CB norm is usually more important than the rst level (Bana h spa e) norm. The simplest example where the Bana h spa e and CB norms dier is the transpose map on M2 . The map 

a

b

d



7!



a

b

d



is isometri , but its ompletely bounded norm is 2. We have now de ned the basi on epts surrounding operator spa es. Next we want to hara terize these spa es abstra tly. This requires a separation theorem whose proof is a bit te hni al but whi h is quite useful. It involves separating points from ompa t onvex sets. However, the desired result does not just a

omplish this on a single level (that

ould be done using lassi al separation theorems); rather, we onsider

onvex sets on all levels. We need the following de nition. DEFINITION 8.2.6

matrix onvex set

over

su h that (a)

X

2

Km

and

Y

K

 Kn

implies

0

X

0

Kn



Y

2

Mn

Km

+

n,

and

)1 , B 2 (Mn;m )1 , and X 2 Kn , then AX B 2 Km . K is losed if ea h Kn is losed in Mn (V ), giving Mn (V ) the natural topology arising from the topology on V (nets in Mn (V ) onverge if and

(b) if

A

2(

2

Let V be a lo ally onvex TVS. A balan ed V is a sequen e = ( ) of subsets of (V )

Mm;n

only if they onverge entrywise).

2 p 0 p

Observe that ea h Kn must be onvex, for if X; Y (s; t  0) then sX

+ bY

p =[

sIn

p

tIn

℄



sIn

X

0

Kn

Y

tIn

and s + t = 1

 ;

where In is the n  n identity matrix, belongs to Kn as well. Property (b) also implies that ea h Kn is balan ed. We now pro eed to prove the desired separation theorem. A fun tion f on a onvex set K is aÆne if (

f tx

+ (1

) ) = tf (x) + (1

t y

) ( )

t f y

for all x; y 2 K and all t 2 [0; 1℄. A one is a subset of a ve tor spa e that is losed under addition of ve tors and multipli ation by nonnegative real numbers.

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Chapter 8: Operator Spa es

172 LEMMA 8.2.7

Let K be a ompa t onvex subset of a TVS and let E be a one of real

ontinuous aÆne fun tions on K . Suppose that for ea h  2 E there exists x 2 K su h that (x)  0. Then there exists x0 2 K su h that (x0 )  0 for all  2 E . PROOF By ompa tness it will suÆ e to show that for any nite set of fun tions 1 ; : : : ; n 2 E there exists x 2 K su h that i (x)  0 for 1  i  n. Suppose this L fails for some 1 ; : : : ; n . Let  : K ! Rn be the dire t sum map  = i . Then (K ) is a ompa t onvex subset of Rn , and by assumption it does not interse t the losed onvex set [0; 1)n . So by a standard separation theorem there is a linear fun tion n ! : Rn ! P R su h that ! ((K )) < 0 and ! ([0; 1) )  0. We have !Æ = ai i where ai = ! (ei )  0 (ei being the ith basis ve tor in Rn ), so that ! Æ  2 E . But then ! ((K )) < 0 ontradi ts the hypothesis of the lemma, so we have rea hed a ontradi tion. LEMMA 8.2.8

Let K = (K ) be a losed balan ed matrix onvex set over a lo ally

onvex TVS V . Let X0 2 M (V ) and suppose X0 62 K . Then there is a ontinuous linear map F : M (V ) ! C and a pair of states ! and  on M su h that n

n

n

n

n

for all X 2 K and

jF (X )j  1 < jF (X0 )j

n

jF (AXB )j  (!(AA )(B  B ))1 2 , B 2 M , and X 2 K (m 2 N). =

for all A 2 M

n;m

m;n

m

PROOF

The existen e of F su h that jF (X )j  1 < jF (X0 )j for all X 2 Kn follows from a standard separation theorem and the fa t that Kn is a losed balan ed onvex set in Mn (V ). Let Sn be the set of states on Mn . Then Sn2 = Sn  Sn is a ompa t

onvex set. Given any X 2 Km and any A 2 Mn;m and B 2 Mm;n , de ne a real ontinuous aÆne fun tion  = A;X;B on Sn2 by (!; ) = ! (AA ) + (B  B )

2Re F (AXB ):

Let E be the set of all su h fun tions A;X;B . We have aE  E for all a  0 sin e aA;X;B = A0 ;X;B 0 where A0 = a1=2 A and B 0 = a1=2 B . We also have E + E  E A1 ;X1 ;B1 + A2 ;X2 ;B2 = A;X;B where   be ause B1 , and X = X1  X2 2 Km1 +m2 . Thus E is a A = [ A1 A2 ℄, B = B2

one.

© 2001 by Chapman & Hall/CRC

173 Moreover, for ea h  2 E there exists a point (!; ) 2 Sn2 su h that (!; )  0. To see this x A; X; B and nd states ! and  on Mn su h that ! (AA

 ) = kAA k = kAk2

Then

A;X;B (!; )

sin e

and

= kAk2 + kB k2



(B B )

= kB  B k = kB k2 :

2Re F (AX B )  0

Re F (AX B )  jF (AX B )j  kAkkB k:

Thus Lemma 8.2.7 applies, and we on lude that there exist states ! and  on Mn su h that A;X;B (!; )  0 for any A 2 Mn;m , B 2 Mm;n , and X 2 Km . For this hoi e of ! and  we have 2Re F (AX B )  ! (AA ) + (B  B ) for all A 2 Mn;m , B 2 Mm;n , and X 2 Km . Repla ing A by B by b 1=2 B where b = ! (AA ) 1=2 (B  B )1=2 then yields

b1=2 A

and

Re F (AX B )  ! (AA )1=2 (B  B )1=2 : Finally, repla ing A with (jF (AX B )j=F (AX B ))A yields the desired inequality. A state ! on a C*-algebra is faithful if ! (A A) = 0 implies

A

= 0.

LEMMA 8.2.9 Under the hypotheses of Lemma 8.2.8, we an obtain the same on lusion with

!

and



faithful.

PROOF

Let F , ! , and  be as in Lemma 8.2.8. The normalized tra e = n1 tr on Mn is faithful, so for any  2 (0; 1) the states ! 0 = (1 )! + and 0 = (1 ) +  are faithful. Let F 0 = (1 )F ; for suÆ iently small  we still have 

for all have

X

2K

n

jF 0 (X )j  1 < jF 0 (X0 )j , and for A 2 M , B 2 M , and X 2 K

jF 0 (AX B )j  (1  21 (1

© 2001 by Chapman & Hall/CRC

n;m

m;n

m

 )1 2 (B  B )1 2   )(! (AA ) + (B B ))

)! (AA

=

=

we also

174

Chapter 8:

Operator Spa es

1 0  0  2 (! (AA ) +  (B B )): Then repla ing A and B by b1=2 A and b 1=2 B as in the proof of Lemma 8.2.8 yields F 0 (AXB ) (!0 (AA )0 (B  B ))1=2 ; as desired. 

j

j 

THEOREM 8.2.10 Let

= (Kn )

K

be a losed balan ed matrix onvex set over a lo ally

X0

onvex TVS V . Let

Mn ( )

: Mn

(n) (X0 ) > 1

2

a ontinuous linear map

m2

N

and

X

2

Km

V

and suppose

V !

but k

X0

su h that k

k

62

Kn .

Then there is

(m) (X )

1

k 

for all

.

PROOF Let F , !, and  be as in Lemma 8.2.8 with ! and  faithful. Apply the GNS onstru tion (Theorem 5.6.6) to ! and  to get representations  : Mn B ( ! ) and  : Mn B (  ). Faithfulness of ! and  implies that  and  are -isomorphisms. Let v0 = In ! and w0 = In , so that ! ( A ) =  ( A ) v ; v and  ( A ) =  ( A ) w  0 0 0 ; w0 for all A Mn . For any 1 n s alar matrix A = [ a1 : : : an ℄ let A~ be the n n s alar matrix 3 2a a an 1 2 ~ = 664 0.. 0.. . . 0.. 775 : A . . . . 0 0 0 Let E be the set of all n n matri es of this form, and de ne =  (E )v0 = (E )w0 ! and  . We have dim( ) = dim( ) = n. For x de ne a sesquilinear form on by ~ )w0 ; (A~)v0 = F (A xB ):  (B Then let (x) : be the linear map for whi h w; v = (x)w; v . The set of linear maps from into an be ompletely isometri ally identi ed with Mn , so we may regard as a map from into Mn . Let ei be P the standard basis of Cn = M1;n . For X = [xij ℄ Mn ( ) we have X = ij ei xij ej , so !

H

!

H



2 H

2 H

h

i

h

i

2






















 H

H

K

 H

H

2 V

K

K H

f

g

K ! H

f

K

g

h

i

H

V

f



g

F (X ) =

where v1

X

h

2

(xij )(~ej )w0 ; (~ei )v0 = (n) (X )w1 ; v1 i

3 2  (~ e1 )v0 = 64 .. 75

.

 (~ en )v0

© 2001 by Chapman & Hall/CRC

and

w1

h

3 2 (~ e1 )w0 = 64 .. 75

.

(~ en )w0

i

V

175 (both in Cn ). A short omputation shows that kv1 k = kw1 k = 1. Sin e jF (X0 )j > 1, this shows that k (n)(X0 )k > 1 also. Finally, let X 2 Km and x v and w in Cmn . Write 2 ~ 3 3 2 ~  (A1 )v0 (B1 )w0 7 6 6 .. 7 .. v=4 and w=4 5 . 5 . ~ ~  (Am )v0 (Bm )w0 2

for some 1  n matri es Ai and Bi . Then kvk2 = !(A A) and kwk2 = (B  B ) where 2A 3

A=4

1

.. 5 .

Am

and

2B 3 1

B

= 4 ... 5 Bm

are m  n matri es. Thus

jh m (X )w; vij = jF (A XB )j  (!(A A)(B  B )) = = kvkkwk: As v and w were arbitrary, this shows that k m (X )k  1. (

)

1 2

(

)

The abstra t hara terization of operator spa es is an easy onsequen e of the pre eding result. An L1 -matrix norm on a ve tor spa e V is a matrix norm whi h satis es



 0 



0 H = max(kk; kH k) for  2 Mm(V ) and H 2 Mn(V ). It is easy to see that the matrix norm on B (H) is an L1 -matrix norm, and hen e the same is true of any operator spa e. The onverse is also true; this is known as Ruan's theorem. THEOREM 8.2.11

Let

V

be a omplete

L1 -matrix

normed spa e. Then

V

is ompletely

isometri to an operator spa e.

PROOF

Let T be the dire t sum of all omplete ontra tions from

V into Mn (n 2 N). It is lear that T is a omplete ontra tion, and to show that it is a omplete isometry we must nd, for ea h  2 Mn (V ) with k k > 1, a omplete ontra tion : V ! Mm for some m 2 N su h that k n ( )k > 1. But the sequen e of unit balls Km = 0

0

( )

0

© 2001 by Chapman & Hall/CRC

176

Chapter 8: Operator Spa es

(M (V ))1 is a losed balan ed matrix onvex set over V , so Theorem 8.2.10 implies the existen e of a ontinuous linear map : V ! M su h that k ( )()k  1 for all  2 K (m 2 N) but k ( ) (0 )k > 1. That is, is a omplete ontra tion, and we on lude that T is a omplete isometry. Finally, we have assumed that V is a Bana h spa e, so T (V ) is losed and hen e is an operator spa e. m

n

m

8.3

n

m

Duality

Every operator spa e V has a dual V  whi h is also an operator spa e. These behave mu h like Bana h spa e duals; for example, V ompletely isometri ally embeds in V  (Proposition 8.3.3). We will use operator spa e duals to formulate a de nition of \dual matrix unit balls" whi h are the operator spa e analog of the dual unit balls dis ussed in Se tion 8.1. Let V be an operator spa e. Its dual is the Bana h spa e dual V  , with matrix norms de ned by taking the norm of an element of Mn (V  ) to be the ompletely bounded norm of the map it indu es from V into Mn . Thus, for X = [xij ℄ 2 Mn (V  ) we have

DEFINITION 8.3.1

kX k = supfkX ()k : m 2 N and  2 (M (V )) g = supfk[x ( )℄k : m 2 N and  = [ ℄ 2 (M (V )) g: (m)

ij

1

m

kl

kl

1

m

When n = 1 this norm agrees with the Bana h spa e norm on V  . To see this let  2 (M (V ))1 , let x 2 V  , and let v; w 2 C  = M 1 . Then n

n

n;

jhx ()v; wij = jx(w v)j  kxkkvkkwk; (n)

whi h shows that kx( ) ()k  kxk. Thus the operator spa e norm of x is at most its Bana h spa e norm, and the reverse inequality is lear. The dual of an operator spa e is itself an operator spa e. One an show this by means of Ruan's theorem (Theorem 8.2.11), but it is easier to give a dire t proof. n

PROPOSITION 8.3.2

Let V be an operator spa e. Then there is a ompletely isometri and weak* homeomorphi isomporphism of its dual spa e V  with a weak*

losed subspa e of some B (H).

PROOF

Let  be the set of all pairs (m; ) su h that m 2 N and

© 2001 by Chapman & Hall/CRC

177  2 (Mm(V ))1 , and let

H= Then de ne T : V  we have

M (m;)2

Cm :

! B(H) by T (x) = L x(m)().



M

( m ) ( n )

X ()

= sup T (X ) =

(m;)2

(m;)2

k

k

For X

2 Mn(V )

kX (m)()k = kX k b;

so the matrix norms on V  and T (V ) agree, and hen e T is a omplete isometry. If (x ) is a bounded net in V  and x ! x weak*, then x(m) () ! x(m) () in ea h entry, for ea h  2 (Mm (V ))1 . It follows that T x ! T x weak* in B (H). Therefore, sin e T is an isometry, the unit ball of T (V ) is weak* ompa t, and hen e by the Krein-Smulian theorem T (V ) is a weak* losed subspa e of B (H). Moreover, this shows that for bounded nets the weak* topology on V  agrees with the indu ed weak* topology on T (V ), whi h implies that the two weak* topologies are equal. Next we prove that any operator spa e ompletely isometri ally embeds in its double dual. This result is an easy onsequen e of the separation theorem proven in Se tion 8.2. PROPOSITION 8.3.3 Let

V

V V

be an operator spa e.

isometri ally embeds

in

Then the natural map

 .



7! ^

ompletely

A straightforward omputation shows that the map  7! ^ is a omplete ontra tion. To prove that it is a omplete isometry, let  = [ij ℄ 2 Mm (V ) and suppose kk > 1; we must show that k^ k = k[^(ijm℄k) > 1. To do(m)this it will suÆ e to nd X 2 Mm(V ) su h that k^ (X )k = kX ()k > 1. But taking Kn = (Mn(V ))1 in Theorem 8.2.10 yields pre isely this.

PROOF

Next we onsider the abstra t hara terization of dual operator spa es. Con retely, Proposition 8.3.2 shows that these are pre isely the weak*

losed subspa es of B (H). Abstra tly, one might expe t that any operator spa e whi h is a dual Bana h spa e should be a dual operator spa e; however, a slightly stronger assumption is needed.

© 2001 by Chapman & Hall/CRC

178

Chapter 8: Operator Spa es

PROPOSITION 8.3.4

Let V be an operator spa e whi h is a dual Bana h spa e and suppose that the unit ball of Mn (V ) is weak* losed for all n. Then V is a dual operator spa e.

We may assume that V = W  for some Bana h spa e W . Then W isometri ally embeds in V  , and inherits an operator spa e stru ture from this spa e. We must show that V = W  as operator spa es. Let  be the set of all pairs (m; ) su h that m 2 N and  2 (Mm (W ))1 , and apply the onstru tion of Proposition 8.3.2 to get a weak* ontinuous omplete ontra tion T : V ! B (H). It will suÆ e to show that T is a omplete isometry. To do this, let X 2 Mn(V ) and suppose kX k > 1; we must nd  2 Mm (W ) su h that kk  1 but kX (m)()k > 1. De ne Kn = (Mn (V ))1 and apply Theorem 8.2.10 to V equipped with the weak* topology. Then we get a matrix of weak*

ontinuous maps from V into C su h that k k  1 but k (n) (X )k > 1. However, every weak* ontinuous map from V into C is obtained by pairing with an element of W , so we have = ^ for some  2 (Mm (W ))1 , as desired. PROOF

Now we are ready to present the matrix version of the dual unit balls dis ussed in Se tion 8.1. A dual matrix unit ball is a balan ed matrix = (Kn ) su h that ea h Kn is ompa t.

DEFINITION 8.3.5

onvex set K

Sin e K is a balan ed matrix onvex set, ea h Kn must be a balan ed

onvex set (see the omment following De nition 8.2); as we are now also assuming that the Kn are ompa t, it follows that ea h Kn is a dual unit ball in the sense of De nition 8.1.1. The prototypi al example of a dual matrix unit ball is the sequen e of matrix unit balls over a dual operator spa e. Example 8.3.6

Kn = (Mn (V  ))1 be the sequen e  of matrix unit balls of the dual operator spa e. Giving V the weak* topology, ea h Kn is ompa t, and K = (Kn ) is a dual matrix unit Let

V

be an operator spa e and let

ball.

If K and L are dual matrix unit balls, we will use the term ompletely linear map from K to L to mean a linear map  : K1 ! L1

© 2001 by Chapman & Hall/CRC

179 su h that (n) (Kn )  Ln for all n. We will say that  is a omplete homeomorphism if ea h (n) is a homeomorphism. PROPOSITION 8.3.7 Let

K = (Kn ) be a dual matrix unit ball.

Then

K is ompletely linearly

homeomorphi to the sequen e of matrix unit balls of some dual operator spa e.

PROOF As we noted above, ea h Kn is a dual unit ball. Let V = span(K ). Condition (b) of balan ed matrix onvexity implies that ea h entry of any element in Kn belongs to K1 , so Mn (V ) = span(Kn ) for all n. De ne a matrix norm on V by letting the unit ball of Mn (V ) be Kn . This does de ne a norm on ea h Mn (V ) by Proposition 8.1.3, and it makes V into an operator spa e by Theorem 8.2.11. As the unit ball of ea h Mn (V ) is Kn , whi h is weak* ompa t, it follows from Proposition 8.3.4 that V is a dual operator spa e. So K is identi ed with the sequen e of matrix unit balls over a dual operator spa e.

We now have a way of passing from operator spa es to dual matrix unit balls, and vi e versa, just as we an pass between Bana h spa es and dual unit balls. As in the Bana h spa e ase, morphisms between the two types of obje ts also orrespond. Example 8.3.8

V and W be operator spa es and let Kn = (Mn (V )) and Ln = (Mn (W )) be the orresponding dual matrix unit balls. If T : V ! W is a omplete ontra tion then omposition with T de-

Let

1

1

nes a ontinuous ompletely linear map from

PROPOSITION 8.3.9 Let

V

and

W

be operator spa es and let

dual matrix unit balls.

:L!K to

W

K

and

L to K .

L be the orresponding

Then any ontinuous ompletely linear map

is given by omposition with a omplete ontra tion from

V

.

PROOF By Proposition 8.1.4 and Example 8.1.5 there is a ontra tion T : V ! W su h that  is omposition with T . If T fails to be a

omplete ontra tion then for some n it does not take the unit ball of Mn (V ) into the unit ball of Mn (W ) for some n, and then omposition with T annot take Ln into Kn , ontradi ting omplete linearity of . So T is a omplete ontra tion.

© 2001 by Chapman & Hall/CRC

180 8.4

Chapter 8: Operator Spa es Matrix-valued fun tions

The notion of a dual matrix unit ball, introdu ed in the previous se tion, provides the basis for a dierent approa h to operator spa es whi h relates them to C*- and von Neumann algebras. The basi idea is this. If V is a Bana h spa e then we an identify V with V (K ), and as the dual unit ball K is a ompa t Hausdor spa e we an write V (K )  C (K )  l1 (K ). For operator spa es there will be an analogous onstru tion in whi h C (K ) and l1 (K ) are repla ed by non ommutative C*- and von Neumann algebras. We give the von Neumann algebra onstru tion rst. DEFINITION 8.4.1 Let K be a dual matrix unit ball. We de ne 1 (K ) to be the set of sequen es f = (f ) of the matrix l1 -spa e lmat n bounded fun tions from Kn into Mn su h that

(a) supn kfn k1  1; (b) X 2 Kn and Y 2 Km implies

fn+m and ( ) if



X 0

0

Y

  =

fn (X ) 0



0 fm(Y ) ;

X 2 Kn and A 2 Mn is unitary then fn (A XA) = A fn (X )A.

This is supposed to be a matrix version of l1 appropriate to the

ontext of dual matrix unit balls. In fa t, the fun tions at the rst level do onstitute pre isely l1 (K1 ). The higher level fun tions take values in matrix algebras rather than C and are also required to satisfy the

ompatibility onditions (b) and ( ) whi h relate to the stru ture of K . 1 (K ) has a natural algebrai stru ture with operaObserve that lmat tions de ned pointwise. It is not ommutative, of ourse, be ause the fun tions fn take values in matrix algebras. For any dual matrix unit ball K let Vmat (K ) 1 (K ) onsisting of those sequen es f = (f ) su h lmat n

DEFINITION 8.4.2

be the subspa e of that

(a) f1 : K1 ! C is linear and ontinuous and (b) fn [xij ℄ = [f1 (xij )℄ for all n 2 N and [xij ℄ 2 Kn , i.e.,

fn = f1(n).

We think of Vmat (K ) as the spa e of \ ontinuous linear fun tions" in

1 (K ). lmat

1 (K ). Now we present the basi fa ts about Vmat (K ) and lmat

© 2001 by Chapman & Hall/CRC

181 LEMMA 8.4.3

K

Let

be a dual matrix unit ball and suppose

there exists

f

X

2 Vmat (K ) su h that fn (X ) 6= 0.

2 Kn is nonzero. Then

PROOF Sin e X is nonzero it must have a nonzero entry xij . Let f1 : K1 ! C be a ontinuous linear map su h that f1 (xij ) 6= 0. Then de ning f = (fn ) by fn = f1(n) immediately yields fn (X ) 6= 0, and it is 1 (K ) and hen e in V straightforward to verify that f 2 lmat mat (K ). THEOREM 8.4.4

V be an operator spa e and let K be the orresponding dual matrix

Let

unit ball. Then

1 (K ) is a von Neumann algebra; (a) lmat

Vmat (K )

(b)

is an operator spa e whi h is ompletely isometri to

and ( )

V;

1 (K ) as a von Neumann algebra. Vmat (K ) generates lmat

PROOF

L

1 (K ) as a subalgebra of B (H) where H = (a) Regard lmat Cm, the sum being taken over all m 2 N and all X 2 Km as in the proof of 1 (K ) is a unital -subalgebra Proposition 8.3.2. It is easy to see that lmat of B (H), and the onditions (b) and ( ) of De nition 8.4.1 are learly 1 (K ) is a von Neupreserved by bounded weak operator limits, so lmat mann algebra. (b) This follows from Proposition 8.3.3. ( ) Call an element X 2 Kn  Mn (V  ) redu ible if there is a nontrivial proje tion P 2 Mn su h that P X = XP ; also, say that X; Y 2 Kn are unitarily equivalent if there exists a unitary matrix U 2 Mn su h that Y = U  XU . 1 (K ) generated by V Let M be the W*-subalgebra of lmat mat (K ). We rst laim that for any irredu ible X 2 Kn , the C*-algebra ffn (X ) : f 2 Mg equals Mn . Suppose not; then by the double ommutant theorem (Theorem 6.5.7) there is a nontrivial proje tion P 2 Mn su h that P fn (X ) = fn (X )P for all f 2 M, and hen e for all f 2 Vmat (K ). By linearity this implies fn (P X XP ) = 0 for all f 2 Vmat (K ), and the lemma then yields P X XP = 0, ontradi ting irredu ibility. This proves the rst laim. We next laim that if X 2 Km , Y 2 Kn are irredu ible and (in

ase m = n) unitarily inequivalent then the C*-algebra A = ffm (X )  fn (Y ) : f 2 Mg equals Mm  Mn . To see this, onsider the natural -homomorphisms 1 : A ! Mm and 2 : A ! Mn ; note that they

© 2001 by Chapman & Hall/CRC

182

Chapter 8: Operator Spa es

are surje tive by the rst laim, and the interse tion of their kernels is zero. If m 6= n, this implies that A = Mm  Mn ( f. Theorem 11.1.2 and Proposition 6.4.1). If m = n then it is also possible that A = Mm . By Proposition 6.4.7, there then exists a unitary matrix U 2 Mm su h that 2 = U  1 U , i.e. fm (Y ) = U  fm (X )U for all f 2 M. But this implies fm (Y U  XU ) = 0 for all f 2 Vmat (K ), and hen e Y U  XU = 0 by the lemma. That is, X and Y are unitarily equivalent, ontradi ting our assumption. This proves the se ond laim. 1 (K ); we will show that Now let f be any self-adjoint element of lmat 1 f 2 M. This is suÆ ient to verify that lmat (K ) = M. For any nite set  of irredu ible, unitarily inequivalent elements Xi ,  = fXi 2 Kn : 1  i  rg; i

it is a onsequen e of the se ond laim that there exists f  2 M su h that fn (Xi ) = fn (Xi ) for 1  i  r. By taking its real part we may assume f  is self-adjoint, and by trun ation using fun tional al ulus we may then assume kf  k  kf k. 1 (K ) It follows from onditions (b) and ( ) in the de nition of lmat that f is determined by its values on irredu ible, unitarily inequivalent elements of K . Thus the net (f  ) eventually agrees with f on any nite subset of K , where this net is ordered by setting   0 if every element of  is unitarily equivalent to some element of 0 . Now

onsider 1 (K ) on the Hilbert spa e H = L Cm whi h the natural a tion of lmat appeared in the proof of part (a) of this theorem. We have just shown that hf  v; wi ! hfv; wi for any elements v and w of the algebrai dire t sum of the Cm , and sin e this is a dense subspa e of H and the net (f  ) is bounded, we therefore have f  ! f weak*. Thus f belongs to the von Neumann algebra generated by Vmat (K ), and this ompletes the proof. i

i

Next, we de ne a matrix version of C (K ). DEFINITION 8.4.5 For any dual matrix unit ball K , let Cmat (K ) 1 (K ). be the unital C*-algebra generated by Vmat (K ) in lmat

1 (K ) su h that ea h Note that this is not the same as the set of f 2 lmat fn is ontinuous. Indeed, it is easy to see that for any f 2 Cmat (K ) the sequen e (fn ) must be uniformly ontinuous. Even this ondition is not suÆ ient to imply f 2 Cmat (K ), however. But the next theorem does give us an abstra t hara terization of Cmat (K ). If A and B are unital C*-algebras, we use the notation hom(A; B) to denote the set of unital -homomorphisms from A into B. Note that if

© 2001 by Chapman & Hall/CRC

183

B = C then this is just the spe trum of A. THEOREM 8.4.6

Let

V be an operator spa e and let K be the orresponding dual matrix

unit ball. Then

(a) Cmat (K ) is the universal unital C*-algebra whi h ompletely isomet, and

ri ally ontains (b)

V

hom(Cmat (K ); Mn ) is anoni ally homeomorphi to Kn

PROOF

for all n.

(a) By de nition, Kn onsists of all ompletely ontra tive linear maps from V into Mn . Re alling the identi ation of V with Vmat (K ) in Theorem 8.4.4 (b), we see that Cmat (K ) is just the C*-algebra generated by the image of V in the dire t sum of all of its nite dimensional ompletely

ontra tive representations. Thus, we need to know that any C*-algebra whi h ompletely isometri ally ontains V is residually nite in the following sense. Given any

ompletely ontra tive representation T : V ! B (H) of V , the norm of any element of the -algebra generated by T (V ) must be approximated by the norms of the orresponding operators in nite dimensional representations of V . In other words, if p is a -polynomial in elements of V then kT pk (interpreting this notation in the obvious way) must be approximated by kT pk for ompletely ontra tive representations T : V ! Mn . Fix a ompletely ontra tive representation T : V ! B (H) and a polynomial p in the elements of V . Given  > 0, hoose a unit ve tor v 2 H su h that kT p(v )k  kT pk . Let K be the nite dimensional subspa e of H spanned by v and the ve tors T p (v ) as p ranges over all subpolynomials of p; let P be the orthogonal proje tion onto K; and let T = P T j be the indu ed ompletely ontra tive representation of V on K. Then T p(v ) = T p(v ), and hen e kT pk  kT pk . Thus the norm of p in the arbitrary representation T is approximated by its norm in the nite dimensional representation T , as we needed to show. (b) For any X 2 Kn let X^ : Cmat (K ) ! Mn be evaluation at X , i.e., X^ (f ) = fn (X ). Then de ne  : Kn ! hom(Cmat (K ); Mn ) by ^ . It is easy to see that this map is one-to-one and ontinuous. ( X ) = X As hom(Cmat (K ); Mn ) is learly Hausdor,  is a homeomorphism, so we only need to show surje tivity. Let  : Cmat (K ) ! Mn be any -homomorphism. Then the restri tion of  to Vmat (K ) is a ompletely ontra tive linear map. Hen e there exists X 2 Kn su h that  agrees with X^ on Vmat (K ). But Vmat (K ) generates Cmat (K ), so  = X^ . This shows that  is surje tive. 0

0

0

0

K

0

0

0

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0

184

Chapter 8:

Operator Spa es

COROLLARY 8.4.7

Let

V and W be operator spa es and let K and L be the orresponding

dual matrix unit balls. Then any ompletely ontra tive linear map from

V to W extends uniquely to a unital -homomorphism from Cmat (K ) -homomorphism from 1 (K ) to lmat 1 (L). lmat to Cmat (L) and to a weak* ontinuous unital

PROOF

The rst statement follows immediately from the universality of Cmat (K ) proven in part (a) of the theorem. (It an also be shown by the following argument.) Let T : V ! W be a ompletely ontra tive linear map and let T # : L ! K be the adjoint ontinuous ompletely linear map given by ompo1 (K ) ! lmat 1 (L) by T~f (X ) = f (T #X ) sition with T . Then de ne T~ : lmat 1 for f 2 lmat (K ) and X 2 L. It is now straightforward to he k that T~ is a weak* ontinuous unital -homomorphism whi h extends T . Unique1 (K ). ness follows from the fa t that Vmat (K ) generates lmat 8.5

Operator systems

As we have seen, operator spa es are quantum versions of dual unit balls, i.e., ompa t balan ed onvex sets. In this se tion we will dis uss the quantum version of a ompa t onvex set, whi h is alled an operator system. The lassi al analogs of operator systems are fun tion systems, whi h are de ned as follows. DEFINITION 8.5.1

A

real ordered ve tor spa e

is a real ve tor

V equipped with a partial ordering for whi h    implies  +    +  and a  a for all  2 V and a  0. Let V be a real ordered ve tor spa e. An Ar hemedian order unit for V is an element " 2 V su h that (a) for ea h  2 V there is a real number a > 0 su h that a  ", and (b) if   a" for all a > 0 then   0. spa e

A (real) fun tion system is a real ordered ve tor spa e together with a distinguished Ar himedian order unit ", su h that the

k k = inf fa  0 : is omplete.

a"

order norm

   a"g

V is a positive (i.e., order preserving) ! R su h that k!k = 1. The state spa e of V is the set S (V ) of all states on V . A

state

on a fun tion system

linear fun tional !

:V

© 2001 by Chapman & Hall/CRC

185 Observe that the Ar himedian property implies

k k"    k k" for all  2 V . It is also possible to de ne omplex fun tion systems. To do this, we start with a omplex ve tor spa e V equipped with an antilinear involution  7!   . Then we require that the real part of V , that is, the set Re V of self-adjoint elements of V , satisfy the previous de nition. We norm V by setting k k = sup kRe a k; jaj=1

where the norm on the right side is the order norm on Re V . It is easy to see that the omplexi ation of any real fun tion system is a

omplex fun tion system, and there is a perfe t equivalen e between the two de nitions. We will work with real fun tion systems be ause in the present setting the omplex part is super uous; however, when we pass to operator systems there is a very good reason for using omplex s alars (see the omment following De nition 8.5.7). Example 8.5.2

Let S be a ompa t onvex subset of a lo ally onvex TVS. Then the spa e A(S ) of real ontinuous aÆne fun tions on S is a fun tion system. Example 8.5.3

The self-adjoint part of any unital C*-algebra A is a fun tion system, and the states on this fun tion system are pre isely the restri tions of the states on A in the sense of De nition 5.6.3.

As per the omment following De nition 5.6.3, states an equivalently be de ned as positive linear fun tionals ! su h that ! (") = 1. Moreover, any linear fun tional ! : V ! R whi h satis es k! k = ! (") = 1 must be positive and hen e a state, sin e   0 implies

k k



! ( )

= ! (k k"

)

 !(k k") = k k

and therefore ! ( )  0. Thus, of the three onditions that ! be positive, that k! k = 1, and that ! (") = 1, any two imply the third. The next result justi es the term \fun tion system," and together with Example 8.5.2 it also establishes a orresponden e between fun tion systems and ompa t onvex sets.

© 2001 by Chapman & Hall/CRC

Chapter 8: Operator Spa es

186 PROPOSITION 8.5.4

Let V be a fun tion system. Then S = S (V ) is a ompa t onvex subset of V , and the natural map  7! ^ is an isometri order isomorphism of V onto the spa e A(S ) of real ontinuous aÆne fun tions on S . PROOF It is straightforward to he k that S (V ) is ompa t and

onvex. Also, the map  7! ^ is learly ontra tive and preserves order. For any  2 V de ne

j j

= supfa 2 R : a"   g

and

j j+ = inf fa 2 R :   a"g: Then for any j j  a  j j+ , a short omputation shows that the linear fun tional !0 de ned on span(; ") by setting !0 (") = 1 and !0 ( ) = a satis es k!0 k = 1. So !0 extends to a state ! on V by the HahnBana h theorem and the omment whi h pre eded this proposition. In parti ular, taking a to be either k k or k k (one of them must lie between j j and j j+ ), this shows that k^k  k k. So the map  7! ^ is isometri . Also, if  2 V and  6 0, then we an take a < 0 and obtain ^ 6 0, and this shows that  7! ^ is an order isomorphism. We must show that this map is onto. Let f 2 A(S ). Then de ne T : V  ! R by setting T (a!

b)

= af (! )

bf ()

for all a; b  0 and !;  2 S . This map is well-de ned by the following argument. Suppose a! b = a0 ! 0 b0 0 . Applying both sides to " yields a b = a0 b0 , and we also have a0 ! 0 a0

+ b +b

2 S:

Thus af (! ) + b0 f (0 ) = (a + b0 )f

= (a0 + b)f

 a! + b0 0  b0  a0a!+ 0 + b 

a0 + b = a0 f (! 0 ) + bf ();

and we on lude that af (! ) bf () = a0 f (! 0 ) b0 f (0 ). So T is wellde ned. Next, we laim that T is de ned on all of V  . To see this let S0

= ft!

(1

© 2001 by Chapman & Hall/CRC

t) : !; 

2 S and t 2 [0; 1℄g:

187 This set is weak* ompa t be ause it is a ontinuous image of the ompa t set S  S  [0; 1℄, and it is also onvex. So for any ! 2 V  not belonging to S 0 , there exists  2 V su h that ( )  1 for all  2 S 0 but ! ( ) > 1. Using the fa t that the map  7! ^ is isometri , and sin e S  S 0 = S 0 , we obtain that k k  1, so we must have k! k > 1. Thus S 0 ontains the unit ball of V  (in fa t the two are equal), and so T is de ned everywhere on V  . Now T is weak* ontinuous on S , so it is weak* ontinuous on S 0 and therefore on all of V  by the Krein-Smulian theorem. Thus there exists  2 V su h that T (! ) = ! ( ) for all ! 2 V  . In parti ular, for any ! 2 S we have ^(! ) = ! ( ) = T (! ) = f (! ): So

f

= ^, and we have shown that the map



7! ^ takes V onto A(S ).

There is also a orresponden e between aÆne maps of ompa t onvex sets and positive unital maps between fun tion systems. Example 8.5.5

V

W

V !W

Let and be fun tion systems and let T : be a positive unital map. Then omposition with T de nes a weak* ontinuous aÆne map from S ( ) into S ( ).

W

V

Example 8.5.6

Let S and S 0 be ompa t onvex sets in lo ally onvex TVSs and let :S S 0 be a ontinuous aÆne map. Then omposition with  de nes a positive unital map from A(S 0 ) into A(S ).

!

Now we de ne the Hilbert spa e analog of lassi al fun tion systems. DEFINITION 8.5.7

If

E

is a ve tor spa e equipped with an invo-

lution, we say that a subspa e of

E

is

self-adjoint

if it is stable under

the involution. An

operator system is a self-adjoint, unital operator spa e V . In other V  B (H) su h that I 2 V , and  2 V

words, it is a losed subspa e implies



2V

.

positive if

V!W V ) ! Mn(W )

T : T (n) : Mn (

A linear map

between operator systems is is positive for all

n

2N

ompletely

.

Unlike the ase of lassi al fun tion systems, here there is no equivalent real version of the de nition be ause positive elements of Mn (V ) may ontain non self-adjoint entries for n > 1, and so the notion of om-

© 2001 by Chapman & Hall/CRC

188

Chapter 8: Operator Spa es

plete positivity involves non self-adjoint elements of V in an essential way. The same example whi h distinguishes boundedness from omplete boundedness also distinguishes positivity from omplete positivity: the transpose map on M2 is positive but not ompletely positive. We on lude this se tion with an abstra t hara terization of operator systems. If the ve tor spa e V is equipped with an antilinear involution  (i.e., a map  : V ! V su h that (a ) = a   and   =  for all  2 V ), then we de ne a orresponding involution on M (V ) by taking the transpose of a matrix and applying  to ea h entry. n

DEFINITION 8.5.8 A matrix ordered spa e is a omplex ve tor spa e V equipped with an antilinear involution  7!   together with a ve tor spa e ordering on the self-adjoint part of ea h Mn (V ), su h that if  2 Mn (V ) and A 2 Mn;m then   0 implies A A  0. It is unital if there is a distinguished Ar himedian order unit " 2 V su h that 3 2 "


0 . . . .. 5 En = 4 .. . .

0


" is an Ar himedian order unit for M (V ), for all n. The matrix order norm on a unital matrix ordered spa e V is de ned n

by

kk = inf





a

for  2 Mn (V ).

 0 : aE 

n





aEn

0



Complete positivity for maps between matrix ordered spa es is de ned just as for maps between operator systems. The above de nition of the matrix order norm is based on the fa t that it gives the orre t norm in B (H) (and hen e in any operator system), as a short omputation shows. Thus: Example 8.5.9 Every operator system is a unital matrix ordered spa e.

For self-adjoint elements of V , the matrix order norm of De nition 8.5.8 also agrees with the order norm on M (V ) given in De nition  E 8.5.1. For if  E  0 then n

n

n

2(E

n

) = [ I

© 2001 by Chapman & Hall/CRC

n

In ℄



En





En



In In



0

189 and

2(E + ) = [ I n

and hen e 

2 E

n

En



n



n

In In

En





 E ; while onversely, if n

E

 = E

In ℄





In In

n

En + 

0

En

0

In In

En



 0;    E then





n

In In

In In



 0:

Also, observe that if V is a matrix ordered spa e and ! : V ! C is positive, then ! is ompletely positive. To see this let  2 M (V ) and suppose   0. Then for any v 2 C  = M 1 we have n

n

n;

h! ()v; vi = !(v v)  0; ()  0. (n)

and therefore !(

n)

THEOREM 8.5.10 Let

V

be a unital matrix ordered spa e whi h is omplete for the matrix

order norm.

Then

V

is ompletely order isomorphi to an operator

system via a ompletely isometri unital map.

PROOF

Let T be the dire t sum of all ompletely positive unital maps from V into M for n 2 N. It is lear that T is unital and that it preserves order. We must show that it is a omplete order isomorphism; the fa t that it is ompletely isometri will then follow be ause order and the unit determine the norm. Thus, let  2 M (V ) and suppose  6 0. We must nd a ompletely positive unital map T0 : V ! M su h that T0( )() 6 0. M (V ) is a omplex fun tion system, so Proposition 8.5.4 (together with the rst omment pre eding this theorem) implies the existen e of a state : M (V ) ! C su h that () 6 0. By the se ond omment pre eding this theorem, is ompletely positive. Say = [! ℄ with ~ : V ! M by ~ ( ) = [! ( )℄. We laim that ~ is ! 2 V  and de ne

also ompletely positive. To see this, let H = [ ℄ 2 M (V ) be positive and de ne H~ = [~ ℄ 2 M 2 (V ) by n

n

m

n

n

n

ij

ij

n

ij

ij

ikp;jlq

~ikp;jlq

m

mn



= 0

i;j

if k = p and l = q otherwise;

then H~ is positive be ause we have H~ = M is the matrix

A HA

where

m;mn

ai;jlq

© 2001 by Chapman & Hall/CRC

n = 1 if i = j and l = q 0 otherwise.

A

= [a

i;jlq

℄2

190

Chapter 8: Operator Spa es

So ~ ( )(H ) = ( ) (H~ )  0 in M , and this shows that ~ is ompletely positive. Moreover, in the ase m = n we have m

mn

mn

2

~ ( ) () = ( ) (A A) = ()A A 6 0: n

n

Thus ~ is a ompletely positive map from V to M and ~ ( ) () 6 0. Finally, ~ (") is a positive matrix in M , so there is a matrix B 2 M su h that B  ~ (")B is the identity in M , where m is the rank of ~ ("). Then T0 = B  ~ B is the desired ompletely positive unital map. n

n

n

n;m

m

8.6

Notes

See [22℄ for more on operator spa es generally. Ruan's theorem (Theorem 8.2.11) was proven in [62℄. The separation theorem presented in Theorem 8.2.10 is taken from [23℄. There is also a Hahn-Bana h theorem for ompletely bounded maps of operator spa es into B (H), known as the Arveson-Wittsto k extension theorem; see [77℄. For more on the dual of an operator spa e see [9℄. Proposition 8.3.4 is from [44℄; that paper also ontains an example of an operator spa e whi h is a dual Bana h spa e but not a dual operator spa e. Se tion 8.4 is based on [70℄. Our treatment of fun tion systems follows [21℄. The abstra t hara terization of operator systems is from [11℄.

© 2001 by Chapman & Hall/CRC

Chapter 9

Hilbert Modules 9.1

Continuous Hilbert bundles

Measurable Hilbert bundles were introdu ed in Chapter 2 (De nition 2.4.8), and they played an important role in Chapters 3, 5, and 6. We begin this hapter by des ribing a ontinuous version of the onstru tion; next we revisit the measurable version; and then we introdu e Hilbert modules over C*- and von Neumann algebras, whi h are the quantum analogs of Hilbert bundles. DEFINITION 9.1.1 Let X be a ompa t Hausdor spa e. A overing spa e of X is a topologi al spa e Y together with a ontinuous open surje tion p : Y ! X . A ontinuous Hilbert bundle over X is then a overing spa e X su h that Hx = p 1 (x) is equipped with a Hilbert spa e stru ture for ea h x 2 X , and satisfying the following onditions:

(a) the map y 7! ky k is ontinuous from X to R; (b) the map (y; z ) 7! y + z is ontinuous from X  X to X ; ( ) for ea h a 2 C, the map y 7! ay is ontinuous from X to X ; and (d) for any neighborhood O of the origin of Hx in X there exists a neighborhood O0 of x in X and an  > 0 su h that

fy 2 X : p(y) 2 O0 and kyk < g  O: A se tion of X is a fun tion  : X ! X su h that (x) 2 Hx for all x 2 X . The set of all ontinuous se tions of X is denoted S (X ). The prototypi al examples of ontinuous Hilbert bundles are geometri . Let X be a smooth manifold; then at ea h point x of X we have a real ve tor spa e Tx whi h onsists of all tangent ve tors at x. This is the tangent spa e at x, and the S disjoint union of all of the tangent spa es is the tangent bundle T X = x Tx . A smooth manifold is Riemannian 191 © 2001 by Chapman & Hall/CRC

Chapter 9: Hilbert Modules

192

if ea h tangent spa e is equipped with an inner produ t and these vary in an appropriately smooth manner as x varies. Now ea h tangent spa e is a real Hilbert spa e, so omplexifying T X gives rise to a ontinuous Hilbert bundle over X . Example 9.1.2

Let X be a ompa t Riemannian manifold and let X = T X + iT X be the omplexi ation of the tangent bundle of X . Then X is a

ontinuous Hilbert bundle.

The spa e of ontinuous se tions of a ontinuous Hilbert bundle has a spe ial stru ture. It is a ve tor spa e, as se tions an be multiplied by s alars and added pointwise. But in fa t a ontinuous se tion  2 S (X )

an be multiplied pointwise by any ontinuous fun tion f 2 C (X ), and the produ t is again a ontinuous se tion. Thus S (X ) is a module over C (X ). S (X ) has even more stru ture. For any ontinuous se tions  and of X , we an take their inner produ t pointwise, and the result will be a ontinuous fun tion on X . That is, we de ne h

;

i

(x) = h(x); (x)i ;

then h; i 2 C (X ) for any ; 2 S (X ). Continuity follows from the fa t that the map (y; z ) 7! hy; z i is ontinuous on X  X , whi h follows from property (a) of ontinuous Hilbert bundles by polarization. This motivates the following de nition.

Let X be a ompa t Hausdor spa e and let be a C (X )-module. A C (X )-valued pseudo inner produ t on E is a map h
;
i : E  E ! C (X ) satisfying (a) hf1 + g2 ; i = f h1 ; i + g h2 ; i; (b) h; i = h ; i; and ( ) h; i  0

DEFINITION 9.1.3 E

for all f; g 2 C (X ) and 1 ; 2 ; ; 2 E . We write jj = h; i1=2 and de ne kk to be the supremum norm of jj in C (X ). If kk = 0 implies  = 0 then h
;
i is a C (X )-valued inner produ t, and a Hilbert module over C (X ) is a C (X )-module equipped with a C (X )-valued inner produ t whose orresponding norm k
k is omplete. Observe that if h
;
i is a C (X )-valued pseudo inner produ t then the map (; ) 7! h; i(x) is a pseudo inner produ t for ea h x 2 X . It follows that kk = sup h; i1=2 (x) is a pseudonorm. Thus k
k really is a norm if h
;
i is a C (X )-valued inner produ t.

© 2001 by Chapman & Hall/CRC

193 Example 9.1.4

Let X be a ontinuous Hilbert bundle over a ompa t Hausdor spa e

X . Then S (X ) is a Hilbert module over C (X ).

This example has a onverse: every Hilbert module is the module of

ontinuous se tions of a ontinuous Hilbert bundle. The onstru tion goes as follows. Given a Hilbert module E over C (X ) and an element x 2 X , let Ix = ff 2 C (X ) : f (x) = 0g and de ne Hx = E =Ix E . Here Ix E is the losed span of the set of elements f  with f 2 Ix and  2 E . The C (X )-valued inner produ t on E then passes to an ordinary inner produ t on Hx , and Hx is the quotient of a Bana h spa e by a losed subspa e, so it is omplete. Thus ea h Hx is a Hilbert spa e. Also, for ea h  2 E let x =  + Ix be its proje tion in Hx . Now we need the following proposition. PROPOSITION 9.1.5

Let X be a ompa t spa e, let E be a Hilbert module over S HHausdor C (X ), and let X = x be the disjoint union of the Hilbert spa es Hx de ned above. Then there is a unique topology on X whi h makes it a

ontinuous Hilbert bundle over X and su h that the map x 7! x is a

ontinuous se tion for all  2 E . The desired topology is hara terized by the ondition that j(x) ! 0. A basis for this topology is given by the sets

PROOF

 x 

! x if and only if x ! x and j fx 2 X :  2 E ; x 2 O;

and

kx

x

k  g;

where O is an open subset of X , 2 E , and  > 0. It is routine but tedious to verify that this is a basis for a topology and that it makes X into a ontinuous Hilbert bundle with the stated property. We write B (E ) for the ontinuous Hilbert bundle provided by Proposition 9.1.5. The following result shows that there is a perfe t orresponden e between ontinuous Hilbert bundles and Hilbert modules. The proof is rather long and we therefore omit it. THEOREM 9.1.6

Let X be a ompa t Hausdor spa e. (a) S (B (E )) is anoni ally isomorphi to E , for any Hilbert module over C (X ).

© 2001 by Chapman & Hall/CRC

E

194

Chapter 9: Hilbert Modules

(b) B (S (X )) is anoni ally isomorphi to X , for any ontinuous Hilbert bundle X over X . 9.2

L

Hilbert

1

-modules

It is typi ally the ase that the natural L (or von Neumann algebra) version of a C (X ) (or C*-algebra) onstru tion involves dual spa es and weak* topologies in pla e of Bana h spa es and norm topologies. In the

ase of Hilbert modules, the appropriate requirement is self-duality in the following sense. 1

Let X be a  - nite measure spa e and let E be a Hilbert module over L1 (X ). Then the dual module E 0 is the set of bounded L1 (X )-linear maps from E into L1 (X ). The module stru ture of E 0 is de ned by (f
)() = f(); 1 0 for f 2 L (X ),  2 E , and  2 E . For any  2 E , let ^ 2 E 0 be the map ^( ) = h ; i. If every element of E 0 is of the form ^ for some  2 E , then we say that E is self-dual.

DEFINITION 9.2.1

It is fairly easy to show that any self-dual Hilbert module over L (X ) is a dual Bana h spa e. We will prove this in greater generality in Proposition 9.4.2. Anyway under suitable separability assumptions we

an ompletely hara terize the stru ture of self-dual Hilbert modules over L (X ), and duality follows easily from this hara terization. The measurable Hilbert bundles used in earlier hapters give rise to examples of self-dual Hilbert L -modules. First we present the onstru tion, and then we prove that it satis es the ondition of self-duality. 1

1

1

S

DEFINITION 9.2.2 X

= (Xn

Let X be a  - nite measure spa e and let

n ) be a (separable) measurable Hilbert bundle over

 H

X , as in De nition 2.4.8. The Hilbert module of L1 se tions of X is the set S L1 (X ; X ) of weakly measurable essentially bounded fun tions  : X ! Hn with the property that (x) 2 Hn for all x 2 Xn .

Re all that \weakly measurable" means that for ea h the fun tion x (x); v is measurable on Xn .

v 2 Hn

7! h

PROPOSITION 9.2.3

n

2

N and

i

S

Let X be a  - nite measure spa e and let X = (Xn  Hn ) be a measurable Hilbert bundle over X . Then L1 (X ; X ) is a self-dual Hilbert module over L1 (X ).

© 2001 by Chapman & Hall/CRC

195 PROOF It is straightforward to he k that L (X ; X ) is a Hilbert module over L (X ). To verify self-duality, let  2 L (X ; X ) . Assume X = X  H where H is separable and in nite dimensional; the nite dimensional ase is similar, and passing to a disjoint union of su h bundles is easy. Let (en ) be an orthonormal basis of H. De ne fn = (1X en ). Then fn 2 L (X ), and for any g1 ; : : : ; gn 2 L (X ) we have 1

1

1

1

0

1

 P

n X i=1

!

gi ei

=

n X i=1

gi fi

=

*

n X i=1

+

gi ei ; n

where n = n1 fi ei . In parti ular, we have (n ) = jn j2 , so boundedness of  implies that the sequen e (n ) is bounded in norm. But jn j is in reasing, so it follows that jn j(x) is Cau hy for almost every x 2 X . Thus  = lim n exists almost everywhere, is weakly measurable, and is essentially bounded, and we have ( ) = h ; i for any whi h an be written as a nite L (X )-linear ombination of the se tions 1X en . It is not yet lear that this equality holds for all 2 L (X ; X ), be ause nite linear ombinations of the se tions 1X en are in general not dense. (For example, they annot approximate in norm a se tion whi h for all n onstantly takes the value en on some positive measure set.) However, for any 2 L (X ; X ) and any  > 0 we an nd a measurable partition (Xm ) of X and a sequen e of se tions m whi h are nite L (X )-linear

ombinations P of the se tions 1X en , su h that m is supported on Xm and k m k  . Then 1

1

1

1

X 

DX

E

i =  Xm
i;  P P for all m, and hen e ( i ) = h i ; i. It now follows by ontinuity that ( ) = h ; i for all 2 L (X ; X ), and we on lude that L (X ; X ) is self-dual. Xm




i

= ( m ) = h

m;  1

1

Now we want to show that under reasonable separability assumptions all self-dual Hilbert L -modules arise in the above fashion from measurable Hilbert bundles. The appropriate ondition is that the module should be separable for the weak topology indu ed by the inner produ t, that is, the weakest topology on E su h that for all 2 E the map  7! h; i is ontinuous as a map into L (X ) with the weak* topology. Thus, if ( ) is a bounded net in E then  !  weakly if and only if h ; i ! h; i weak* in L (X ) for all 2 E . We will show in Proposition 9.4.2 that on bounded sets this is a tually a weak* topology. 1

1

1

THEOREM 9.2.4

Let X be a  - nite measure spa e and let E be a self-dual Hilbert module

© 2001 by Chapman & Hall/CRC

196 over

Chapter 9:

L1 (X )

E = L1 (X ; X )

whi h is weakly separable.

Hilbert bundle

X

over

X

su h that

Hilbert Modules

Then there is a measurable .

PROOF

Any  - nite measure an be repla ed by a probability measure with whi h it is mutually absolutely ontinuous, so without loss of generality suppose X is a probability measure spa e and let  be the measure on X . De ne an inner produ t [
;
℄ on E by setting R [; ℄ = h; i d and let H be the Hilbert spa e ompletion of E for this inner produ t. If n !  weakly then [n ; ℄ ! [; ℄ for all 2 E , whi h implies that H is weakly separable and hen e separable. Let k
k2 denote the norm on H. For ea h measurable subset S of X , let E (S ) be the losure in H of S
E . It is straightforward to verify that E is an H-valued spe tral measure on X . By Corollary 3.4.3 there is a probability measure 0 on X , a measurable Hilbert bundle X over X , and an isometri isomorphism U : L2 (X ; X )  = H su h that U (L2 (S ; XjS )) = E (S ) for every measurable set S  X . If S is null with respe t to  then E (S ) = 0, so 0 is absolutely ontinuous with respe t to . Possibly letting X be zero on a positive measure set, we an therefore assume that the two measures are mutually absolutely ontinuous. Having done this, it is not hard to see that by modifying U we an a tually take 0 = . Next we show that U 1 takes E isometri ally into L1 (X ; X ). First, the Cau hy-S hwarz inequality applied pointwise shows that

jh; ij(x)  jj(x)j j(x)  jj(x)k k almost everywhere, and hen e kh; ik2  k kkk2 for any ; 2 E . It follows that for all 2 E the map  7! h; i from H to L2 (X ) has norm at most k k. Conversely, for any  > 0 let S be a positive measure set on whi h j j  k k ; then taking  = S yields kk

2  2

while

Z

=

S

h

;

Z

kh; ik22 = jh S

;

i  k k2(S ); ij2  (k k

4 ) (S );

and we on lude that the map  7! h; i from H to L2 (X ) has norm exa tly k k. The same argument shows that the L1 norm of U 1 equals the norm of the map U 1  7! hU 1 ; U 1 i. So the fa t that U is isometri for L2 norms implies that U 1 takes E isometri ally into L1 (X ; X ).

© 2001 by Chapman & Hall/CRC

197 It is lear that U 1 : E ! L1 (X ; X ) respe ts L1 (X )-module stru ture. For any  2 E let f = h; i and g = hU 1 ; U 1 i. By L1 (X )linearity and the fa t that U 1 is an isometry for L1 norms, we have kf jS k1 = kgjS k1 for any S  X . Sin e f and g are both positive, this implies f = g almost everywhere; polarization then implies that h; i = hU 1 ; U 1 i for all ; 2 E . Thus U 1 is ompatible with the L1 (X )-valued inner produ t. Finally, to show U 1 maps E onto L1 (X ; X ), let 2 L1 (X ; X ). Then the map  7! hU 1 ; i is a bounded L1 (X )-linear map from E into L1 (X ), so by self-duality there exists 0 2 E su h that h; U i = 0 is orthogonal (in H) to hU 1 ; i = h; 0 i for all  2 E . Then U 0 = 0. every element of E , and sin e E is dense in H it follows that U 1 0 1 1 Thus = U , and we have shown that U maps E onto L (X ; X ). Conversely, every Hilbert module onstru ted as in De nition 9.2.2 is weakly separable. So Theorem 9.2.4 exa tly hara terizes weakly separable self-dual Hilbert L1 -modules.

9.3

Hilbert C*-modules

There is a C*-algebrai version of the Hilbert C (X )-modules introdu ed in Se tion 9.1; in fa t, due to non ommutativity there are two versions, left Hilbert modules and right Hilbert modules. There is no essential dieren e between the two ases, but in various ir umstan es one or the other may be more onvenient. The basi on i t lies in the fa t that Hilbert spa e inner produ ts are antilinear in the se ond variable, but if operators a t from the left then it is natural to have s alars a t from the right, whi h a

ords better with linearity in the se ond variable. We use the symbols  and to denote elements of an abstra t C*algebra. DEFINITION 9.3.1 Let A be a C*-algebra and let E be a left Amodule. An A-valued pseudo inner produ t on E is a map h
;
i : EE ! A satisfying

(a) h1 + 2 ;  i = h1 ;  i + h2 ;  i; (b) h;  i = h;  i; and ( ) h;  i  0

for all ; 2 A and 1 ; 2 ; ;  2 E . We write j j = h;  i1=2 and de ne k k to be the norm of j j in A. If k k = 0 implies  = 0 then h
;
i is an A-valued inner produ t and a left Hilbert module over A is a left A-module equipped with an A-valued

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198

Chapter 9: Hilbert Modules

inner produ t whose orresponding norm k
k is omplete. Right Hilbert modules are de ned similarly, with the axiom (a') h; 1  + 2 i = h; 1 i + h; 2 i in pla e of (a).

This de nition is ontingent on our verifying that k
k really is a pseudonorm, whi h we will do momentarily. It is easy to he k that if A+ is the unitization of A (as in De nition 5.3.1) and E is a Hilbert module over A then E is also a Hilbert module over A+ via the obvious a tion of A+ on E . Left modules an be onverted into right modules in the following way. Let E be a left A-module. Then de ne E op to be the set E , denoting by  the element of E op orresponding to  2 E , with addition as de ned  =   . It is easy to in E but with module multipli ation given by  op see that E is a right A-module, and an A-valued pseudo inner produ t  i = h;  i . Similarly, on E is onverted to one on E op by setting h; right modules an be onverted into left modules, so there is no essential dieren e between left and right Hilbert modules. Thus we will feel free to use whi hever version is more onvenient at any given moment. Our immediate burden is to prove that k
k is a pseudonorm on E ; this amounts to proving the triangle inequality, whi h is a onsequen e of the following version of the Cau hy-S hwarz inequality. PROPOSITION 9.3.2

Let A be a C*-algebra and let E be a left A-module equipped with an -valued pseudo inner produ t h
;
i. Then

A

; ih;  i  j j2 kk2

h

for all ;  2 E .

PROOF Let ! be a state on A. Then the map (; ) 7! !(h; i) is a pseudo inner produ t on E and it therefore satis es the ordinary Cau hy-S hwarz inequality. Thus, letting  = h;  i, we have !( ) = !(h;  i) 1=2 !(h;  i)1=2  ! (h;  i) 2  1=2 = ! (j j  ) ! (j j2 )1=2  1=2  k k! ( ) !(j j2 )1=2 ; whi h implies ! ( )  k k2 ! (j j2 ). Sin e this is true for any state, it follows that h;  ih;  i =   j j2 k k2 .

© 2001 by Chapman & Hall/CRC

199 In parti ular, we have kh;  ik  k kk k, and this implies the triangle inequality for k
k in the usual way. So k k = kh;  ik1=2 is indeed a pseudonorm. It also follows, just as in the s alar ase, that fa toring out null ve tors onverts an A-valued pseudo inner produ t into an Avalued inner produ t, and ompleting an A-valued inner produ t yields a Hilbert module. So there is no obstru tion to making a Hilbert module out of any A-module equipped with an A-valued pseudo inner produ t. Example 9.3.3

A

A

P

(a) Let be a C*-algebra and X a set. De ne l2 (X ; ) to be the set of fun tions f : X su h that x f (x)f (x) onverges in . 2 Then l (X ; ) is a left Hilbert -module with the obvious left a tion of and the inner produ t

A

A

!A

A

A

hf; gi = X f (x)g(x): x2X

The proof that this expression is well-de ned is similar to the s alar

ase (Example 2.1.7), as is the proof that l2 (X ; ) is omplete. (b) Now let X be a  - nite measure spa e. The simplest way to de ne L2 (X ; ) is to take a ompletion of the set of all simple fun tions from X to whose support has nite measure. This set has an -valued pseudo inner produ t de ned by linear extension of the equation

A

A A

A

hS ; T i = (S \ T ) ;

for ;

2 A and S and T nite measure subsets of X .

Next we des ribe the Hilbert module analog of B (H). Let E be a Hilbert module over a C*-algebra . A bounded A-linear map A : E ! E is adjointable if there exists a bounded A-linear map A : E ! E su h that hA;  i = h; A  i for all ;  2 E . We denote the set of all bounded adjointable A-linear maps from E to itself by B (E ). DEFINITION 9.3.4

A

Adjointability is not automati , even in the ommutative ase. For example, let E = C [0; 1℄  C0 (0; 1℄, where we identify C0 (0; 1℄ with the

ontinuous fun tions on C [0; 1℄ whi h vanish at 0. The a tion h(f  g ) = hf  hg and the inner produ t hf1  g1 ; f2  g2 i = f1 f2 + g1 g2 make E into a left Hilbert C [0; 1℄-module. Now onsider the map A : E ! E de ned by A(f  g ) = g  0. This is learly a bounded A-linear map, but hA(f  g ); 1  0i = g;

© 2001 by Chapman & Hall/CRC

200 so that if

Chapter 9: Hilbert Modules A

were adjointable we would have

hf  g; A (1  0)i = g for all f and g . But there is no possible value of A (1  0) for whi h this

an hold. Adjointability is automati in the von Neumann algebra setting, however: see the omment following De nition 9.4.1. LEMMA 9.3.5

E1 and E2 be left Hilbert modules over a C*-algebra A and let ! E2 be a bounded A-linear map. Then jA j2  kAk2 j j2 for all 2 E1 .

Let A 

: E1

Fix  > 0 and de ne f 2 C0 (R) by f (t) = 0 for t  , = 1 for t  2, and f (t) = (t )= for   t  2. Let  = f (j j) . Then, working momentarily in the unitization of A (whi h we an do by the omment following De nition 9.3.1), we have

PROOF

f (t)

j

j = (IA f (j j))j j ! 0 as  ! 0. It follows that  !  , and hen e A ! A , as  ! 0, so it will suÆ e to show that jA j2  kAk2 j j2 . (This is be ause jA j2 = jA + A( )j2 = jA j2 + 2RehA; A(  )i + jA(  )j2 ; and the last two terms go to zero in norm as  !  .) Now x g 2 C0 (R) su h that g (0) = 0 and g (t) = 1=t for t  , and de ne  = f g (j j) . We have j j = f (j j)  IA , so k k  1; therefore kA k  kAk, whi h implies 2 2 2 f (j j)jA j f (j j) = j jjA j j j  kAkj j : Hen e jA j2  kAk2 j j2 , as desired. PROPOSITION 9.3.6



Let E be a left Hilbert module over a C*-algebra abstra t C*-algebra.

A.

Then B (E ) is an

PROOF Without loss of generality assume A is unital. For ea h state ! on A let H! be the Hilbert spa e formed from the pseudo inner produ t ! (h
;
i) on E by fa toring out null ve tors and ompleting. Let

© 2001 by Chapman & Hall/CRC

201

H = L H! be the dire t sum of all of these Hilbert spa es. Then de ne a map  : B (E ) ! B (H) by setting  (A)(! ) = (A )! for A 2 B (E ), where ! is the element of H! orresponding to  2 E . Applying an arbitrary state ! to both sides of the inequality jA j  kAk j j proven in the lemma, we nd that  is well-de ned, (A) 2 B (H), and k(A)k  kAk. Conversely, for any  2 E and  > 0 we an nd a state ! on A su h that ! (jA j )  kA k ; then 2

2

2

2

k(A)! k

2

2

= k(A )! k2 = ! (jA j2 )  kA k2

:

But k! k2 = ! (j j2 )  k k2 , so this shows that k (A)k  kA k=k k. As this is true for all  , we on lude that  is an isometry. It is lear that B (E ) is omplete in norm and  is a -homomorphism. So B (E ) is -isomorphi to a C*-subalgebra of B (H). As a rst illustration of the way Proposition 9.3.6 an be used, we now present a version of the GNS onstru tion for ompletely positive maps between C*-algebras. Let A and B be C*-algebras and let T : A ! B be ompletely positive (De nition 8.5.7). Regard the algebrai tensor produ t A B as a right B -module via the a tion ( ) =  . We de ne a B -valued pseudo inner produ t on A B by setting

h ;  i =  T (  ) : 1

1

2

2

1

1

2

2

Complete positivity of T omes P into the veri ation of positivity of the inner produ t. Namely, if n1 i i is an arbitrary element of A B then we have

*X i

i i ;

X i

+ X

i i =

= B  T (n) (A A)B

where

A = [ 1




n ℄

i;j

i T (i j ) j

2 3 6 B = 4 ... 7 5: 1

and

n

The matrix A A 2 Mn (A) is positive, so T (n) (A A) 2 Mn (B ) is also positive, and hen e B  T (n) (A A)B is positive in B , as desired. Let ET be the Hilbert B -module formed from A B by fa toring out null ve tors and ompleting. The following proposition, whi h generalizes Theorem 5.6.6 ( f. the omment pre eding Theorem 8.5.10), is now a short al ulation.

© 2001 by Chapman & Hall/CRC

202

Chapter 9: Hilbert Modules

PROPOSITION 9.3.7

Let A and B be C*-algebras and let T : A ! B be ompletely positive. Then there is a unique -homomorphism  : A ! B (ET ) su h that ()( ) =  for all ; 2 A and 2 B. If A and B are unital and T (IA ) = IB , then kIA IB k = 1 in ET and

T () = h()(IA IB ); IA IB i for all  2 A.

9.4

Hilbert W*-modules

In the von Neumann algebra setting it is natural to onsider self-dual modules, as we did in Se tion 9.2. The following is the general de nition. DEFINITION 9.4.1 Let E be a left Hilbert module over a C*algebra A. Then the dual module E 0 is the set of bounded A-linear maps from E into A. The module stru ture of E 0 is de ned by

(
)( ) = ( ) for  2 A,  2 E 0 , and  2 E . For any  2 E , let ^ 2 E 0 be the map ^( ) = h;  i. If every element of E 0 is of the form ^ for some  2 E , then we say that E is self-dual. For right Hilbert modules, we de ne (
)( ) =  (x) and ^( ) = h;  i.

Self-duality has many ni e onsequen es. For example, in ontrast to the general ase (see the omment following De nition 9.3.4), here any bounded module map A : E1 ! E2 has a bounded adjoint A . This is easily seen, by essentially the same proof as in the Hilbert spa e ase. Before giving examples of self-dual Hilbert modules we need to develop some ma hinery. First we prove that every self-dual Hilbert module over a von Neumann algebra is a dual Bana h spa e. In proving this result we need the fa t that weak* ontinuous linear fun tionals are abundant on any von Neumann algebra M. For this it is suÆ ient to note that for any v; w 2 H the map A 7! hAv; wi is weak* ontinuous on B (H), and hen e on any von Neumann algebra M  B (H ). PROPOSITION 9.4.2

Let E be a self-dual left Hilbert module over a von Neumann algebra . Then E an be identi ed with the dual of a Bana h spa e in su h a way that a bounded net ( ) in E onverges weak* if and only if the net h ;  i onverges weak* in M for every  2 E . M

© 2001 by Chapman & Hall/CRC

203 De ne a topology on the unit ball of E by saying that !  if and only if h ; i ! h; i weak* in M for all  2 E . In other words, this is the weakest topology whi h makes the maps  7! h;  i

ontinuous with respe t to the weak* topology on M. Let V  E  be the set of bounded linear fun tionals whose restri tion to the unit ball of E is ontinuous for the above topology. It is routine to verify that V is a losed subspa e of E  , so V is a Bana h spa e. Let T : E ! V  be the natural map, i.e., T  (! ) = ! ( ). For every  2 E and every weak* ontinuous linear fun tional  on M, the map  7! (h; i) belongs to V . For any  2 E and any  > 0 we an hoose  so that kk = 1 and j(h;  i)j  k k2 . This implies that kT  k  k k, and as T is learly nonexpansive we on lude that it

PROOF



is an isometry. To show that T is surje tive, let ! belong to the unit ball of V  . We

an extend ! to a linear fun tional !~ on E  with norm at most 1, and then we an nd a net ( ) in the unit ball of E whi h onverges weak* to !~ in E  . Passing to a subnet, we may suppose that h;  i onverges weak* in M for all  2 E ; then the map  7! lim h;  i belongs to E 0 , so by self-duality there exists  2 E su h that h;  i = lim h;  i for all  2 E . From the original de nition of V it is lear that ! ( ) ! ! ( ) for all ! 2 V , so therefore T  ! T  weak* in V  . But T  ! ! weak* in V  as well, so we must have ! = T  . Thus T is a surje tive isometri isomorphism from E onto V  . If (h ;  i) ! (h;  i) for every weak* ontinuous linear fun tional  on M then h ;  i ! h;  i weak*. This implies that the restri tion of T 1 to the unit ball of V  is ontinuous for the topology on E des ribed in the statement of the theorem. But the unit ball of V  is ompa t and the unit ball of E is Hausdor, so T 1 : (V  )1 ! (E )1 must be a homeomorphism. This shows that the weak* topology on the unit ball of E is as stated. As a onsequen e of the pre eding theorem, we get the von Neumann algebra version of Proposition 9.3.6. COROLLARY 9.4.3

Let E be a self-dual left Hilbert module over a von Neumann algebra M. Then B (E ) is an abstra t von Neumann algebra.

PROOF

The proof is similar to the proof of Proposition 9.3.6. For ea h weak* ontinuous state ! on M, form the orresponding Hilbert spa e H! ; let H be their dire t sum; and let  : B (E ) ! B (H) be the natural map. It is lear that  is a -homomorphism. Existen e of suÆ-

© 2001 by Chapman & Hall/CRC

204

Chapter 9:

Hilbert Modules

iently many weak* ontinuous states to ensure that  is an isomorphism follows from the observation made just prior to Proposition 9.4.2. We must show that (B (E )) is weak* losed in B (H). Thus, let (A ) be a bounded net in (B (E )), and suppose A ! A weak*. We must show A 2 (B (E )). Let B =  1 (A ); then (B ) is a bounded net in B (E ), and by passing to a subnet we may suppose that B  onverges weak* in E for every  2 E . Then B = lim B  de nes an operator B 2 B (E ) whi h satis es (B ) = A. So (B (E )) is weak* losed in B (H), and hen e B (E )  = (B (E )) is a von Neumann algebra. Every Hilbert module E over a von Neumann algebra M an be ompleted to a self-dual module. The simplest way to do this is by embedding E in its dual module E 0 , giving E 0 a Hilbert module stru ture, and then showing that E 0 is self-dual. The natural embedding is the map  7! ^. We will de ne an inner produ t on E 0 by ontinuous extension with respe t to the following topology. We say that a net ( ) in E 0

onverges weakly to  if  ( ) ! ( ) weak* in M, for every  2 E . It will turn out that E 0 is a dual spa e (via Proposition 9.4.2), and by the following lemma this topology therefore agrees with the weak* topology on bounded sets. The main step in the onstru tion of an M-valued inner produ t on E 0 is showing density of E in E 0 . We address this rst. LEMMA 9.4.4 Let

E

be a left Hilbert module over a von Neumann algebra

the map



7! ^ E0

unit ball of

takes the unit ball of

E

M

. Then

into a weakly dense subset of the

.

Let  2 E 0 and suppose kk  1. Fixing 1 ; : : : ; n 2 E , it will suÆ e to nd a bounded sequen e (k )  E su h that ^k (i ) ! (i ) weak* in M for 1  i  n. Fix k, let  = 1=k, and for  2 E write j j = (h;  i + IM )1=2 . Observe that j j is invertible and j j
j j 1  IM . Let 10 = j1 j 1 1 and indu tively de ne i0 by an approximate Gramm-S hmidt pro ess, setting i0+1 = ji+1 j 1 i+1 where

PROOF

i+1

= i+1 hi+1 ; 10 i10


hi+1 ; i0 ii0 :

Finally we de ne k

= (10 )10 +


+ (n0 )n0 :

One an prove by indu tion on i that the M-linear span E of f10 ; : : : ; n0 g

© 2001 by Chapman & Hall/CRC

205 is independent of ; that 1 ; : : : ; n 2 E ; that hi ; j i weak* onverges as k ! 1 to a proje tion in M if i = j and to 0 if i 6= j ; and that h; k i ! ( ) weak* in M as k ! 1, for all  2 E . Thus ( ) has the desired property. 0

0

LEMMA 9.4.5

Let E1 and E2 be left Hilbert modules over a von Neumann algebra M and let A : E1 ! E2 be a bounded M-linear map. Let ( ) be a bounded net in E1 . Then h ;  i ! h;  i weak* in M for all  2 E1 implies hA ;  i ! hA;  i weak* in M for all  2 E2 .

We may suppose h ; i ! 0 weak* in M for every  2 E1 . Fix a weak* ontinuous state ! on M and let H1 and H2 be the

orresponding Hilbert spa es onstru ted as in Proposition 9.3.6 for E1 and E2 respe tively. Then !(h ; i) ! 0 for all  2 E , whi h implies that  ! 0 weakly in H1 . Applying ! to the result of Lemma 9.3.5 shows that A is bounded as a map from H1 to H2 , and bounded maps between Hilbert spa es preserve weak ontinuity, so A ! 0 weakly in H2 . Thus !(hA ;  i) ! 0 for all  2 E2 . As this holds for any weak*

ontinuous state !, we on lude that hA ;  i ! 0 weak* in M for all  2 E2 .

PROOF

THEOREM 9.4.6

Let E be a Hilbert module over a von Neumann algebra M. Then the dual module E 0 arries a natural M-valued inner produ t whi h makes it a self-dual Hilbert module, and E embeds as a weak* dense submodule of E 0 .

Given ; H 2 E , let ( ) be a bounded net in E su h that ^ !  weakly. Su h a sequen e exists by Lemma 9.4.4. Then de ne an M-valued inner produ t on E by setting h; H i = lim H ( ). Applying Lemma 9.4.5 to the map H : E ! M shows that this limit exists and is independent of the hoi e of ( ). We verify self-duality. Let T : E ! M be a bounded module homomorphism. Its restri tion to E is an element  of E ; then T = ^ on E  E , and this implies T = ^ by Lemmas 9.4.4 and 9.4.5. The weak* topology on E was hara terized in Proposition 9.4.2, and that result, together with Lemma 9.4.4, implies that E is weak* dense in E .

PROOF

0

0

0

0

0

0

0

Lemma 9.4.4 also implies the following onverse to Proposition 9.4.2.

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206

Chapter 9: Hilbert Modules

Let E be a left Hilbert module over a von Neumann algebra M. Suppose E an be identi ed with the dual of a Bana h spa e in su h a way that a bounded net ( ) in E onverges weak* if and only if the net h ;  i

onverges weak* in M for every  2 E . Then E is self-dual. PROPOSITION 9.4.7

Let  2 E . By Lemma 9.4.4 there exists a bounded net ( ) in E su h that ^ !  weakly. By passing to a subnet, we may suppose that  !  weak*; then for any  2 E we have 0

PROOF

h; i = lim h; i = ();

so ^ = . Thus E is self-dual.

Now we an des ribe the basi examples of self-dual Hilbert modules over von Neumann algebras. Example 9.4.8

M M

(a) Let be a von Neumann algebra and let X be a set. De ne l2 (X ; ) to be the set of fun tions f : X su h that  f ( x ) f ( x ) is bounded. It is easy to see that boundedness of this x sum implies that it onverges weak* in . Note that this de nition is not the same as the one given in Example 9.3.3 (a). The proof that l2 (X ; ) is a left Hilbert module runs along familiar lines. Self-duality an be proven as follows. Given any F in the dual module, let f (x) = F (IM x ) ; then g; f = F (g ) for all nitely supported g : X , and the fa t that F = f^ then follows from Lemma 9.4.5. (b) Now let X be a  - nite measure spa e. We de ne L2 (X ; ) to be the dual of the ompletion of the simple fun tions with nite measure support ( f. Example 9.3.3 (b)). This is self-dual by Theorem 9.4.6. Again, this de nition does not agree with Example 9.3.3 (b).

P

M

M

!M

h i

!M

M

We now know that if E is a Hilbert module over a von Neumann algebra M then E is a self-dual Hilbert M-module and E embeds in E . Also, B(E ) is a C*-algebra and B(E ) is a von Neumann algebra. The nal result we need in this dire tion says that B (E ) embeds in B (E ). This is useful be ause dualization is the usual method employed to onstru t self-dual Hilbert modules, and often one is interested in operators whi h are initially de ned only on the original module. 0

0

0

0

E be a left Hilbert module over a von Neumann algebra M and let 2 B(E ). Then there is a unique extension A~ 2 B(E ) of A. The map

PROPOSITION 9.4.9

Let

A

© 2001 by Chapman & Hall/CRC

0

207 A 7! A~ is a -isomorphism of B (E ) into B (E 0 ).

First, de ne A# : E ! E 0 by A#  () = h; Ai; then de ne ## 0 ~ A=A : E ! E 0 by A~( ) = h; A#  i. It is straightforward to he k ~ that A is a left module map and that kA~k  kAk. Moreover, A~ extends ~k = kAk. Uniqueness follows from Lemmas 9.4.4 and 9.4.5. A, so that kA This shows that the map A 7! A~ from B (E ) into B (E 0 ) is well-de ned and isometri . The fa t that it preserves sums, produ ts, and adjoints follows from uniqueness; for example, A~B~ is an extension of AB , and hen e must equal (AB )~.

PROOF

We on lude this se tion with a des ription of the stru ture of selfdual Hilbert modules over von Neumann algebras. The key tool is the following lemma, whi h generalizes Theorem 2.2.5. Here, given E0  E we write E0? for the set of  2 E su h that h; i = 0 for all  2 E0 . LEMMA 9.4.10

Let E be a self-dual left Hilbert module over a von Neumann algebra M and let E0 be a weak* losed submodule of E . Then E = E0  E0? .

PROOF

Let  2 E ; we must nd 1 2 E0 and 2 2 E0? su h that

= 1 + 2 . By Proposition 9.4.7, E0 is self-dual for the Hilbert module stru ture it inherits from E . Thus the identity map A : E0 ! E has an adjoint A : E ! E0 by the omment following De nition 9.4.1. Let P = AA : E ! E , and de ne 1 = P  and 2 =  P  . It is routine to verify that 1 2 E0 and 2 2 E0? . 

L P

L L

Let M be a von Neumann algebra. For any family (E ) of self-dual Hilbert modules, write 1 E for the set of sequen es  su h that  2 E for all  and j j2 is bounded. Equivalently, 1 E is the dual of the norm losure of the algebrai dire t sum of the modules E . Either way it is a self-dual Hilbert M-module. Observe also that if I is a weak* losed left ideal of M then the inner produ t h; i =  and the natural left a tion of M make I into a left Hilbert module over M. Moreover, I is self-dual by Proposition 9.4.7. The following theorem hara terizes the stru ture of self-dual Hilbert modules over von Neumann algebras. THEOREM 9.4.11

Let

E

be a self-dual left Hilbert module over a von Neumann algebra

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208

Chapter 9: Hilbert Modules

( )

L

M. Then there is a family I of weak*

losed left ideals of 1I . that E is isometri ally isomorphi to 

M su h

PROOF Suppose rst that E is singly generated in the sense that there exists  2 E su h that no proper weak* losed submodule of E

ontains . We laim that E is isometri ally isomorphi to a weak*

losed left ideal of M. To see this, let n = (jj + n1 IM ) 1 and observe that knk  1. Thus the sequen e (n ) has a weak* luster point . We have jj = , so  also generates E . Also j jh; i1=2 = h;  i1=2 = j j; whi h implies that jj is the proje tion onto the losure of ran(jj). Let I = Mjj. For any ; 2 I we have h; i = jj2  =   ; this shows that the map  7!  from I into E is a Hilbert module isomorphism, and sin e  generates E it is surje tive. This proves the

laim. The remainder of the proof is an easy appli ation of Lemma 9.4.10 and Zorn's lemma. In identally, this proof shows that every weak* losed left ideal of a von Neumann algebra M is of the form Mp for some proje tion p 2 M ( f. Proposition 6.5.5). 9.5

Crossed produ ts

We will now use the ma hinery of Hilbert modules to present an important general onstru tion of C*- and von Neumann algebras, the redu ed rossed produ t onstru tion. This an be arried out without using Hilbert modules, but it then be omes substantially more te hni al. We will need two basi fa ts about topologi al groups, that is, groups that are equipped with a topology whi h is ompatible with the produ t and inverse operations in the natural way. First, every lo ally ompa t group has a unique (up to a positive s alar multiple) regular Borel measure h whi h is nite on ompa t sets, stri tly positive on nonempty open sets, and invariant under left translation by any group element. Thus Z Z f (xy) dh(y) = f (y) dh(y) G G for any f 2 C (G) and x 2 G. This is the Haar measure. Se ond, on every lo ally ompa t group G there is a homomorphism
: G ! R+ into the multipli ative group of positive reals su h that Z Z 1 f (x ) dh(x) = f (x)
(x) 1 dh(x) G

© 2001 by Chapman & Hall/CRC

G

209 for any f 2 C (G). This is the modular fun tion. For the sake of simpli ity it is possible to restri t the following dis ussion to dis rete groups, when Haar measure is just ounting measure and the modular fun tion is onstantly 1. Outside the dis rete ase, the most important examples are the groups Rn and Tn , for both of whi h Haar measure is Lebesgue measure and the modular fun tion is again

onstantly 1. DEFINITION 9.5.1

(a) A C*-dynami al system is a triple (A; G; ) su h that A is a C*algebra, G is a topologi al group, and  is a homomorphism from G into the group of -automorphisms of A whi h is ontinuous for the pointnorm topology, meaning that if x ! x in G then (x )() ! (x)() in norm, for every  2 A. (b) A W*-dynami al system is a triple (M; G; ) su h that M is a von Neumann algebra, G is a topologi al group, and  is a homomorphism from G into the group of -automorphisms of M whi h is ontinuous for the point-weak* topology, meaning that if x ! x in G then (x )() ! (x)() weak*, for every  2 M. We write x for (x). Intuitively, the redu ed rossed produ t of a C*-dynami al system (A; G; ) is a C*-algebra generated by the elements of A and G in an appropriately \smoothed out" manner, and similarly for W*-dynami al systems. An important spe ial ase arises when A = C and G a ts trivially; then the rossed produ t onstru tion produ es the redu ed group C*-algebra Cr (G) and von Neumann algebra Wr (G). The former is the C*-algebra generated by C (G) (the ontinuous fun tions on G with ompa t support) a ting by onvolution on L2 (G), and the latter is the weak* losure of the former. General redu ed rossed produ ts are de ned as follows. DEFINITION 9.5.2

(a) Let (A; G; ) be a C*-dynami al system and let C (G; A) be the set of all ontinuous fun tions from G into A with ompa t support. This is a right A-module via pointwise right multipli ation of elements of A, and it has an A-valued inner produ t de ned by Z hf; g i = f (x) g(x) dh(x) G

( f. Example 9.3.3). Let E be the Hilbert module obtained by ompleting C (G; A) for the resulting norm.

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210

Chapter 9: Hilbert Modules

For ea h  2 C (G; A) de ne a twisted by Z

onvolution operator L on

E

( ) = x 1 ((y)) f (y 1x) dh(y): G The redu ed rossed produ t of by G, denoted  G, is the C*subalgebra of B ( ) generated by the operators L for  C (G; ). (b) Now let ( ; G; ) be a W*-dynami al system. De ne and L ~  B( 0 ) be as B ( ) as above. Let 0 be the dual module and let L given in Proposition 9.4.9. Then the redu ed rossed produ t of by 0 G, denoted  G, is the W*-subalgebra of B ( ) generated by the ~  for  C (G; ). operators L One an prove using Minkowski's inequalityRfor integrals that for f C (G; ) we have L f a f where a = (x) dh(x). Thus L does extend to a bounded module operator on . The term \redu ed" is used to distinguish this rossed produ t, whi h is represented on L2(G; ), from the \full" rossed produ t, whi h arises from a universal representation of C (G; ). The redu ed rossed produ t is generally more important. In fa t,  G (  G) is really just the norm losure (weak*

losure) of the set of operators L (L~ ). That is be ause this set is

losed under sums, produ ts, and adjoints. Indeed, simple al ulations show that L + L = L+ , LL = L where L f x




A

A 

E

2

M

A

E

E

E

2

2

E

M

M 

E

2

M

2

A

k

k 

k

k

k

k

E

A

A

A 

M 

(

)(x) =



Z G

( ) ( (y

 y y

1 x)) dh(y);

and L = L where  x

( ) = x((x 1 ))
(x) 1 : The quantum plane and tori an be viewed as rossed produ t algebras. Example 9.5.3

(a) Let h > 0 and de ne an a tion  of Z on C (T) by letting n be translation by nh. That is, n f (t) = f (t nh). Now de ne ; : Z ! C (T) by




 = 1T f1g

Then we have (  )(n) =

X

© 2001 by Chapman & Hall/CRC

and

(m)m ( (n

= eit
f0g :




m)) = ei(t h ) f1g (n)

211 and (

 )(n) =

X

(m)m ((n




m)) = eit f1g (n):

Thus L L = e ih L L . Similar omputations show that L and are unitary. Finally, it is not too hard to show that C (T)  Z is generated by L and L , so if h  is an irrational multiple of  then Corollary 5.5.9 implies that C (T)  Z  = C h (T2 ). In fa t, this is 1 2 true for all h  > 0. Likewise, L (T)  Z  = Lh1  (T ). (b) Similarly, for h  > 0 we de ne an a tion  of R on C0 (R) by letting t be translation by  ht. For f1 ; f2 2 C 1 (R) let  be the fun tion from R to C0 (R) de ned by (x)(t) = f1 (x)f2 (t); then the operators L generate C0 (R)  R, and there is a -isomorphism from C0 (R)  R onto C0h (R2 ) whi h takes L in the sense of De nition 9.5.2 to Lf1 f2 2 in the sense of De nition 5.4.1. Likewise, L1 (R)  R  = Lh1  (R ). L

9.6

Hilbert

-bimodules

By Theorems 9.1.6 and 9.2.4, it is reasonable to regard Hilbert C*and W*-modules as the quantum analog of ontinuous and measurable Hilbert bundles, that is, bundles for whi h ea h ber has a omplex Hilbert spa e stru ture. In geometri appli ations, however, we are often more interested in real Hilbert bundles ( f. Example 9.1.2). Their quantum version involves modules with an involution  , whi h is analogous to onjugation on the omplexi ation of a real bundle (see Example 9.6.4). The self-adjoint part of su h a module orresponds to the

ontinuous se tions of a bundle of real Hilbert spa es. The existen e of an involution has a surprising onsequen e: given a left module with involution we an use the equation ( ) =    to de ne a right a tion, and vi e versa. Thus it is natural to work with bimodules, that is, modules whi h possess ommuting left and right a tions. Interestingly, bimodule stru ture is also ru ial in geometri appli ations; see Se tion 10.3. The pre ise de nition of the relevant lass of modules goes as follows. Let

DEFINITION 9.6.1

bimodule

over

A

is an

A

be a C*-algebra.

A-A-bimodule E

; : E  E ! A su h that (a) h;  i = h;  i (b) h;  i = h;  i ( ) h; i = h;  i ear map

h

i

and an antilinear involution (d)

h



; i = h ;   i

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:E

! E

A

pre-Hilbert

-

together with a omplex lin-

su h that

212

Chapter 9: Hilbert Modules

(e) ( ) =    (f) h;   i  0

for all  2 A and ;  2 E . We de ne left and right A-valued pseudo inner produ ts h;  il = h;   i and h;  ir = h  ;  i and left, right, and middle seminorms k k2l = kh;  il k, k k2r = kh;  ir k, and k km = max(k kl ; k kr ). A Hilbert -bimodule over A is a pre-Hilbert -bimodule over A for whi h k km is a omplete norm. We all h
;
i a bilinear form. The left and right pseudo inner produ ts h
;
il and h
;
ir that it gives rise to satisfy the appropriate versions of De nition 9.3.1; thus k
kl and k
kr are indeed both pseudonorms. It is

important to note that these pseudonorms not only typi ally disagree, they often are not even omparable (though we always have k kl = k kr if  is self-adjoint). Thus, in general it will not be possible to omplete k
kl or k
kr without destroying the -bimodule stru ture: if one fa tors out null ve tors and ompletes for k
kl , for example, the result will only be an ordinary left Hilbert module. One an fa tor out null ve tors and

omplete for k
km , though (Proposition 9.6.3). The following lemma is basi . LEMMA 9.6.2

Let A be a C*-algebra and E a pre-Hilbert -bimodule over A. Then for any  2 A and  2 E we have

k kl ; kkl  kkk kl

PROOF

and

k kr ; kkr  kkk kr :

We have

k k2l = kh; ( ) ik = kh;   i k  kk2 k k2l ; so k kl  kkk kl . For the se ond inequality, observe that  implies h ;  il  h ;  il sin e h ( );  il = h ( )1=2 ;  ( )1=2 il  0: Without loss of generality suppose kk  1; then letting =  we have 2  and 0  h  ;   il = h;  il 2h ;  il + h 2 ;  il  h;  il h ;  il ; so that kk2l = kh ;  il k  k k2l , as desired. Taking adjoints yields the same inequalities for the seminorm k
kr .

© 2001 by Chapman & Hall/CRC

213 Using this lemma, we an give a Hilbert -bimodule version of the te hnique of fa toring out null ve tors and ompleting. PROPOSITION 9.6.3

Let E be a pre-Hilbert -bimodule over a C*-algebra A and de ne = f 2 E : k km = 0g. Then E0 is a sub-bimodule of E and the inner produ t and involution on E des end to E =E0 and extend to the

ompletion of E =E0 . The ompletion of E =E0 is a Hilbert -bimodule over A. E0

PROOF By Lemma 9.6.2, E0 is a sub-bimodule of E and the left and right a tions of A on E =E0 extend ontinuously to its ompletion for k
km . The inner produ t des ends to E =E0 and then extends to its ompletion by the Cau hy-S hwarz inequality for ordinary Hilbert modules (Proposition 9.3.2), and the orresponding assertions for the involution are trivial.

Next we present some simple examples of Hilbert -bimodules. The rst generalizes Example 9.1.2, and the se ond is the -bimodule version of Example 9.3.3. Example 9.6.4

Let X be a ompa t Hausdor spa e and let X be a real ontinuous Hilbert bundle over X ; this is de ned as in De nition 9.1.1, but using real s alars. If S (X ) is the spa e of ontinuous se tions of X , then S (X )  iS (X ), with involution (f + ig) = f ig and bilinear form hf; gi de ned by pointwise omplexi ation of the inner produ t on bers, is a pre-Hilbert -bimodule over C (X ). It an be ompleted to a Hilbert -bimodule by Proposition 9.6.3. Example 9.6.5

P

P

2 (a) Let A be a C*-algebra and X a set. Let lsym (X ; A) be the set of fun tions f : X ! A su h that x f (x)f (x) and x f (x) f (x) both onverge in A. Then the natural a tions and the bilinear form

hf; gi =

X f (x)g(x)

x2 X

2 make lsym (X ; A) a Hilbert -bimodule. (b) Now let X be a  - nite measure spa e and de ne L2sym (X ; A) to be the ompletion for k
km of the set of all simple fun tions from X to A whose support has nite measure ( f. Example 9.3.3 (b)). This is also a Hilbert -bimodule over A.

© 2001 by Chapman & Hall/CRC

Chapter 9: Hilbert Modules

214

The following is a sort of onverse to Example 9.6.4, as well as a two-sided version of Theorem 9.1.6 (a). THEOREM 9.6.6

Let X be a ompa t Hausdor spa e and let E be a Hilbert -bimodule over C (X ). Suppose that any inner produ t of self-adjoint elements of E is a real-valued fun tion in C (X ). Then there is a real ontinuous Hilbert bundle X over X su h that E  = S (X )  iS (X ).

PROOF

We want to show that the left and right a tions of C (X ) on E oin ide. Fix x 2 X . For any f 2 C (X ) su h that f  0 and f (x) = 0, and any  2 E su h that  =  , set g = f 1=2 ; then

hg + g; g + gi(x) = hg; gi(x); so hf; i(x)  0. But also

hg + g; ig

ig i(x) = ihg; gi(x);

sin e both g + g and ig ig are self-adjoint, so is their inner produ t, so this shows that hf; i(x) must be purely imaginary. But we already showed that it is real, so it follows that hf; i(x) = 0. By linearity, we have hf; i(x) = 0 for any f 2 C (X ) su h that f (x) = 0. Now let f 2 C (X ) and  2 E and suppose f is real and  =  . Sin e the quantity

hf + f; if if i = ihf 2 ; i if 2h; i is real, it follows that Rehf 2 ; i = Re f 2 h; i. Therefore hf f; f f i = Rehf f; f f i = 2Re(f hf; i f 2 h; i); and evaluating this expression at x yields 2Re(f (x)h(f f (x)); i(x)), whi h is zero by the last paragraph. Sin e this is true for all x 2 X , we

have f f = 0 as desired. Taking linear ombinations, we on lude that this is true for any f 2 C (X ) and  2 E . De ne Esa = f 2 E :  =  g, so that E = Esa + iEsa . Then the real version of Theorem 9.1.6 (a) implies that Esa  = S (X ) for some real Hilbert bundle X over X . Thus E  S ( X ) + i S ( X ). =

The left and right a tions do not oin ide in general in the ommutative ase; a ounterexample is given by taking E to be M2 (C) and A to be the diagonal matri es in M2 (C), and for A; B 2 E de ning hA; B i 2 A to be the diagonal part of AB .

© 2001 by Chapman & Hall/CRC

215 Also, the reality ondition on inner produ ts in Theorem 9.6.6 an fail even if the left and right a tions agree. This is shown by taking A = C, 2  E = C , and de ning (a; b) = ( b; a) and h(a; b); ( ; d)i = b . We on lude this se tion with a dis ussion of Hilbert -bimodules whi h satisfy a sort of lo ality ondition. This turns out to be the key property that enables us to prove a version of the stru tural hara terization of self-dual Hilbert modules given in Theorem 9.4.11. Although we formulate only the von Neumann algebra version of the ondition, there is an obvious C* version as well. First we de ne the von Neumann algebra version of a Hilbert -bimodule. DEFINITION 9.6.7 Let E be a Hilbert -bimodule over a von Neumann algebra M. We de ne the -weak topology on E to be the weakest topology su h that the maps  7! h;  i and  7! h;  i are ontinuous from E into M for all  2 E , with respe t to the weak* topology on M. We say that E is a dual bimodule if it is a dual Bana h spa e in su h a way that the weak* topology and the -weak topology agree on the unit ball. If for any bounded, weak* onvergent net  !  in M and any  2 E we have   !  -weakly, then we say that E is normal. Finally, if E is both normal and dual we all it a W* Hilbert -bimodule.

In fa t, -weak ompa tness of the unit ball of E is suÆ ient to imply that E is a dual bimodule. This an be established by the argument used to prove Proposition 9.4.2. The desired lo ality ondition is formulated as follows. DEFINITION 9.6.8 Let E be a W* Hilbert -bimodule over a von Neumann algebra M. The enter of E is the set

Z (E ) = f 2 E :  =  for all  2 Mg: We say that E is entered if M
Z (E ) is weak* dense in E , and we say that is lo al if it is entered and the inner produ t of any two self-adjoint elements of Z (E ) is self-adjoint in M. E

The main onsequen e of lo ality is given in the following lemma, whi h is analogous to Lemma 9.4.10. Here we say that two subspa es E1 ; E2  E are orthogonal if h;  i = h;  i = 0 for all  2 E1 and  2 E2 . We also use the notation Z (M)sa and Z (E )sa for the set of self-adjoint elements in Z (M) and Z (E ). LEMMA 9.6.9

Let E be a lo al W* Hilbert -bimodule over a von Neumann algebra M

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216

Chapter 9: Hilbert Modules

and let E1 be a entered, weak* losed, self-adjoint sub-bimodule of E . Then there is another entered, weak* losed, self-adjoint sub-bimodule E2 of E whi h is orthogonal to E1 , su h that E = E1  E2 .

PROOF Let V be a sub Z (M)sa -module of Z (E1 )sa whi h is nitely generated as a module over Z (M)sa . We laim that we an nd a nite set f1 ; : : : ; n g whi h spans V over Z (M)sa su h that hi ; j i = 0 whenever i 6= j . To see this let f1 ; : : : ; n g be any nite set whi h spans V and assume indu tively that hi ; j i = 0 for i; j  n 1, i 6= j . Sin e h; i = h; i = h; i = h; i = h; i = h; i for any ;  2 Z (E1 ) and  2 M, it follows that the inner produ t of any two elements of Z (E1 ) is in Z (M). Thus Z (E1 ) satis es the hypothesis of Theorem 9.6.6 as a Hilbert -bimodule over Z (M). Using the on lusion of Theorem 9.6.6 it is easy to verify that

 = n

X hn; ii i

n 1 i=1

hi ; i i

is well-de ned and orthogonal to i (i  n 1) and that f1 ; : : : ; n 1 ;  g spans V . This proves the laim. Now x  2 Z (E )sa . For any V  Z (E1 )sa as above, let f1 ; : : : ; n g verify the laim and de ne

V =

Xn h; ii i 2 V : i=1

hi ; i i

This expression is sensible and kV k  k k by Theorem 9.6.6. Dire t the subspa es V by in lusion and let P  be a luster point of the net (V ); then P  2 Z (E1 )sa and h P ;  i = 0 for all  2 Z (E1 )sa , and as there is at most one element of Z (E1 )sa whi h an have this property P is a well-de ned proje tion from Z (E )sa onto Z (E1 )sa . Let E2 be the weak* losure of the set M
ker(P ). It is lear that E2 is orthogonal to E1 . Also Z (E1 )sa  Z (E2 )sa = Z (E )sa , so E1  E2 is weak* dense in E . So for any  2 E we an nd a bounded net ( + 0 ) su h that  2 E1 and 0 2 E2 and  + 0 !  weak*. By orthogonality, the nets ( ) and (0 ) are also bounded and so they have luster points  2 E1 and 0 2 E2 , and  + 0 =  by ontinuity. So E = E1  E2 .

L

L P

Let (I ) be a family of W*-ideals of a von Neumann algebra M. We denote by 1 I the set of sequen es  ( 2 I ) with the property that both of the sums   and   onverge weak* (or equivalently, that both sums are bounded). This is a W* Hilbert

© 2001 by Chapman & Hall/CRC

P

217 -bimodule, and its enter is L1 Z (I ), so it is entered and lo al. The following result shows that this onstru tion is general. THEOREM 9.6.10

=L

Let E be a lo al W* Hilbert -bimodule over a von Neumann algebra M. 1I . Then there is a family I of W*-ideals of M su h that E  

( )

PROOF As in the proof of the lemma, regard Z (E ) as a lo al W* Hilbert -bimodule over Z (M). Observe that for any  2 Z (E )sa the sequen e (h; i1=2 + n 1IM ) 1 is bounded, and if  is a weak* luster point of this sequen e then h; i is a proje tion in Z (M). Now let fg be a maximal family of orthogonal elements of Z (E )sa with the property that h ; i is a proje tion in Z (M). It follows from the lemma and the pre eding observation that M
spanf g is weak* dense in E . Also the sub-bimodules M are pairwise orthogonal. So it suÆ es to show that ea h M is isomorphi to I = pM where p = h ; i. But the kernel of the map  7!  is a weak* losed ideal of M, and hen e is of the form qM for some proje tion q 2 Z (M), and learly q = IM p. Also, for any ; 2 I we have  = h ;  i and ( ) =  . So indeed M = I as Hilbert -bimodules. 9.7

Notes

For more on Hilbert modules over C*-algebras see [43℄. The orresponden e between ontinuous Hilbert bundles and Hilbert modules is given a very thorough treatment in [26℄. Hilbert bundles over C*-algebras were originally onsidered in [56℄ and [59℄, and these papers are still a good sour e on the topi . The material in Se tion 9.4 is all from [56℄. The approa h to rossed produ ts presented in Se tion 9.5 is taken from [41℄. For more on Hilbert -bimodules see [74℄. There is also a quantum analog of Bana h bundles, viz., operator modules; see [57℄.

© 2001 by Chapman & Hall/CRC

© 2001 by Chapman & Hall/CRC

Chapter 10

Lips hitz Algebras 10.1

The algebras

X)

Lip0 (

The quantum version of a metri spa e is analogous to the quantum versions of topologi al and measure spa es. In ea h ase there is a Hilbert spa e generalization of the algebra of s alar-valued maps on su h an obje t whi h preserve the relevant stru ture. The topologi al and measure theoreti ategories involve the algebras C (X ) and L1(X ); in the metri

ase the appropriate algebra is Lip0(X ). DEFINITION 10.1.1

(a) Let X and Y be metri spa es. A map f : X

Y is Lips hitz if its Lips hitz number  (f (x); f (y))  L(f ) = sup : x; y X; x = y (x; y ) is nite. (We use the generi notation ( ; ) for distan e in any metri 2

!

6





spa e.) (b) Let X be a omplete metri spa e with nite diameter (that is,
(X ) = supf(x; y) : x; y 2 X g is nite) and let e 2 X be a distinguished element. Then Lip0 (X ) is the set of Lips hitz fun tions f : X ! C whi h satisfy f (e) = 0.

We all a metri spa e with a distinguished element pointed. In part (b) there is no loss of generality in requiring that X be omplete, as s alar-valued Lips hitz fun tions always extend to the ompletion of a spa e without in reasing their Lips hitz number. It is straightforward to verify that Lip0(X ) is a Bana h spa e for the norm L( ) and is losed under produ ts. (The latter uses the fa t that the diameter of X is nite; spe i ally, we have the bound L(f g) 2
(X )L(f )L(g).) After this the most important fa t about Lip0 (X ) is that it is a dual spa e.




219 © 2001 by Chapman & Hall/CRC

220

Chapter 10: Lips hitz Algebras

PROPOSITION 10.1.2

Let X be a omplete pointed metri spa e with nite diameter. Then Lip0 (X ) is a dual Bana h spa e in su h a way that on bounded sets its weak* topology agrees with the topology of pointwise onvergen e.

PROOF

The argument is similar to the proof of Proposition 9.4.2. For every x 2 X the map x^ : f 7! f (x) satis es

jx^(f )j = jf (x)

f (e)j  L(f )
(x; e)

and hen e belongs to Lip0 (X ) . Let V  Lip0 (X ) be the losed span of the set fx^ : x 2 X g and let T : Lip0 (X ) ! V  be the natural map, i.e., T f (! ) = ! (f ). For distin t x; y 2 X we have kx^ y^k  (x; y ). Sin e T f (^x y^) = f (x) f (y ), it follows that kT f k  jf (x) f (y )j=(x; y ). Taking the supremum over x; y 2 X , we dedu e that kT f k  L(f ), and the reverse inequality is easy. So T is an isometry. To show that T is surje tive, let ! belong to the unit ball of V  . De ne a fun tion f : X ! C by setting f (x) = ! (^x); then it is straightforward to verify that f 2 Lip0 (X ) (with L(f ) = k! k) and T f = ! . Thus Lip0 (X )  = V . By evaluating on x^, it is lear that f ! f weak* implies f (x) ! f (x). Conversely, sin e the unit ball of Lip0 (X ) is weak* ompa t and the topology of pointwise onvergen e is Hausdor, the two topologies must agree on the unit ball. We now want to prove results analogous to those given in Se tions 5.1, 6.1, and 6.2, and establish orresponden es between metri properties of X and algebrai properties of Lip0 (X ). As in Chapter 6, use of the weak* topology will be key. The main tool is the following version of the Stone-Weierstrass theorem. LEMMA 10.1.3

Let X be a omplete pointed metri spa e with nite diameter and let M be a W*-subalgebra of Lip0 (X ). Then for any set of real-valued fun tions (f )  M+ = M + C
1X whi h is bounded in both Lips hitz and supremum norm, we have sup f ; inf f 2 M+ .

PROOF Let f 2 M+ be real-valued. Then f (e) 2 R, so f f (e) 2 M is also real-valued. Let a = kf k1. For any fun tion g 2 C 1 [ a; a℄, we have g Æ f 2 Lip0 (X )+ C
1X sin e L(g Æ f )  L(f )L(g ) = L(f )kg 0 k1 ; we laim that g Æ f 2 M+ . To see this, nd a sequen e of polynomials hn su h that khn g 0 k1 ! 0. Let gn be the inde nite integral of hn whi h

© 2001 by Chapman & Hall/CRC

221 satis es gn(0) = g(0), and let fn = gn Æ f . Then fn is a polynomial in f , so fn 2 M+ , and L(fn g Æ f )  L(f )L(gn g ) = L(f )kgn g k ! 0: Thus (fn fn(e)) is a bounded sequen e in M and onverges to g Æ f g Æ f (e), so we on lude that the latter fun tion also belongs to M, and hen e g Æ f 2 M+. Now let (gn) be a sequen e of C 1 fun tions whi h onverges pointwise to the absolute value fun tion t 7! jtj on [ a; a℄ and is uniformly bounded in Lips hitz norm. By the laim, gn Æ f 2 M+ for all n, and gn Æ f ! jf j pointwise, so jf j 2 M+ by weak* losure of M. It follows that for any real-valued f1; f2 21M+ the fun tions max(f1; f2) = 12 (f1 +f2 +jf1 f2j) and min(f1; f2) = 2 (f1 + f2 jf1 f2j) both belong to M+. Finally, for any Lips hitz and sup norm bounded set of real-valued fun tions in M+ we an take a pointwise limit of the max (respe tively, min) of every nite subset to get the sup (respe tively, inf) of the set, and this shows that Re M+ is losed under suprema and in ma of bounded sets. We say that a subspa e E  Lip0 (X ) separates points uniformly if there exists C  1 su h that for any x; y 2 X there exists f 2 E with L(f )  C and jf (x) f (y )j = (x; y ). 0

0

1

THEOREM 10.1.4

Let X be a omplete pointed metri spa e with nite diameter and let

M be a W*-subalgebra of Lip0 (X ) whi h separates points uniformly. Then M = Lip0 (X ).

PROOF Let f 2 Lip0 (X ) be real-valued and suppose L(f )  1. It will suÆ e to show that f 2 M+ = M + C
1X . Fix C  1 as in the de nition of uniform separation of points. For ea h x; y 2 X nd g 2 M su h that L(g)  C and jg(x) g(y)j = (x; y ). Multiplying g by the onstant (f (x) f (y ))=(g (x) g (y )), we may assume that g(x) g(y) = f (x) f (y). Then adding the real

onstant f (x) g(x) = f (y) g(y) to g, we get a fun tion hxy 2 M+ su h that hxy (x) = f (x), hxy (y) = f (y), L(hxy )  C , and khxy k  kgk + jf (x)j + jg(x)j  (2C + 1)
(X ): Finally, we an repla e hxy with its real part without ae ting the previous fa ts, and the lemma then implies that f = inf sup hxy 2 M+ ; x X 1

1

2

© 2001 by Chapman & Hall/CRC

y X 2

222

Chapter 10: Lips hitz Algebras

as desired. Now we an establish the expe ted relationships between subalgebras of Lip0 (X ) and quotients of X and between ideals of Lip0 (X ) and subsets of X . As usual, given  : X ! Y we let C : Lip0 (Y ) ! Lip0 (X ) be the

omposition map C f = f Æ . Example 10.1.5

Let X and Y be omplete pointed metri spa es with nite diameter and let  : X Y be a ontra tion su h that (eX ) = eY . Assume that C : Lip0 (Y ) Lip0 (X ) is an isometry. Then C (Lip0 (Y )) is a W*-subalgebra of Lip0 (X ).

!

!

PROPOSITION 10.1.6

Let X be a omplete pointed metri spa e with nite diameter and let M be a W*-subalgebra of Lip0 (X ). Then there is a omplete pointed metri spa e Y with nite diameter and a ontra tion  : X ! Y su h that (eX ) = eY , C : Lip0 (Y ) ! Lip0 (X ) is isometri , and M = C (Lip0 (Y )). Set x  y if f (x) = f (y ) for all f

ompletion of X=  for the metri

PROOF

M ([x℄; [y ℄) = supfjf (x)

f (y )j : f

2 M, and let Y

be the

2 M; L(f )  1g:

Let eY = [eX ℄ and let  : X ! Y be the quotient map. It is lear that C takes Lip0 (Y ) nonexpansively into Lip0 (X ). Conversely, it is easy to see that any f 2 M des ends to a fun tion f~ on Y with L(f~)  L(f ). Thus, the map f 7! f~ is an isometry from M into Lip0 (Y ). Sin e M is weak* losed in Lip0 (X ), it is losed under pointwise onvergen e of bounded nets, and this remains true of its image in Lip0 (Y ), so C 1 (M) is a W*-subalgebra of Lip0 (Y ). But by the de nition of the metri on Y , C 1 (M) separates points uniformly. Thus C 1 (M) = Lip0 (Y ), whi h is enough. Example 10.1.7

Let X be a omplete pointed metri spa e with nite diameter and let K be a losed subset of X whi h ontains e. Then f Lip0 (X ) : f K = 0 is a W*-ideal of Lip0 (X ).

j

g

f 2

The onverse of this example requires two lemmas. We use the following notation: X is a omplete pointed metri spa e with nite diameter,

© 2001 by Chapman & Hall/CRC

223

I is a W*-ideal of Lip0 (X ), and K = fx 2 X : f (x) = 0 for all f We also de ne a pseudometri I (x; y )

= supfjf (x)

2 Ig.

j : f 2 I ; L(f )  1g

f (y )

as in the proof of Proposition 10.1.6. LEMMA 10.1.8 Let

2

x X . If I (x; K ) < a then for (x; y ) < a and I (y; K ) < .

any

>

0

there exists

y

2X

su h

that

PROOF

(x; K )

If (x; K ) < a then the on lusion is trivial, so assume

 a. Let  > 0 and de ne f (y )

= min(1; I (y; K )=)

g (y )

= max(0; a

and (x; y )):

Sin e e 2 K and (x; K )  a, we have f (e) = g (e) = 0, so f; g 2 Lip0 (X ). From the onstru tion of X=  in the proof of Proposition 10.1.6 with M = I , it is seen that the fun tion f des ends to a Lips hitz fun tion on the quotient X= , so that result implies f 2 I . Thus f g 2 I . Now if the lemma fails then f must take the value 1 everywhere g is nonzero, whi h implies g = f g 2 I ; but this implies g jK = 0 and I (x; y )

 jg(x)

j=a

g (y )

for all y 2 K , a ontradi tion. Thus the lemma holds. LEMMA 10.1.9 We have

(x; K )

= I (x; K )

for all

x

2X

.

Suppose I (x; K ) < (x; K ) for some x 2 X . Let Æ = Take a = (x; K ) Æ = I (x; K ) + Æ and  = Æ=3 in Lemma 10.1.8; we get an element y1 2 X su h that (x; y1 ) < a (and hen e (y1 ; K ) > Æ ) and I (y1 ; K ) < Æ=3. Indu tively, we an

onstru t a sequen e (yn ) su h that (yn ; yn+1 ) < Æ=3n , (yn ; K )  (1 + 31 n )Æ=2, and I (yn ; K ) < Æ=3n . Thus yn ! y for some y whi h satis es (y; K )  Æ=2 but I (y; K ) = 0, whi h is impossible. This

ontradi tion establishes the lemma.

PROOF

1 ((x; K ) 2

I (x; K )).

© 2001 by Chapman & Hall/CRC

224

Chapter 10: Lips hitz Algebras

PROPOSITION 10.1.10

Let X be a omplete pointed metri spa e with nite diameter and let I be a W*-ideal of Lip0 (X ). Then there is a losed subset K of X whi h

ontains e su h that I = ff 2 Lip0 (X ) : f jK = 0g. Let K = fx 2 X : f jK = 0 for all f 2 Ig. It is lear 2 I implies f jK = 0. Conversely, let f 2 Lip0 (X ) and suppose f jK = 0; we must show that f 2 I . We have jf (x) f (y )j  L(f )(x; y ) and jf (x) f (y)j  jf (x)j + jf (y)j  L(f )((x; K ) + (y; K )) for all x; y 2 X . We laim that

PROOF

that

f

I (x; y )

 min((x; y); 12 ((x; K ) + (y; K )));

this will imply that jf (x) f (y )j  2L(f )I (x; y ), and we may then

on lude that f 2 I by Proposition 10.1.6. To prove the laim, x x; y 2 X . Without loss of generality suppose (x; K )  (y; K ), and x 0 <  < (y; K ). De ne g (z )

= max(0; (x; K )

(x; z )

):

This fun tion vanishes in the -neighborhood of K , so by Lemma 10.1.9 we have gh = g where h(z )

= min(1; I (z; K )=) = min(1; (z; K )=):

But h 2 I by Proposition 10.1.6, so we therefore have g 2 I . If (x; y )  (x; K ) then g (x) = (x; K )  and g (y ) = 0, so

jg(x)

g (y )

j = (x; K )



 12 ((x; K ) + (y; K ))

:

Otherwise (x; y ) < (x; K ) and we have

jg(x) g(y)j  (x; y) : In either ase, sin e g 2 I and L(g )  1, we on lude I (x; y )

 min((x; y); 21 ((x; K ) + (y; K )))

:

Taking  ! 0 ompletes the proof.

We an use the pre eding result to hara terize quotients of Lip0 (X ). This also requires a version of the Tietze extension theorem, whi h we present rst.

© 2001 by Chapman & Hall/CRC

225 LEMMA 10.1.11

Let X be a omplete metri spa e, let K  X , and let f : K ! R be a Lips hitz fun tion. Then there is an extension f~ : X ! R su h that L(f~) = L(f ).

PROOF

De ne

f~(x) = inf (f (y ) + L(f )(x; y )):

y 2K

A short omputation veri es that f~jK = f and L(f~) = L(f ). PROPOSITION 10.1.12

Let X be a omplete pointed metri spa e with nite diameter and let I be a W*-ideal of Lip0 (X ). Then Lip0 (X )=I is isomorphi to Lip0 (K ) where K = fx 2 X : f (x) = 0 for all f 2 Ig, and the isomorphism is isometri on real-valued fun tions.

PROOF

Let  : Lip0 (X ) ! Lip0 (K ) be the restri tion map. It is

lear that  is nonexpansive, and ker( ) = I by Proposition 10.1.10. For any real-valued f 2 Lip0 (K ), the lemma implies that there exists f~ 2 Lip0 (X ) su h that L(f~) = L(f ) and  f~ = f , so  must be surje tive and isometri on real-valued fun tions. This is enough.

Next we turn to the spe trum, and show that the metri spa e X

an be re overed from the algebra Lip0 (X ). Let x^ 2 Lip0 (X ) be the evaluation map at the point x. Sin e 1X 62 Lip0 (X ), the weak* spe trum de ned in De nition 6.1.1 is not appropriate here. Instead, we let sp0 (Lip0 (X )) be the set of all weak*

ontinuous -homomorphisms, in luding the zero map, from Lip0 (X ) to C. We give sp0 (Lip0(X )) the metri it inherits from Lip0(X ). PROPOSITION 10.1.13

Every omplete pointed metri spa e X with nite diameter is isometri to sp0 (Lip0 (X )) via the orresponden e x $ x^.

PROOF

Let X be a omplete pointed metri spa e with nite diameter. The map x 7! x^ is learly nonexpansive, and applying x^ y^ to the fun tion f (z ) = (x; z ) (x; e) shows that kx^ y^k = (x; y ). To prove surje tivity, let ! : Lip0 (X ) ! C be a weak* ontinuous homomorphism. If ! = 0 = e^ we are done. Otherwise its kernel is a odimension one W*-ideal, so Proposition 10.1.12 implies that there

© 2001 by Chapman & Hall/CRC

226

Chapter 10: Lips hitz Algebras

exists x 2 X , x 6= e, su h that ker(! ) = ff 2 Lip0 (X ) : f (x) = 0g. It follows that ! = ax^ for some nonzero a 2 C; hoosing f 2 Lip0 (X ) su h that f (x) = 1, we then have

a = ! (f ) = ! (f 2 ) = ! (f )2 = a2 ; so that a = 1. Thus ! = x^. The desired orresponden e between Lips hitz fun tions and weak*

ontinuous -homomorphisms follows easily. Example 10.1.14

Let X and Y be omplete pointed metri spa es with nite diameters. Let  : Y ! X be Lips hitz and suppose (eY ) = eX . Then the map C : f 7! f Æ  is a weak* ontinuous -homomorphism from Lip0 (X ) into Lip0 (Y ). The norm of C is L(). PROPOSITION 10.1.15

Let X and Y be omplete pointed metri spa es with nite diameters and let  : Lip0 (X ) ! Lip0 (Y ) be a weak* ontinuous -homomorphism. Then there is a Lips hitz map  : Y ! X su h that (eY ) = eX and  = C . For any y 2 Y the map y^ Æ  belongs to sp0 (Lip0 (X )), so there is a unique x 2 X su h that y^ Æ  = x^. De ne (y ) = x. For any f 2 Lip0 (X ) and any y 2 Y we then have

PROOF

f ((y )) = (y )^(f ) = y^(f ) = f (y ); so f Æ  = f . It is lear that (eY ) = eX . To see that  is Lips hitz, let x; y 2 Y and de ne f 2 Lip0 (X ) by

f (z ) = ((x); z )

((x); eX ):

Then

jf (x) f (y)j = jf ((x)) f ((y))j = ((x); (y)): Sin e L(f )  k kL(f ) = k k, this implies ((x); (y ))  k k(x; y ). So  is Lips hitz.

10.2

Measurable metri s

There is a measurable version of the notion of a metri and a orresponding version of the spa e of Lips hitz fun tions. In the spe ial ase

© 2001 by Chapman & Hall/CRC

227 of ounting measure su h \measurable metri s" redu e to ordinary metri s. However, we need to examine general measurable metri s before passing to the Hilbert spa e setting, be ause they, not pointwise metri s, represent the general ommutative ase. Measurable metri s are de ned as follows. Let

DEFINITION 10.2.1

X

be a  - nite measure spa e and let

+ be the family of positive measure subsets of X modulo null sets. A measurable pseudometri is a map  : 2+ ! [0; 1℄ whi h satis es (a) (S; S ) = 0 (b) (S S; T ) = (T ; S ) ( ) ( S ; T ) = inf (S ; T ) (d) (R; T )  supf(R; S ) + (S ; T ) : S  S g for all R; S; T ; S 2 + . A measurable metri is a measurable pseudometri with the property i

i

0

0

0

i

that the measurable  -algebra is generated up to null sets by the sets S 2 + with the property that, for every T 2 + , S \ T = ; implies (S; T 0 ) > 0 for some T 0  T .

Intuitively, the quantity (S; T ) is supposed to represent the minimum distan e between the sets S and T , modulo null sets. Indeed, if we are given a genuine pointwise metri on a - nite measure spa e X , a areful interpretation of the pre eding senten e does produ e a measurable metri (or at least a pseudometri ). The onverse is true in a sense: given any measurable metri on a - nite measure spa e, we an repla e the original spa e with a measurably equivalent spa e su h that the orresponding measurable metri on the new spa e arises from a pointwise metri in the manner suggested above. We will not prove this result here. In nite distan es are permitted in measurable metri spa es. This is done to a

omodate general onstru tions su h as the one in Theorem 10.3.6 whi h sometimes produ e su h a result. We will also nd it more onvenient to work with measurable pseudometri s rather than measurable metri s. The following is a fundamental lemma that we will need later. LEMMA 10.2.2

Let X be a  - nite measure spa e, let  be a measurable pseudometri on X , and let S and T be positive measure subsets of X . Let  > 0. Then there exist positive measure sets S0  S and T0  T su h that

(

 S; T

© 2001 by Chapman & Hall/CRC

)  (S ; T ) < (S; T ) +  0

0

228

Chapter 10: Lips hitz Algebras

for all S 0  S0 and T 0  T0 . PROOF For any S 0  S and T 0  T we have (S; T )  (S 0 ; T 0 ) by axiom ( ) of De nition 10.2.1 applied su

essively to S = S [ S 0 and T = T [ T 0 . To prove the remainder, we will nd S0  S su h that S 0  S0 implies (S 0 ; T )  (S; T ) + ; applying the same argument to T then produ es the desired pair S0 ; T0 . Without loss of generality assume the measure  on X is nite, and let a = supf(R) : R  S and (R; T )  (S; T ) + g:

Then nd a S sequen e (Ri ) su h that (Ri ; T )  (S; T ) +  and (Ri ) ! Let R = Ri ; then we have (R; T )  (S; T ) + , and evidently R

ontains, up to a null set, every positive measure subset of S with this property. Thus S0 = S R is a positive measure set (sin e (S; T ) 6= (R; T )) and we have (S 0 ; T ) < (S; T ) +  for any S 0  S0 , as desired. a.

Next we de ne Lips hitz spa es for measurable pseudometri s. Re all the de nition of the essential range of a map from Example 3.2.2 (b). DEFINITION 10.2.3 Let X be a  - nite measure spa e, let  be a measurable pseudometri on X , and let f : X ! C be measurable. For positive measure sets S; T  X we write f (S; T ) for the distan e in C between the essential ranges of f jS and f jT . The Lips hitz number of f is then f (S; T ) L(f ) = sup ; (S; T )

taking the supremum over all positive measure S; T  X su h that we say f is Lips hitz if L(f ) < 1. is the subspa e of L1 (X; ) onsisting of those fun tions f with the property that L(f ) < 1. We de ne the Lips hitz norm on this spa e to be kf kL = max(kf k1 ; L(f )).

(S; T ) > 0; Lip(X; )

In the atomi ase, where  is ounting measure, Lip(X; ) is a variation of the spa e Lip0 (X ) dis ussed in the last se tion. Our rst nonatomi example is the following. Example 10.2.4

Let X be a  - nite measure spa e. De ne a measurable metri  on X by setting  \ T) > 0 (S; T ) = 02 ifif ((S S \ T ) = 0.

© 2001 by Chapman & Hall/CRC

229 L(f )  kf k1 for any f 2 L1 (X ), so we therefore 1 have Lip(X; ) = L (X ), both as sets and as Bana h spa es.

One an he k that

Just as for Lip0 spa es, every Lip(X; ) is a dual Bana h spa e. LEMMA 10.2.5

Let X be a  - nite measure spa e and let  be a measurable pseudometri on X . Then the unit ball of Lip(X; ) is ompa t for the weak* topology inherited from L1 (X ).

PROOF Let (f ) be a net in Lip(X; ) whi h onverges weak* to a fun tion f in L1 (X ), and suppose kf kL  1 for all . We must show that kf kL  1; it is enough to he k that L(f )  1. Let S; T  X be positive measure sets and x  > 0. Find S0 and T0 as in Lemma 10.2.2. By passing to subsets of S0 and T0 , we may assume that jf (x1 ) f (x2 )j   and jf (y1 ) f (y2 )j   for almost every x1 ; x2 2 S0 and y1 ; y2 2 T0 . Then Z



 (f (S0 ; T0) 2)(S0 )(T0 )  (f (S; T ) 2)(S0 )(T0 ): R R There also exists  su h that j S0 (f f )j  
(S0 ) and j T0 (f 
(T0 ); thus S0 T0

Z

S0 T0

(f (x)

(f (x)

f (y ))



f (y ))

Z



(f (x) +

Z



Z

f (x)) +

(f (y )

f (y ))

(f (x)

j

f)

f (y ))

 (3 + (S; T ))(S0 )(T0 ): Combining the two inequalities yields f (S; T )  (S; T ) + 5, whi h is enough.

PROPOSITION 10.2.6

Let X be a  - nite measure spa e and let  be a measurable pseudometri on X . Then Lip(X; ) is a dual Bana h spa e in su h a way that on bounded sets its weak* topology agrees with the weak* topology inherited from L1 (X ). The proof of this result pro eeds along familiar lines. We refer the reader to the proof of Proposition 9.4.2. We also have a Stone-Weierstrass theorem for Lip(X; ). This involves a measurable version of uniform separation of points.

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230

Chapter 10: Lips hitz Algebras

THEOREM 10.2.7

Let X be a  - nite measure spa e, let  be a measurable pseudometri on X , and let M be a unital W*-subalgebra of Lip(X; ). Suppose there exists C  1 su h that for all positive measure sets S; T  X there exists f 2 M su h that L(f )  C and f (S; T ) = (S; T ). Then M = Lip(X; ). Let f 2 Lip(X; ) be positive and satisfy L(f )  1. It will suÆ e to show that f 2 M. To do this, x a positive measure set S  X . For every positive measure set T  X , we an nd g 2 M su h that L(g )  C and g (S; T ) = (S; T ). Sin e the essential ranges of g jS and g jT are ompa t, we an nd representative elements a and b su h that ja bj = (S; T ). Subtra ting a from g , we may assume a = 0, and multiplying g by jbj=b and taking the real part, we may assume g is real valued. Then g jS  0 and g jT  (S; T ). Next, by exa tly the same argument as in Lemma 10.1.3, but here with the simpli ation that we an take a = 0, we an show that the real part of M is losed under suprema and in ma of bounded sets. Thus, repla ing g su

essively with max(g; 0) and min(g; (S; T )) we

an assume that g jS = 0 and g jT = (S; T ) onstantly. De ne gT = min(g; kf k1 ) and

PROOF

hS

= supfgT : (T ) > 0g:

Then hS jS = 0 and hS jT  min((S; T ); kf k1 ) for all T  X . Also, hS 2 M and L(hS )  1. Now for ea h positive measure set S  X , let aS = kf jS k1 . We laim that f

= inf fhS + aS : (S ) > 0g;

this will imply that f 2 M, as desired. First, suppose f  hS + aS +  on some positive measure set some S  X . Then f

jT  hS jT + aS +   (S; T ) + aS + 

T,

for

while f jS  aS by the de nition of aS , so we dedu e that f (S; T ) (S; T ) + , ontradi ting the assumption that L(f )  1. Thus f hS + aS almost everywhere, for all S . Conversely, on any positive measure set S we have f

jS 6 hS jS + aS



= aS

 



for any  > 0. This shows that for any  > 0 we annot have f  inf fhS + aS : (S ) > 0g  on any positive measure set. We on lude that f = inf fhS + aS : (S ) > 0g 2 M as laimed.

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231 As an immediate orollary we dedu e that unital W*-subalgebras of Lip(X; ) are themselves of the form Lip(Y;  ). COROLLARY 10.2.8

Let X be a  - nite measure spa e, let  be a measurable pseudometri on X , and let M be a unital W*-subalgebra of Lip(X; ). Then M = Lip(Y;  ) for some measurable pseudometri on the measure spa e (Y;  ) = (X; ).

PROOF De ne a measurable pseudometri on (Y;  ) = (X; ) by setting (S; T ) = supff (S; T ) : f 2 M; L(f )  1g. It is lear that M  Lip(Y;  ) isometri ally, and the theorem then implies M = Lip(Y;  ). The stru ture of W*-ideals is less transparent; we do not have a simple analog of Propositions 5.1.5, 6.1.7, and 6.2.6. For example, give [0; 1℄ Lebesgue measure and let I = ff 2 Lip[0; 1℄ : f (0) = 0g. This is a proper W*-ideal but it does not vanish on any positive measure set. 10.3

The derivation theorem

Generally speaking, a derivation is a linear map d whi h satis es the Leibniz identity d( ) = d( ) + d() . There is a remarkable onne tion between the algebras Lip(X; ) and derivations whi h states that the Lip(X; ) are pre isely those spa es whi h arise as the domains of a ertain type of derivation. This hara terization of ommutative Lips hitz algebras has an immediate non ommutative (Hilbert spa e) generalization. In order for the derivation ondition to make sense, we must be able to multiply elements of the range of the derivation on either side by elements of its domain; thus, the range must be a bimodule over the domain. Our examples in this se tion will involve L1 spa es, and in order to make a spa e L1 (Y ) a bimodule over L1 (X ), we will require a pair of weak* ontinuous -homomorphisms from the latter into the former. Here, for the rst time, it really is onvenient for us to use spa es whi h are not  - nite. The next best ondition is that X be lo ally nite, whi h means that it an be written as a disjoint union X = X su h that ea h X is a nite measure spa e, a subset of X is measurable if and only if its interse tion with ea h X is measurable, and (S ) = (S \ X ) for all measurable subsets S  X . This lass of measure spa es ontains all dis rete spa es and all  - nite spa es, and mu h of what one an prove

S

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P

232

Chapter 10: Lips hitz Algebras

for - nite spa es a tually holds for lo ally nite spa es with only minor modi ations in the proofs. For instan e, we have L1 (X )  = L1(X ) for 1 any lo ally nite X . (Indeed, the spa es L (X ) for X lo ally nite are pre isely, up to -isomorphism, the abelian von Neumann algebras.) Let X be a  - nite measure spa e. An L1 of L1 (Y ) for some lo ally nite measure spa e Y , together with a pair of weak*

ontinuous unital -homomorphisms l ; r : L1 (X ) ! L1 (Y ) su h that l (f )g; r (f )g 2 E whenever f 2 L1 (X ) and g 2 E . We write fl = l (f ) and fr = r (f ). An (unbounded) L1 -derivation from L1 (X ) into an L1 -bimodule E is a linear map d : L ! E su h that

DEFINITION 10.3.1

bimodule over L1 (X ) is a weak* losed self-adjoint subspa e E

(a) L is a weak* dense, unital -subalgebra of L1 (X ); (b) d satis es d(f) = d(f ) and d(fg ) = fl d(g ) + d(f )gr for all f; g 2 L; and ( ) the graph of d, d = ff  df : f 2 Lg, is a weak* losed subspa e of L1 (X )  E . We give L = dom(d) the graph norm kf kD

= max(kf k1; kdf k1):

When onstru ting L1-derivations it is always possible to take E = However, the greater generality given in the de nition will be

onvenient. The basi example of an L1-derivation to keep in mind is dierentiation on the unit interval (or on the unit ir le). Here d is the map d(f ) = f 0 . There are several natural domains for this map: it an be de ned on C 1 [0; 1℄ or on C 1 [0; 1℄, for example. However, with these domains its graph is not weak* losed. In order to have a weak* losed graph, its domain must be expanded to Lip[0; 1℄. Thus, the derivative map d : L1 [0; 1℄ ! L1[0; 1℄ with domain L = Lip[0; 1℄ is an L1derivation. In higher dimensions, the range of the analogous map will no longer equal the domain, but will still be an L1-bimodule over it (see Example 10.4.1). Next we show that every Lip(X; ) is the domain of an L1 -derivation. This is where we need to use non - nite sets Y . L1 (Y ).

Example 10.3.2

Let X be a  - nite measure spa e and let  be a measurable pseudometri on X . For S; T  X positive measure sets de ne + (S; T ) = supf(S 0 ; T 0 ) : S 0

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 S; T  T g: 0

233 Then for ea h f 2 Lip(X; ) and ea h pair of positive measure subsets S; T  X su h that + (S; T ) > 0, de ne df 2 L1 (S  T ) by

df (x; y ) =

f (x) f (y ) : + (S; T )

(Note: this is not a pseudometri .) Also de ne l f (x; y ) = f (x) and r f (x; y ) = f (y ). Letting Y be the disjoint union of the spa es S  T , the maps l ; r : L1 (X ) ! L1 (Y ) make L1 (Y ) an L1 bimodule over L1 (X ), and the map d : L1 (X ) ! L1 (Y ) with domain Lip(X; ) is an L1 -derivation. Moreover, kdf k1 = L(f ) for all f 2 Lip(X; ).

Now we pro eed to prove the onverse of this example, namely, that the domain of any L1 -derivation is of the form Lip(X; ). Fix the following notation. X is a - nite measure spa e, L1 (Y ) is an L1bimodule over L1 (X ) (re all that we an assume E = L1 (Y )), d : L1 (X ) ! L1 (Y ) is an L1 -derivation, and L = dom(d). Observe that sin e L is isometri to the weak* losed subspa e d of the dual spa e L1 (X )  L1 (Y ), it is itself a dual spa e, with weak* topology de ned by the ondition that f ! f weak* in L if and only if both f ! f weak* in L1 (X ) and df ! df weak* in L1 (Y ). LEMMA 10.3.3

Let f 2 L be real-valued and let g 2 Lip[ a; a℄ where a = kf k1 . g Æ f 2 L and kd(g Æ f )k1  L(g )
kdf k1 .

If g is a polynomial then it is lear that g Æ f derivation identity then yields

PROOF

d(f n ) = (fln

Say g(t) =

1

2 L.

Then

The

+ fln 2 fr +


+ frn 1 )df:

P antn and de ne h 2 Lip([ a; a℄ ) by X a (sn + sn t +


+ tn h(s; t) = 2

n

1

2

1

):

We have h(s; t) = (g(s) g(t))=(s t) for s 6= t, so khk1 = L(g), but also d(g Æ f ) = h(fl ; fr )
df , so kd(g Æ f )k1  L(g)
kdf k1 as desired. Next suppose g 2 C 1 [ a; a℄ and let gn be a sequen e of polynomials su h that gn0 ! g0 uniformly on [ a; a℄ and gn (0) = g(0) for all n. Then gn ! g uniformly on [ a; a℄, so gn Æ f ! g Æ f uniformly in L1 (X ), and the sequen e (d(gn Æ f )) is Cau hy in L1 (Y ) be ause

kd(gm Æ f )

d(gn Æ f )k1  L(gm gn )
kdf k1 0 g0 k1
kdf k1 : = kgm n

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234

Chapter 10:

Lips hitz Algebras

Thus, norm losure of d implies that g Æ f 2 L and d(gn Æ f ) ! d(g Æ f ). Sin e L(gn ) = kgn0 k1 ! kg0k1 = L(g), the desired bound on kd(g Æf )k1 follows. Finally, for any g 2 Lip[ a; a℄ we have g0 2 L1 [ a; a℄, so we an nd a sequen e gn 2 C 1 [ a; a℄ su h that gn (0) = g(0), kgn0 ! g0 k1 ! 0, and kgn0 k1 ! kg0k1 . This implies that gn ! g uniformly, and we have kd(gn Æ f )k1  sup kgn0 k1
kdf k1 , so we an nd a subnet of (gn Æ f ) whi h onverges weak* in L. Sin e gn Æ f ! g Æ f uniformly, we on lude that g Æ f 2 L and kd(g Æ f )k1  L(g)
kdf k1. LEMMA 10.3.4 Let

f

2L

suppose

f

be real-valued, let


S = 0

. Then

S

be a positive measure subset of

(S )l df (S )r = 0.

X,

and

Let fn = 1 e nf . Then fn 2 L by Lemma 10.3.3, and a short omputation shows that L(f
fn )  2. Thus kd(f
fn )k1  2kdf k1. But f
fn ! f uniformly, and hen e weak* in L, and so d(f
fn ) ! d(f ) weak* in L1 (Y ) by weak* losure of d. Thus

PROOF

2

0 = (S )l [fl d(fn ) + d(f )(fn )r ℄(S )r = (S )l d(f
fn)(S )r ! (S )l df (S )r : So (S )l df (S )r = 0. LEMMA 10.3.5 Let

f; g

2L

be real-valued. Then

max(f; g); min(f; g) 2 L and

kd(max(f; g))k1 ; kd(min(f; g))k1  max(kdf k1; kdgk1): The fun tion t 7! jtj is Lips hitz on R, so max(f; g) = (f + g + jf gj)=2 and min(f; g) = (f + g jf gj)=2 belong to L by Lemma 10.3.3. We laim that jd(max(f; g))j  max(jdf j; jdgj) almost everywhere on Y ; this implies the desired inequality for d(max(f; g)), and the inequality for d(min(f; g)) an be proven similarly. Let S = fx 2 X : f (x)  g(x)g and T = fx 2 X : g(x)  f (x)g, and for  > 0 let S = fx 2 X : f (x)  g(x) + g and T = fx 2 X : g (x)  f (x) + g. Then de ne sets YS ; YT ; Y ; Y0  Y by the equations YS = (S )l (S )r , YT = (T )l (T )r , Y = (S )l (T )r , and Y = (T )l (S )r . We have 0 Y = YS [ YT [ Y [ Y ;

PROOF

0

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[

[







235 so it will suÆ e to prove the desired inequality separately on ea h of these sets. First, observe that (S )l d(max(f; g ))(S )r = (S )l df (S )r by Lemma 10.3.4. This proves that jd(max(f; g ))j  max(jdf j; jdg j) on YS , and the same is true in YT by similar reasoning. We will on lude the proof by verifying this on Y ; the argument for Y0 is the same. De ne  f g   S h = min max ;0 ;1 and



= min max

g



f

  ;1

:

Then hS ; hT 2 L and hS + hT is identi ally 1 on Lemma 10.3.4, we have

S

T

h




dk Y



;0

[ T .

Thus, using

= [d(k (hS + hT ))℄
Y = [kl (dhS ) + (dk )hSr + (hT )l (dk ) + (dhT )kr ℄
Y = [kl (dhS ) + kr (dhT )℄
Y = h(kl kr )

for any k 2 L, where h = (dhS )
Y . Examination of ases shows that jfl gr j  max(jfl fr j; jgl gr j) on Y , and hen e, with a double appli ation of the result of the last paragraph, we get

jd(max(f; g))j = jhj
j(max(f; g))l (max(f; g))r j = jhj
jfl gr j  jhj
max(jfl fr j; jgl gr j)  max(jdf j; jdgj) on Y . This ompletes the proof. THEOREM 10.3.6

Let (X; ) be a  - nite measure spa e, let E be an L1 -bimodule over L1 (X ), let d : L1 (X ) ! E be an L1 -derivation, and let L = dom(d). Then there exists a measurable pseudometri on X su h that L = Lip(X; ) isometri ally and L(f ) = kdf k1 for all f 2 L.

PROOF

De ne (S; T )

= supff (S; T ) : f

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2 L; kdf k  1g: 1

236

Chapter 10: Lips hitz Algebras

It follows from Lemma 10.3.5 and the fa t that L is a dual spa e that suprema and in ma of k
kD -bounded sets of real-valued fun tions in L also belong to L. Using this fa t, it is straightforward to verify that  is a measurable pseudometri , that L  Lip(X ), and that L(f ) = kdf k1 for all f 2 L. The desired on lusion follows from Theorem 10.2.7. On the basis of the pre eding result, we make the following de nition. Let M be a von Neumann algebra. A W*is a weak* losed self-adjoint subspa e E of a von Neumann algebra N together with a pair of weak* ontinuous unital homomorphisms l ; r : M ! N su h that l (); r () 2 E whenever  2 M and  2 E . We write l = l () and r = r (). An (unbounded) W*-derivation from M into a W*-bimodule E is a linear map d : L ! E su h that DEFINITION 10.3.7

bimodule over

M

(a) L is a weak* dense, unital -subalgebra of M; (b) d satis es d( ) = d() and d( ) = l d( ) + d() r for all ; 2 L; and ( ) the graph of d, d = f  d :  2 Lg, is a weak* losed subspa e of M  E . We give L = dom(d) the graph norm

f kD = max(kk; kdk):

k

A (non ommutative) Lips hitz algebra is the domain of some W*derivation. 10.4

Examples

In this se tion we des ribe various lasses of examples of W*-derivations and their asso iated Lips hitz algebras. We begin with the most important lass of ommutative examples. Example 10.4.1

Let X be a ompa t, onne ted Riemannian manifold. Thus X is a smooth manifold, and the tangent spa e at ea h point is equipped with an inner produ t. Let X be the omplexi ed tangent bundle over X as in Example 9.1.2 and regard X as a measurable Hilbert bundle. This an be done by writing X minus a null set as a disjoint union of nitely many open sets Xi (1  i  k) ea h of whi h is dieomorphi to an open subset of Rn . Then X  = (X1  Cn ) [


[ (Xk  Cn ). 1 Let L (X ; X ) be the Hilbert module of L1 se tions of X as in De nition 9.2.2; this spa e an be identi ed with the omplexi ation

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237 of the spa e of bounded measurable ve tor elds on X . Thus we may de ne d : Lip(X ) ! L1 (X ; X ) by d(f ) = rf , the gradient ve tor eld of f . We regard L1 (X; X ) as a W* Hilbert -bimodule. It an be realized as an L1 -bimodule as follows. Let Y be the set of all pairs (x; v ) su h that x 2 X and v is a omplex tangent ve tor at x of norm 1. That is, Y = (X1  S 2n 1 ) [


[ (Xk  S 2n 1 ) where S 2n 1 is the unit sphere of Cn . Then L1 (Y ) is an L1 -bimodule over L1 (X ) via the a tions fl g (x; v ) = gfr (x; v ) = f (x)g (x; v ). Finally, de ne T : L1 (X ; X ) ! L1 (Y ) by T (x; v ) = h(x); v i; this is an isometri embedding of L1 (X ; X ) in L1 (Y ) whi h respe ts the -bimodule stru ture.

The fa t that l = r is an important feature of the pre eding example; one says that L1 (X ; X ) is a monomodule over L1 (X ). This is related to the dierentiable hara ter of the metri spa e X . The same relationship persists in the non ommutative ase. Monomodule derivations an also be onstru ted by dierentiating one-parameter unitary groups, as in the next example. Example 10.4.2

2 (a) Re all the von Neumann algebra Lh1  (R ) of the quantum plane from De nition 6.6.1 and the automorphism s;t of Lh1 (R2 ). De ne 2 1 (R2 ) (i = 1; 2) by unbounded maps di : Lh1 ( R ) ! L  h 

d1 (A) = lim

1

!0 s

s

and

d2 (A) = lim

1

(s;0 (A)

A)

( ;t (A)

A);

!0 t 0 2 with domains the set of A 2 L1  (R ) for whi h the limits exist in the t

weak* sense. If f 2 S (R2 ) we have d1 (Lf ) = Lf =x and d2 (Lf ) = Lf =y ; this follows from Proposition 5.4.5. Informally, we an write d1 (A) = hi [P ; A℄ and d2 (A) = hi [Q; A℄ (see Se tion 4.4). 2 1 2 1 2 Now de ne d : Lh1  (R ) ! Lh  (R )  Lh  (R ) by setting d(A) = 2 d1 (A)  d2 (A), with domain Liph (R ) = dom(d1 ) \ dom(d2 ) and 2 1 2 range Lh1  (R )  Lh  (R ), regarded as a W* Hilbert -bimodule over 2 Lh1 ( R ) as in Example 9.6.5. Then Liph (R2 ) is a Lips hitz algebra.  (b) The same onstru tion applies to the quantum tori von Neumann 2 ^ algebras Lh1  (T ), using the automorphism s;t in pla e of s;t . We de ne Liph (T2 ) to be the domain of the orresponding derivation. h

In the pre eding example, setting h = 0 redu es to the map df = (f=x; f=y ), i.e., to Example 10.4.1. Putting this example into the format of De nition 10.3.7 requires that we be able to realize W* Hilbert -bimodules as W*-bimodules.

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238

Chapter 10: Lips hitz Algebras

2 Here this an be done in a non-isometri fashion by treating Lh1  (R )  1 2 Lh (R ) as a von Neumann algebra and letting l = r be the map A 7! (A; A). A ompletely general, isometri result will be given in Corollary 10.5.2. We an relate the Lips hitz algebras Liph (T2 ) to the harmoni analysis of quantum tori as developed in Theorem 5.5.7 and Proposition 6.6.4, for example. The relevant notation was given in De nition 5.5.4. 2 ^ h1 Let Liph^(T2 ) be the orresponding Lips hitz algebra inside L  (T ). We start with a straightforward result whose veri ation we omit. (Its rst assertion follows from Proposition 6.6.5, as ontinuity of the map (s; t) 7! ^s;t (A) is ne essary for d1 (A) and d2 (A) to be de ned.) PROPOSITION 10.4.3

We have Liph^(T2 )  C^ h (T2 ). For all s; t 2 R the map ^s;t restri ts to a -isomorphism from Liph^(T2 ) onto itself. This de nes an a tion of R2 by automorphisms of Liph^(T2 ). THEOREM 10.4.4

Let A 2 Liph^(T2 ). Then sN (A) ! A in norm.

PROOF

Let DN be the Diri hlet kernel,

X N

DN (t) =

=

n

eint =

N

sin(n + 1=2)t ; sin(t=2)

and let VN be the de la Valee Poussin kernel, VN = 2K2N +1

KN ;

where KN is the Fejer kernel de ned in the proof of Theorem 5.5.7. Let GN (s; t) = DN (s)DN (t) and WN (s; t) = VN (s)VN (t) and de ne s0 (A) = N

1

ZZ

1

ZZ

(2 )2

Also, observe that sN (A) =

(2 )2

^s;t (A)WN (s; t) dsdt:

^s;t (A)GN (s; t) dsdt:

We have kA s0N (A)k ! 0 by the argument in the proof of Theorem 5.5.7 whi h showed kA N (A)k ! 0, so it will suÆ e to show that ksN (A) s0N (A)k ! 0.

© 2001 by Chapman & Hall/CRC

239 Let BN = sN (A) s0N (A) and let ak;l = ak;l (A) and bk;l = bk;l (BN ) be the Fourier oeÆ ients of A and BN , respe tively. The bk;l have the following properties: (a) jbk;l j  jak;l j for all k; l; (b) bk;l = 0 if jk j; jlj  N ; and ( ) bk;l = 0 if max(jk j; jlj) > 2N + 1.

X

Therefore

jb j 

X

k;l

where XN = f(k; l) : N have k 2 + l2 > N 2 , so

2 N

(k;l)

k;l

ja j k;l

X

max(jk j; jlj)  2N + 1g. For (k; l) 2 XN we


0 there exists m 2 N and  2 Mm (V ) su h that kk = 1 and k(X Y )(m) ()k  kX Y k . Then T (m) 2 Mm (Lipmat (K )) and we have k((X ) (Y ))(m) (T (m))k = k(X Y )(m) ()k  kX Y k ;

whi h shows that k(X ) (Y )k = kX Y k, i.e.,  is an isometry. Surje tivity of  follows as in Theorem 8.4.6 (b) from the fa t that Vmat (K ) generates Lipmat (K ). COROLLARY 10.4.7

Let V and W be operator spa es and let K and L be the orresponding dual matrix unit balls. Then any ompletely ontra tive linear map from V to W extends uniquely to a ompletely ontra tive weak* ontinuous unital -homomorphism from Lipmat (K ) to Lipmat (L).

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242

Chapter 10: Lips hitz Algebras

PROOF Let T : V ! W be a ompletely ontra tive linear map and form T~ : Lipmat (K ) ! Lipmat (L) in the same way as in Corollary 8.4.7. The main point is to estimate the ompletely bounded norm of this map. Thus let F 2 Mn (Lipmat (K )). Then for any m 2 N and X; Y 2 Lm we have kX Y k  kT # X T # Y k, and hen e

kT~ (n)F (X ) kX

T~ (n) F (Y ) Y

k

k  kF (T #X ) kT #X

F (T # Y )

T #Y k

k:

This shows that L(T~ (n) F )  kd(n) F k. Also kT~ (n) F k  kF k from the 1 (K ) to lmat 1 (L) as fa t that T~ extends to a -homomorphism from lmat ( n ) ~ in Corollary 8.4.7. Thus kT F kD  kF kD , and we on lude that T~ is a omplete ontra tion. 10.5

Quantum Markov semigroups

In this se tion we will present a very general onstru tion that produ es W*-derivations of von Neumann algebras into W* Hilbert -bimodules. By analogy with Example 10.4.1 su h stru tures an be thought of as quantum Riemannian manifolds (or perhaps quantum sub-Riemannian manifolds, but we will not elaborate on this). They are the basis of \non ommutative (quantum) geometry." We rst address the Hilbert -bimodule aspe t of the onstru tion. Although in the von Neumann algebra setting the appropriate topologi al onditions on Hilbert -bimodules are that they be normal and dual (i.e., W* Hilbert -bimodules), in pra ti e one often rst onstru ts a Hilbert -bimodule without these onditions and then attempts to omplete it. Doing this requires that the un ompleted bimodule satisfy a

ertain topologi al ondition, but assuming this is the ase one an prove that a ompletion of the desired type does exist (Theorem 10.5.4). We will establish this fa t by using a \linking algebra" onstru tion. This te hnique also shows, in identally, that W* Hilbert -bimodules an be embedded in von Neumann algebras in a manner that is ompatible with their -bimodule stru ture. Thus, they an be viewed as W*-bimodules in the sense of De nition 10.3.7. Let A be a C*-algebra and E a Hilbert -bimodule over A. To produ e a right Hilbert module, let E0 = f 2 E : k kr = 0g and de ne Er to be the ompletion of E =E0 for k
kr . Then h
;
ir extends to an A-valued inner produ t on Er , and we use the same notation for the extension. Let E + = A  Er be the dire t sum of right Hilbert A-modules and let B (E + ) be the spa e of bounded adjointable right A-linear maps from E + to itself, as in De nition 9.3.4; this is a C*-algebra by Proposition 9.3.6. It is alled the linking algebra of the Hilbert module Er .

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243 PROPOSITION 10.5.1

Let A be a C*-algebra and let E be a Hilbert -bimodule over A. Then

()(  ) =    de nes a -isomorphism  : A ! B (E + ) and ( )(  ) = h  ; i   de nes an isometri linear embedding : E ! B (E + ). For every  2 A and ;  2 E we have () ( ) = ( ), ( )() = (), and ( ) = (  ). r

PROOF For any  2 A the map () is bounded by Lemma 9.6.2, and a short omputation shows that () = ( ), so  is a -homomorphism from A into B (E + ). It is learly inje tive. For any  2 E the map ( ) is learly A-linear, and it is bounded be ause h ( )(   ); ( )(   )i = h;   i h  ;  i + h ;  i  k  k2 h;  i + k k2   k k2 (h;  i +  ): This a tually shows that k ( )k  k k , and the onverse inequality follows from the omputations h ( )(h;  i  0); ( )(h;  i  0)i = h;  i3 (hen e k ( )k  k k ) and h ( )(0    ); ( )(0    )i = h  ;   i2 (hen e k ( )k  k  k = k k ). Also, ( ) is adjointable and in fa t ( ) = (  ) by the omputations h ( )(   ); (   )i = hh  ;  i   ;   i = h;   i + h ;  i and h(   ); (  )(   )i = h  ; h;  i    i =  h;  i + h;   i : Finally, it is trivial to he k that () ( ) = ( ) and ( )() = (). r

r

r

r

r

r

r

r

m

m

r

r

r

r

r

r

r

r

l

r

r

r

r

r

r

r

r

r

r

Given a W* Hilbert -bimodule E over a von Neumann algebra, the linking algebra onstru tion an be modi ed by using the dual module

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244

Chapter 10: Lips hitz Algebras

Er0

introdu ed in Se tion 9.4 in pla e of Er ; this has the onsequen e that B (E + ) is then a von Neumann algebra (Corollary 9.4.3), so that the onstru tion of Proposition 10.5.1 makes E a W*-bimodule over M in the sense of De nition 10.3.7. We re ord this fa t: COROLLARY 10.5.2

Let M be a von Neumann algebra and let E be a W* Hilbert -bimodule over M. Then E is a W*-bimodule over M.

Now if E is any Hilbert -bimodule over M, we an repla e it with the weak* losure of (E ) in B (E + ). In this way we an turn an ordinary Hilbert -bimodule into a W* Hilbert -bimodule, provided the original bimodule satis es a version of normality. We formulate this result next. LEMMA 10.5.3

Let M and N be von Neumann algebras and let A be a weak* dense -subalgebra of M. Suppose  : A ! N is a -homomorphism and suppose  ! 0 boundedly and weak* in A  M implies  ( ) ! 0 weak* in N . Then  extends to a weak* ontinuous -homomorphism from M to N .

PROOF Let M0 be the weak* losure of A0 = f  () :  2 Ag in M  N . Then the natural proje tion 1 : M0 ! M has zero kernel and hen e is a -isomorphism, the map 2 Æ 1 1 : M ! N is weak*

ontinuous, and the restri tion of this map to A agrees with . THEOREM 10.5.4

Let M be a von Neumann algebra, let A be a weak* dense -subalgebra of M, and let E be a pre-Hilbert -bimodule over A. Suppose that for any bounded net ( ) in A and any ;  2 E,  ! 0 weak* implies h ;  i ! 0 weak*. Then E densely embeds in a unique W* Hilbert -bimodule over M. In parti ular, if E is a normal Hilbert -bimodule over M then it densely embeds in a unique W* Hilbert -bimodule over M.

PROOF Here we use the on ept of a pre-Hilbert -bimodule over an un ompleted C*-algebra; this auses no diÆ ulties. Let E + = MEr0 be onstru ted as in Proposition 10.5.1, but using the dual module Er0 in pla e of Er , and let 0 : M ! B (E + ) and 0 : E ! B (E + ) be the orresponding maps. For a bounded net (A ) in B (E + ), weak*

onvergen e is equivalent to weak* onvergen e of hA ; ir in M for

© 2001 by Chapman & Hall/CRC

245 all ;  2 E + . Thus the lemma implies that there is a weak* ontinuous extension of 0 jA to M, and the ontinuity hypothesis implies that this extension must be 0. So 0 is weak* ontinuous. De ne E1 to be the weak* losure of 0(E ) in B(E + ). This is a bimodule over M = 0 (M) via operator multipli ation, and normality and duality are trivial. It is also straightforward to he k that the bimodule stru ture of E1 extends that of 0(E ) = E . The bilinear form and adjoint an either be extended from E by ontinuity or de ned dire tly by h; i  0 = 0  ( 0 (IM  0)) and operator adjoints. For uniqueness, let E2 be any other bimodule with the same properties and de ne a map T : E1 ! E2 by T (limE1 ) = limE2  for any bounded weakly onvergent net () in E . This map is well-de ned and ompatible with inner produ ts sin e h lim  ; i = lim h ; i = h lim  ; i E2 M E1

for any  2 E . It follows that E1 is unique up to isometri isomorphism.

We now present a general method for onstru ting W*-derivations. It involves Markov semigroups, whi h we de ne next. In this de nition we use the following terms. Let M be a von Neumann algebra and let  be a linear map from a -subalgebra of M into C. We say that  is a tra e if  ( ) =  ( ) for all ; , and   0 implies  ()  0; it is faithful if  > 0 implies  () > 0; it is normal if  ( ) ! 0 whenever () is a de reasing net that weak* onverges to 0; and it is semi nite if for every   0 in M there exists 2 dom( ) su h that 0   . A faithful, normal, semi nite tra e is alled an fns tra e. M be a von Neumann algebra. A C 0 semigroup of operators on M is a family of linear maps t : M ! M (t  0) su h that s t = s+t for all s; t  0 and the maps t 7! t () and  7! t () are weak* ontinuous (from R into M and from M into M, respe tively). The generator of a C0 -semgroup (t ) is the map Let

DEFINITION 10.5.5

t()
(a) = tlim !0 t



;

with domain the set of  2 M for whi h the limit exists in the weak* topology. A quantum Markov semigroup is a C0 -semigroup for whi h ea h t is ompletely positive and whi h satis es (a) 0 = idM , (b) t (IM ) = IM for all t, and

© 2001 by Chapman & Hall/CRC

246

Chapter 10:

Lips hitz Algebras

A1 = f 2 M :  2 D(
n ) for all ng is a -algebra. It is symmetri if there is an fns tra e  on M su h that dom( ) = A1 and  (t ( )) =  (t () ) for all ; 2 A1 . ( ) the set

The onstru tion of the asso iated W*-derivation goes as follows. Let (t) be a symmetri quantum Markov semigroup of operators on a von Neumann algebra M. For ; ; ; Æ 2 A1 de ne ( ) =   and h ; Æi = 
( )Æ; and extend both linearly to A1 A1 . We do not have h; i  0 on A1 A1 , but this does hold on the sub-A1 -A1 -bimodule E0 = spanf  : ; ; 2 A1 g: To see this, let  2 E0 and write  = Pn1 i i i i i i . Let 2 1 1 3 66 ... 77 6 77 B=6 66 n n 77 64 .1 75

..

and

n

C = [ 1




n

1 1






n n ℄ :

Then C t (BB)C   0 for all t  0 sin e t is ompletely positive, with equality when t = 0; dierentiating at t = 0 therefore yields C
(BB )C   0, whi h, written out, is the desired inequality h;   i  0. Next we apply Theorem 10.5.4. To verify its hypothesis, let  : M ! B (H ) be the GNS representation dedu ed from  . The ondition that  be fns implies that  is a weak* ontinuous -isomorphism. Thus if () is a bounded net in A1 and  ! 0 weak*, then  (   ) = h ; i ! 0 for any ; 2 A1 . Now let  =   and  = 0 0 0 0 0 0 and let Æ; Æ0 2 A1 . We have h(h ; i)Æ0 ; Æi =  (Æ h ; iÆ0 ); writing this out, we get  (Æ h(  ) ; (0 0 0 0 0 0)iÆ0 )

© 2001 by Chapman & Hall/CRC

247 =  (Æ 
(  0 ) 0 0 00Æ00 0 Æ
(  0 00 ) 0 0Æ00 0 Æ 
(   ) Æ + Æ 
(   ) Æ ) =  (  00
( 00 00Æ00Æ) 00 00
( 00Æ00Æ)


( Æ Æ  ) +  
( Æ Æ  )); and the latter onverges to 0 by the previous observation. Sin e this is true for all Æ; Æ0 2 A1 and  is one-to-one, we on lude that h; i ! 0 weak*. Taking linear ombinations, we nd that this holds for all ;  2 E0 , so the hypothesis of Theorem 10.5.4 is veri ed and we get that E0 densely embeds in a unique W* Hilbert -bimodule E over M. E plays the role of the module of bounded measurable 1-forms, and we have an exterior derivative d0 : A1 ! E de ned by d0 () = i(1   1): It is easy to he k that d0 is a derivation and d0( ) = d0() . Moreover, it is weak* to weak* losable be ause if ( ) and (d0 ()) are bounded and  ! 0 weak* then  (Æ hi(1   1); (  )iÆ0 ) = i (
( Æ0 Æ0 )  
( Æ0 Æ0 ) 
() Æ Æ + 
( ) Æ Æ )

onverges to zero for any ; ; ; Æ; Æ0 2 A1 , and similarly for the inner produ t in reverse order, whi h implies that i(1   1) ! 0 weak* in E by the same reasoning as in the last paragraph. Thus the

losure d of d0 is a W*-derivation and its domain is a non ommutative Lips hitz algebra. We summarize this result in the following theorem. THEOREM 10.5.6

( )

Let M be a von Neumann algebra and let t be a symmetri quantum Markov semigroup of operators on M. Then the map d M ! E

onstru ted above is a W*-derivation into a W* Hilbert -bimodule.

:

Example 10.5.7

Let be a ompa t, onne ted Riemannian manifold. Then there is a Lapla e operator on whi h an be written at ea h point as
( ) = 1( )


n( ) where 1 n is an orthonormal basis of the tangent spa e at . A symmetri Markov semigroup t = t
an be obtained by exponentiating the Lapla ian; this is known as the diusion semigroup on . Then A1 = 1 ( ), and arrying out the above onstru tion yields X

X

f x

f

v

f

x

v

v ;:::;v

x

x

e

X

C

X

© 2001 by Chapman & Hall/CRC

Chapter 10: Lips hitz Algebras

248

hdf; dgi = rf
rg for any f; g 2 C 1 (X ). Thus we re over the usual

rst-order exterior derivative on X des ribed in Example 10.4.1. Example 10.5.8

Let  be the weight (a tually an fns tra e) de ned in Example 5.6.7 (b) on the non ommutative plane algebra Lh1 (R2 ). A quantum Markov semigroup (t) an be de ned by exponentiating the Lapla e operator
= d21 d22 , where d1 and d2 are as de ned in Example 10.4.2 (a). Then the above onstru tion re overs the derivation given in Example 10.4.2. The non ommutative tori work similarly; here we take  to be the tra e de ned in Proposition 6.6.7. 10.6

Notes

A thorough treatment of Lips hitz algebras is given in [73℄.

Non om-

mutative geometry is dis ussed in [12℄; for the relation between that approa h and ours see Se tion V of [71℄.

Examples 10.4.1 and 10.4.2

are also treated in greater detail in [71℄. The operator spa e example dis ussed in Se tion 10.4 is from [70℄. The material of Se tion 10.5 follows [74℄. Our onstru tion of a non ommutative Lips hitz algebra from a symmetri quantum Markov semigroup is based on [64℄. See [66℄ for the general theory of fns tra es.

© 2001 by Chapman & Hall/CRC

Chapter 11

Quantum Groups 11.1

Finite dimensional C*-algebras

Quantum groups are the Hilbert spa e analog of topologi al groups, and thus their des ription naturally involves C*-algebras. (There is a von Neumann algebra version of the theory as well; it is more or less equivalent to the C*-algebra version, just as in the lassi al ase where there is an equivalen e between topologi al and measurable groups.) As we are going to dis uss quantum groups rst in the nite dimensional setting, we will begin by giving an expli it des ription of all nite dimensional C*-algebras. Here there is some room for variety, as the following example shows. Example 11.1.1

L

Let n1 ; : : : ; nm be natural numbers and let Mn be the matrix algebra of linear operators on Cn . Then m Mni is a nite dimensional i=1 C*-algebra whi h a ts on m Cni in an obvious way. Thus Mn1 i=1 is naturally realized as a blo k diagonal C*-subalgebra of Mn where n = n1 +


+ nm .

L

L

The onverse of this example is also true: every nite dimensional C*-algebra has the above form. THEOREM 11.1.2

Every nite dimensional C*-algebra is a dire t sum of matrix algebras. Let A be a nite dimensional C*-algebra; then it is also a von Neumann algebra. By Proposition 6.5.5, if A has a proper C*ideal I then it de omposes as A = P A  (IA P )A. We may therefore indu tively redu e to the ase that A is simple, i.e., has no proper C*ideals. PROOF

249 © 2001 by Chapman & Hall/CRC

250

Chapter 11: Quantum Groups

Say A  B (H). Let v be a nonzero ve tor in H; then K = Av = Av : A 2 Ag is a nite dimensional subspa e of H and A(K)  K, so the restri tion map takes A -homomorphi ally into B (K). Sin e A is simple, this map is a -isomorphism. In fa t, we may assume that the representation A ! B (K) is irredu ible, i.e., Aw = K for all nonzero w 2 K. If not, let w be a ounterexample and repla e K with Aw  K. Sin e K is nite dimensional, this pro ess must eventually terminate with a Hilbert spa e K that has no su h ve tors w. We may suppose A  B (K). Let B lie in its ommutant A and suppose B is self-adjoint. Let v be an eigenve tor for B , say Bv = v , and for any w 2 K nd A 2 A so that Av = w. Then f

Bw = BAv = ABv = Av = w so that w is also an eigenve tor of B , with the same eigenvalue. This shows that B = I , and we on lude that A = C
I . Then A = A

= B (K) by Theorem 6.5.7. We an also expli itly des ribe all *-homomorphisms between nite dimensional C*-algebras.

L

Example 11.1.3

L

Let A = pi=1 Mmi and B = qj=1 Mnj be two nite dimensional C*-algebras. For ea h j let ij1 ; : : : ; ijk (here k depends on j ) be a family of possibly repeating indi es su h that nj = mij +


+ mij . Then 1 k Mm j 


 Mm j naturally embeds in Mnj  B , and omposing i i 1

k

with the natural map from A into Mmij

1




 Mm j i k

gives rise to a

unital -homomorphism from A into Mnj . Taking the dire t sum of these maps produ es a unital -homomorphism from A into B.

PROPOSITION 11.1.4

Let A and B be nite dimensional C*-algebras and let  : A ! B be a unital -homomorphism. Then there are realizations of A and B as dire t sums of matrix algebras su h that  is expressed as a map of the form given in Example 11.1.3.

L

It suÆ es to onsider the ase B = Mn . Say A = p1 Mmi and for ea h i let Pi be the identity matrix in Mmi . Then the Pi are

ommuting proje tions and P1 +


+ Pp = IA . Sin e  is a unital -homomorphism, similar relations hold for Qi =  (Pi ). Thus, the Qi are proje tions onto a spanning set of orthogonal subspa es of Cn . Let Hi be the range of Qi and onsider the map i from

PROOF

© 2001 by Chapman & Hall/CRC

251 Mmi into B (Hi ) given by omposing  with the restri tion to Hi . This map is also a unital -homomorphism. Let fv1 ; : : : ; vr g be a maximal set of nonzero ve tors in Hi su h that As in Theorem K1 ; : : : ; Kr are orthogonal, where Kj = i (Mmi )vj . 11.1.2 we may assume that Mmi is irredu ibly represented on ea h Kj , and hen e that i takes Mmi -isomorphi ally onto ea h B (Kj ). Thus Hi = K1 


 Kr realizes Hi in su h a way that i takes the form A 7! A 


 A (r summands) on Mmi . Doing this for ea h i, we obtain a realization of B su h that  is in the desired form.

Finally, we de ne tensor produ ts of nite dimensional C*-algebras. DEFINITION 11.1.5 Let A  Mm and B  Mn be nite dimensional C*-algebras. For any operators A 2 A and B 2 B , de ne A B 2 Mmn by (A B )(v w) = Av Bw. Let A B be the C*algebra generated by all of the operators A B for A 2 A and B 2 B .

It is easy to see that a

ording to this de nition Mm Mn  = Mmn , and more generally

M i

11.2

=1

! 0M 1 0M  M A= M q

p

Mmi



j

=1

nj



mi nj

1 A:

i;j

Finite quantum groups

In this se tion we introdu e quantum groups in the nite dimensional setting. One might think that in this setting topology will be irrelevant, but this is not the ase be ause of the variety of possible C*-algebras of a given nite dimension. Of ourse, there is (up to -isomorphism) only one n-dimensional abelian C*-algebra, namely Cn , re e ting the fa t that there is only one Hausdor topology on an n-element set. But as we saw in the last se tion, there are many more, though not unmanageably many more, nite dimensional nonabelian C*-algebras. Now topology passes from the lassi al to the quantum setting by letting a nonabelian C*-algebra play the role of a \quantum C (X )." So in order to in orporate group stru ture into the pi ture, we must nd a way of expressing the group axioms in terms of fun tions on a group rather than dire tly in terms of its elements. The natural way to do this is to onsider the omposition of ontinuous fun tions with the group operations. For the inverse operation 1 : G ! G this yields a map from C (G) to C (G). Likewise, the group produ t  : G2 ! G gives rise to a map from C (G) to C (G  G)  = C (G) C (G). Finally, the identity

© 2001 by Chapman & Hall/CRC

252

Chapter 11: Quantum Groups

element may be regarded as an operation with no arguments, i.e., a map e : G0 G where G0 = . Composition with this map yields a linear fun tional on C (G). We now des ribe the quantum analog of the pre eding. A linear map T : between C*-algebras is an antihomomorphism if T ( ) = T ( )T () for all ; . Also, let  : be the identity map and (assuming is nite dimensional) let m : be the multipli ation map m( ) =  . !

f;g

A ! B

2 A

A ! A

A

A A ! A



Let A be a nite dimensional C*-algebra. A on A is a unital -homomorphism  : A ! A A whi h satis es the oasso iativity axiom

DEFINITION 11.2.1

oprodu t

( )  = ( ) 

Æ



Æ

as maps from A into (A A) A  = A (A A); A -homomorphism " : A ! C su h that

ounit is a unital

(" )  =  = ( ") ;

Æ



Æ

and an antipode is a unital antihomomorphism  : A ! A su h that

m Æ ( ) Æ  = "
IA = m Æ ( ) Æ : A nite quantum group is a nite dimensional C*-algebra equipped with a oprodu t, a ounit, and an antipode.

Why is the antipode, whi h plays the role of a oinverse, assumed to be an antihomomorphism? Be ause if we took it to be a -homomorphism, the only examples would be ommutative! In fa t, if we assume only that  is a linear map, it follows from the remaining axioms that  must be an antihomomorphism, and it is also automati ally surje tive. Thus, requiring  to also be a homomorphism for es to be abelian. When we pass to the in nite dimensional setting in Se tion 11.3, we will adopt an approa h that may be more satisfying; there  is left out of the axiomatization altogether, and its existen e and antihomomorphi nature is dedu ed only later. However, in the nite dimensional setting it is easier to simply assert the existen e of . The antipode intera ts with adjoints by the equation (() ) = . This an be proven from the above axioms (and it in identally shows that  must be surje tive). Our rst examples of quantum groups arise from ordinary groups. 

A

© 2001 by Chapman & Hall/CRC

253 Example 11.2.2

Let G be a nite group with identity element e. Then A = C (G) is a nite quantum group, with  : C (G) ! C (G G), " : C (G) ! C, and  : C (G) ! C (G) de ned by f (x; y) = f (x  y) "(f ) = f (e) f (x) = f (x 1 ):

And these are the only ommutative nite quantum groups. THEOREM 11.2.3

Let (A; ; "; ) be a nite quantum group and suppose the C*-algebra A is abelian. Then (A; ; "; ) is isomorphi to the quantum group C (G) for some nite group G.

Say A = C (G) where G is a nite set and identify A A with C (G  G). The oprodu t is then a unital -homomorphism from C (G) into C (G  G), and hen e it is given by omposition with a map  : G  G ! G by Proposition 5.1.9. The oasso iativity axiom then yields  Æ (  idG ) =  Æ (idG  ) as maps from G  G  G to G; that is, ((x; y); z ) = (x; (y; z )) for all x; y; z 2 G. Thus  is an asso iative binary operation whi h makes G into a semigroup. Likewise, sin e " is a -homomorphism it is given by evaluation at an element e of G, and a short omputation shows that the ounit axiom implies ex = xe = x for all x 2 G. So e is an identity element of G. Finally, although  is only assumed to be an antihomomorphism (and hen e a homomorphism in the present ase), the omputation (S ) = (2S ) = (S )2 shows that it takes hara teristi fun tions to hara teristi fun tions, and thus by linearity it is a tually a -homomorphism. Sin e we do not assume  is onto, this means that there is a subset S of G (namely, S = (1G )) and a map : S ! G su h that (f )jS = f Æ on S and (f ) is zero o P of S , for all f 2 C (G). Applying the antipode axiom to e then yields (x )y = 1G ; where the sum is taken over all pairs x; y 2 G su h that x  y = e; this implies that for ea h y 2 G there exists x 2 G su h that x  y = e and (y ) = x. From this it follows that S = G and is an inverse operation on G. PROOF

Next, we onsider the group algebra onstru tion. If the underlying group is nonabelian, this will involve a nonabelian C*-algebra and hen e no longer be a quantum group of the pre eding type.

© 2001 by Chapman & Hall/CRC

254

Chapter 11: Quantum Groups Example 11.2.4

Let G be a nite group and for x 2 G de ne a unitary translation operator Tx on l2 (G) by Txf (y) = f (x 1 y). Then Tx Ty f (z ) = Ty f (x 1 z ) = f (y 1 x 1 z ) = f ((xy) 1z) = Txy f (z); so Tx Ty = Txy . Let C  (G) be the C*-algebra generated by the operators Tx for all x 2 G. De ne , ", and  by (x ) = x x , "(x ) = hx ; e i, and (x) = x 1 . This makes C  (G) into a nite quantum group.

The quantum groups of Example 11.2.4 are dual to the quantum groups of Example 11.2.2 in the following sense. Let (A; ; ; ") be a nite quantum group. De ne A^ to be the dual ve tor spa e A with algebra and oalgebra stru ture given by

DEFINITION 11.2.5

!() = (! )(()) !() = !(()) IA^ = "A and

!( ) = !( ) !() = !( ) "(!) = !(IA ) for !;  2 A^ and ; 2 A.

More or less straightforward omputations show that this de nes a algebra stru ture on A and that , , and " satisfy the quantum group axioms. However, it is not obvious that A^ is a C*-algebra. This an be proven using the Haar state h on A, whi h we will onstru t in Se tion 11.4. Letting Hh be the asso iated GNS Hilbert spa e, one an show that the representation (!)() = ( !)() takes A^ -isomorphi ally into B (Hh ). This exhibits A^ as a C*-algebra. In any ase, by Proposition 3.2.4 the norm of a self-adjoint element of a C*-algebra an be omputed algebrai ally, and the equality kk2 = k k then determines the norm of an arbitrary element of the C*algebra. In this way we an determine the norm on the dual quantum group A^ without a tually arrying out the above onstru tion. But the fa t that this does produ e a C*-algebra (in parti ular, that the norm is not really a seminorm) still depends on that argument.

© 2001 by Chapman & Hall/CRC

255 Modulo the fa t that A^ is a C*-algebra, the veri ation of the following result is straightforward but tedious, so we omit it. THEOREM 11.2.6

If A is a nite quantum group then so is A^. There is a natural isomorphism of A with A^ . We also re ord the fa t, mentioned above, that Examples 11.2.2 and 11.2.4 are dual to ea h other. PROPOSITION 11.2.7

Let G be a nite group. Then C  (G) is naturally isomorphi to C (G)^. The proof of this result is straightforward; the desired isomorphism  : C  (G) ! C (G)^ is given by  (Tx )(f ) = f (x). Noti e that if G is abelian then C  (G)  = C (G)^ is also abelian, and ^ ) for some nite group G ^ by Theorem therefore is isomorphi to C (G ^ is the dual group, the set of all homomor11.2.3. In fa t, this group G phisms  : G ! T, with pointwise produ t (i.e.,  (x) = (x) (x)). One an easily verify that if G is a nite abelian group then there is a natural isomorphism between C  (G) and C (G^ ) that relates Tx with evaluation at x. ^ ). But when G is Thus, when G is abelian we have C (G)^  = C (G nonabelian the dual of C (G) is no longer of the form given in Example 11.2.4. So the quantum group setting allows us to generalize the notion of duality in a way that in ludes nonabelian groups. There are even more nite quantum groups besides those dis ussed above. We in lude one example, omitting the tedious veri ation that it a tually is a quantum group. Example 11.2.8

Let A = C  C  C  C  M2 and write a typi al element of A as (e ; a ; b ;  ; A). Also let e = (1; 0; 0; 0; 0), a = (0; 1; 0; 0; 0), b = (0; 0; 1; 0; 0),  = (0; 0; 0; 1; 0), Ue = I2 (the 2  2 identity matrix), and









Ua = 01 0i ; Ub = 0i 10 ; U =



1 0 0 1



:

Give G = fe; a; b; g  = Z=2  Z=2 a group stru ture by setting a2 = b2 = 2 = e and ab = . A is a C*-algebra, and we make it into a quantum group by de ning the ounit by "(e ; a ; b ;  ; A) = e ;

© 2001 by Chapman & Hall/CRC

256

Chapter 11: Quantum Groups

the oinverse by (e ; a ; b ;  ; A) = (e ; a ; b ;  ; AT ); and the oprodu t by X   (y ) = x x 1 y + 12 (uy )im (uy )jn ijmn x2G and

(A) =

X

x2G

x

1

Ux AUx + 21

X

x2G

Ux AUxT

x :

Here T denotes transpose, Uy = [(uy )ij ℄, and Uy = [(uy )ij ℄. 11.3

Compa t quantum groups

General quantum groups present several te hni al diÆ ulties that do not arise in the nite dimensional ase. For instan e, the multipli ation map m : A A ! A whi h appears in the antipode axiom is unbounded if A is in nite dimensional, and in some important examples the ounit and antipode are also unbounded. Consequently, there is room for argument as to what is the best general de nition of quantum groups; more restri tive onditions will ex lude some examples, but will presumably give rise to a more satisfying theory. In the ompa t ase, however, there is a de nition whi h both en ompases all of the most important examples and supports a ri h and satisfying theory. This approa h to ompa t quantum groups takes the oprodu t  as fundamental. Classi ally, this orresponds to starting with a semigroup. We then identify a lassi al ondition whi h for es a semigroup to be a group, and we use the quantum version of this ondition to omplete the de nition of ompa t quantum groups. Re all that an abelian C*-algebra is the algebra of ontinuous fun tions on a ompa t spa e if and only if it has a unit. Thus, the de nition of ompa t quantum groups will involve unital C*-algebras. Also, we will need to use tensor produ ts of C*-algebras. As in De nition 11.1.5, if A  B (H) and B  B (K) are C*-algebras then we de ne A B  B (H K) to be the losure of the span of the elementary tensors A B for A 2 A and B 2 B. This is alled the spatial tensor produ t C*-algebra and is a tually independent of the representations of A and B, although we will not need this fa t. It follows from the StoneWeierstrass theorem that C (X ) C (Y )  = C (X  Y ) for any ompa t Hausdor spa es X and Y . The following is the relevant lassi al ondition whi h guarantees that a semigroup (i.e., a set equipped with an asso iative binary operation) will be a group.

© 2001 by Chapman & Hall/CRC

257 PROPOSITION 11.3.1

Let G be a ompa t semigroup and let

A

= C (G).

Let

:

A ! A A

be omposition with the semigroup operation. Then G is a group if and only if the sets A A.

( ) (1G A




) and ( ) (

A

A




A

1G) are both dense in

PROOF Suppose G is a group. Then ( ) (1G ) is spanned by all ontinuous fun tions of the form (x; y) f (xy)g(y). It follows that this set is a -subalgebra of C (G G). To see that it separates points, let (x1 ; y1 ); (x2 ; y2 ) G G. If y1 = y2 then take f = 1G and

hoose g so that g(y1 ) = g(y2 ). If y1 = y2 and x1 = x2 then we must have x1 y1 = x2 y2 (this is where we use the fa t that G is a group), so let g = 1G and hoose f so that f (x1 y1 ) = f (x2 y2 ). This shows that ( ) (1G ) separates points, so it is dense in = C (G G) by the Stone-Weierstrass theorem. Density of ( ) ( 1G ) is proven similarly. Conversely, suppose the on lusion holds and let x1 ; x2 ; y G. Suppose x1 y = x2 y; then the points (x1 ; y) and (x2 ; y) are not separated by ( ) (1G ). Sin e the latter is dense in C (G G), we must have x1 = x2 . Similarly, from density of ( ) ( 1G) we an prove that xy1 = xy2 implies y1 = y2 . Now x x G and let H be the losed semigroup generated by x. Note that H is abelian. Let K be the interse tion of all losed nonempty semigroups J H su h that H J; J H J . If J and J are two su h semigroups then J J J J , so ompa tness implies that K is nonempty. Let y K ; then yK K , and minimality of K implies that yK = K (using the fa t that H is abelian to verify that yK is a semigroup and H (yK ); (yK )H yK ). Thus ye = y for some e K . For any z G, we then have yez = yz , and therefore ez = z by the last paragraph. Similarly ze = z for all z G. So G has a unit. Also, xe = x implies that x K sin e H K K . So xK = K , and therefore xz = e for some z K . As x was arbitrary, we on lude that G has inverses. So G is a group. A




A

7!





2



6

6

6

6

6

A




A A 

A

A






A

2

A




A



A




A

2



0



0



2

0

\





2

2

2

2



2

This motivates the following de nition of ompa t quantum groups. DEFINITION 11.3.2

algebra

A

A

ompa t quantum group is a unital C*-homomorphism  : su h

together with a unital

that

(



A ! A A

)  = ( )  (i.e.,  is oasso iative) and the sets ( ) (I ) and ( ) (



Æ



A


are dense in

A A.

© 2001 by Chapman & Hall/CRC

Æ

A

A

A
A IA

)

258

Chapter 11:

Quantum Groups

Thus neither a ounit nor an antipode omes into the de nition of

ompa t quantum groups. However, it is possible to dedu e the existen e of su h operations. Just as in the nite dimensional ase (Example 11.2.2), lassi al ompa t groups give rise to quantum ompa t groups for whi h the underlying C*-algebra is abelian. Example 11.3.3

Let be a ompa t group and let A = ( ). De ne  : A ! A A by  ( ) = (  ). Then (A ) is a ompa t quantum group. It satis es oasso iativity be ause  is asso iative, and it satis es the density onditions by Proposition 11.3.1. G

C G

f x; y

f x

y

;

The onverse also follows easily from Proposition 11.3.1: THEOREM 11.3.4

(A; ) (A; ) Let

be a ompa t quantum group and suppose

is isomorphi to

A

is abelian. Then

C (G) for some ompa t group G.

PROOF Say A = C (G) where G is a ompa t Hausdor spa e. Then  is a unital -homomorphism from C (G) to C (G G), and therefore  = C for some ontinuous map  : G  G ! G by Proposition 5.1.9. As in the proof of Theorem 11.2.3 the oasso iativity axiom implies that  is asso iative, so it makes G a semigroup. It then follows from Proposition 11.3.1 that G is a group.

We will now des ribe an interesting example of a ompa t quantum group, the quantum SU (2). It depends on a parameter q 2 (0; 1) and is denoted SUq (2). (A tually, one an also de ne it for other values of q.) The idea is this. The lassi al SU (2) group onsists of those 2  2 omplex unitary matri es with unit determinant. By the Stone-Weierstrass theorem, the algebra of ontinuous fun tions on SU (2) is generated by the \ oordinate" fun tions whi h evaluate at the four matrix entries. These four fun tions satisfy a small number of identities arising from the fa t that the matri es lie in SU (2), and it is possible to show that the algebra C (SU (2)) is the universal abelian C*-algebra generated by four fun tions satisfying these identities. In fa t, sin e two of the entries of any matrix in SU (2) are the omplex onjugates of the other two entries, C (SU (2)) is generated by only two fun tions. To de ne the quantum version of SU (2), we modify these identities in the following way. Let ij be operators for i; j = 1; 2. These orrespond to the fun tions whi h evaluate at the four entries of matri es in SU (2).

© 2001 by Chapman & Hall/CRC

259 In line with the ondition that matri es in SU (2) are unitary, we require 1i 1j + 2i 2j = i1 j1 + i2 j2 = Æij I ; the unit determinant ondition, however, is \twisted" to say X( q)jj  j j (1) (1) (2) (2) = ( q ) I 

where  is either permutation of f1; 2g and the sum is taken over both permutations  of f1; 2g. Here jj is the parity of . If q = 1 this formally redu es to the ondition that det[ij ℄ = 1. We have written these onditions in a way that suggests their generalization to n dimensions. In the ase n = 2 they an be simpli ed. Write  = 11 , = 12 , = 21 , and Æ = 22 . Then it is possible to dedu e that = q  and Æ =  , and the pre eding relations be ome   +  = I  + q2

 = I   = q   = q  

 =  : The C*-algebra SUq (2) is then the universal C*-algebra generated by two elements  and satisfying the above relations. Con retely, SUq (2) an be des ribed in the following way. Let H = l2(N  Z) and de ne operators A and C on H by p Avnk = 1 q2n vn 1;k Cvnk = qn vn;k+1 : Then SUq (2) is the C*-algebra generated by A and C . The quantum group stru ture of SUq (2) is given by setting () =   q  ( ) =  +  ; P whi h spe ializes the general pres ription (ij ) = k ik kj . It is straightforward to verify that the operators () and ( ) satisfy the same relations as  and ; thus, the universal property of SUq (2) implies that there is a unital -homomorphism  : SUq (2) ! SUq (2) SUq (2) taking the pres ribed values on  and . (This an also be he ked dire tly in the l2(N  Z) model.) It is straightforward to he k that  is oasso iative. The universal property of SUq (2) guarantees the existen e of a homomorphism " : SUq (2) ! C taking  to 1 and to 0. This is the

ounit. The antipode  satis es () =  ( ) =  ( ) = q (  ) = q 1  and it is de ned only on the -algebra generated by  and , not on the whole C*-algebra SUq (2).

© 2001 by Chapman & Hall/CRC

260

Chapter 11:

Quantum Groups

To verify the density ondition that is required of quantum groups, observe that ()
(I  ) + q2 (  )
(I ) =  I and ( )
(I  ) q( )
(I ) = I: Thus  and both belong to (SU (2))
(I SU (2)). Taking adjoints in the pre eding omputations shows that  and  belong to this set as well. Moreover, for any ~ ; ~ 2 SU (2) su h that q

X ~ I = (~ )(I ~ )

q

~ I =

and

0

i

q

i

X ( ~ )(I ~ ) 0

j

j

j

i

belong to (SU (2))
(SU (2) I ), we have ~ ~ I = (~ I )( ~ I ) = (~ )(I ~0 )( ~ I ) = (~ )( ~ I )(I ~0 ) q

q

X X X = (~ )( ~ )(I ~ )(I ~ ) X = (~ ~ )(I ~ ~ ); i

i

i

i

i

i

i

0

j

j

0

i

i;j

i

0 0

j

j

i

i;j

so that ~ ~ I 2 (SU (2))
(I SU (2)) as well. It follows that SU (2) I is ontained in the losure of (SU (2))
(I SU (2)), and thus the latter is dense in SU (2) SU (2). This veri es one of the density onditions; the other is similar. We have shown the following. q

q

q

q

q

q

q

THEOREM 11.3.5

SU (2) is a ompa t quantum group. q

11.4

Haar measure

We already mentioned Haar measure in Se tion 9.5. On a ompa t group it an always be normalized to be a probability measure. Thus integration against Haar measure an be des ribed as a left invariant state on C (G). This des ription still makes sense in the setting of quantum groups, where we all it a \Haar state." In this se tion we will prove that every ompa t quantum group has a unique Haar state. The orre t notion of invarian e is given in the following de nition. We in lude the de nition of right invarian e be ause we will prove that,

© 2001 by Chapman & Hall/CRC

261 as in the lassi al ase, the Haar state on a ompa t quantum group is also right invariant. Let (A; ) be a ompa t quantum group and ! is left invariant if ( !) Æ () = !()IA for all  2 A, and it is right invariant if (! ) Æ () = !()IA for all  2 A.

DEFINITION 11.4.1 let

! 2 A .

We say

Haar state

A

is a left invariant state.

To motivate this de nition, onsider the lassi al ase where A =

C (G). Here left invarian e of a measure  is equivalent to the ondition

that

Z

f (xy) d(y) =

Z

f (y) d(y) G G for all f 2 C (G) and x 2 G. Now f (x; y) = f (xy), so this ondition

an be rewritten as

Z

G

f (x; y) d(y) =

Z

G

f (y) d(y):

Instead of regarding this as a separate ondition for ea h x, we an regard both sides as fun tions of x. Then letting ! be integration against , the right side be omes !(f )
1G and the left side be omes ( !)(f ). We pro eed to prove the existen e and uniqueness of a Haar state. Let (A; ) be a ompa t quantum group and de ne a produ t on the dual spa e A by setting !() = (! )(()): LEMMA 11.4.2 Let

! 2 A

PROOF

be a state. Then there is a state

h su h that h! = !h = h.

De ne 1

!n = (! + !2 +


+ !n) n

and let h be a weak* luster point of the sequen e (!n ). Then ea h !n is a state, so h is a state, and k! ! ! k = k!! ! k  2 n

implies h! = !h = h.

© 2001 by Chapman & Hall/CRC

n

n

n

n

262

Chapter 11:

LEMMA 11.4.3

!; h 2 A be states then h = (IA )h.

Let

satisfying

!h = h.

If



Quantum Groups

2 A

and

0!

PROOF Let  2 A and set 0 = ( h)(()). Then ( !)((0 )) = ( !h)(()) = ( h)(()) = 0 ; whi h by a short omputation implies (h !)(((0 ) 0 I ) ((0 ) 0 I )) = 0: Now h ! is a state on A A, so the Cau hy-S hwarz inequality in the asso iated GNS Hilbert spa e implies that (h !)(( )
((0 ) 0 I )) = 0 for all ; 2 A. Repla ing 0 with ( h)(()) yields (h ! h)(( I )
( )()) = !( )(h h)(( I )
()): But ( I )
( )(()) = ( I )
( )(()) = (I I )
( )(( I )
()) by oasso iativity. By the density ondition whi h de nes ompa t quantum groups, we an repla e ( I )
() with I Æ for an arbitrary Æ 2 A in the previous two results to get (! h)(( I )
(Æ)) = !( )h(Æ): Passing to the GNS representation  asso iated to ! and letting Æ0 = ( h)(Æ), the pre eding be omes h(Æ0 )I; ( ) Ii = h(Æ)hI; ( ) Ii; so we must have (Æ0 )I = h(Æ)I. Sin e 0    ! the map ()I 7! () is bounded, so there exists v 2 H! su h that () = h()I; vi for all  2 A. In parti ular, h(Æ) = (Æ0 ) = h(Æ0 )I; vi = h(Æ)hI; vi = (I )h(Æ); and we on lude that h = (I )h. THEOREM 11.4.4 Let

(A; )

state

be a ompa t quantum group. Then there is a unique Haar

h on A.

Moreover, it is right invariant.

© 2001 by Chapman & Hall/CRC

263 PROOF For any ! 2 A , !  0, let S! be the set of states h on A su h that !h = !(I )h. This set is nonempty for any su h ! by Lemma 11.4.2. It is straightforward to he k that S! is weak* ompa t, and Lemma 11.4.3 implies that S!  S if   !. This implies that the interse tion of nitely many S! (1  i  n) ontains S!1 + +! , and hen e is always nonempty. Thus there exists a state h whi h belongs to every S! . So (! h)(()) = !(I )h() for all  2 A and all ! 2 A , and this implies that ( h) Æ  = h
I . So h is left invariant. A similar argument shows that there exists a right invariant state h su h that h ! = !(I )h for all ! 2 A . But then h h = h(I )h = h and (by the same property for h) h h = h (I )h = h, so h = h. This shows that h is right invariant and also shows that h is unique. 

A






i

n

A



A

0

0

A

0



0

0

0

A

0

0

0

A

The Haar state on SUq (2) an be expressed in terms of its representation on H = l2 (N  Z) given in Se tion 11.3. We have h(A~) = (1 q2 )

X q2 hAv~ 0; v 0i 1

n=0

n

n

n

for all A~ 2 SUq (2). 11.5

Notes

The material of Se tion 11.1 an be found in most standard referen es on C*-algebras. Se tion 11.2 is based on [39℄. Our treatment of ompa t quantum groups in Se tions 11.3 and 11.4

losely follows [46℄. The quantum group SUq (2) was introdu ed in [78℄ and most of its properties were established there. The fa t that the representation given on l2(N  Z) is faithful (i.e., that the operators  and de ned there are universal) follows from an analysis of the irredu ible representations of SUq (2); see [67℄. The formula for the Haar state on SUq (2) is proven in [79℄.

© 2001 by Chapman & Hall/CRC

© 2001 by Chapman & Hall/CRC

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© 2001 by Chapman & Hall/CRC

© 2001 by Chapman & Hall/CRC