Strong Limit Theorems in Non-Commutative Probability (Lecture Notes in Mathematics, 1110) 354013915X, 9783540139157

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

1110 Ryszard Jajte

Strong Limit Theorems in Non-Commutative Probability

Springer-Verlag Berlin Heidelberg New York Tokyo 1985

Author

Ryszard Jajte Institute of Mathematics, University of .r.6dz Banacha 22, 90-238 -t.6dz, Poland

AMS Subject Classification (1980): 46L50, 46L55; 28005, 60F15 ISBN 3-540-13915-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13915-X Springer-Verlag New York Heidelberg Berlin Tokyo

This work IS subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140·543210

PREFACE Recently many authors have extended a series of fundamental pointwi se convergence theorems in the theory of probabi 1i ty and ergodic theory to the von Neumann algebra context. They have rr­o v l de d some new tools for mathematical physics and at the same time created interesting techniques in the theory of operator algebras. The main purpose of these notes is to present a self­contained exposition of some ideas and results from this area. We shall confine ourselves to the case of von Neumann algebras ahd shall not touch on the problems concerning C*­algebras. One of the reasons for this is that we are trying to keep the book on a relatively elementary level. The material presented here has been chosen in such a way that only very little knowledge of the theory of operator algebras is needed for reading it. On the other hand, the von Neumann algebras are very natural non­commutative generalizations of Loo­algebras, and their rich structure gives the possibilities to obtain the limit theorems in their "almost sure" versions. In a von Neumann algebra one can introduce the "almost uniform" convergence which, in the classical commutative case of the algebra Loo' is equivalent (via Egoroff's theorem) to the almost sure convergence. This type of convergence will be fundamental for the whole book. Recently, C. Lance proved a non­commutative version of the individual ergodic theorem for *­automorphisms of a von Neumann algebra. From the point of view of applications in quantum dynamics this result is of great importance. Chapter 2 is devoted to the discussion of some results of this kind and their generalizations. In particular, we prove some "individual" ergodic theorems for normal positive maps of a von Neumann algebra, the non­commutative versions of Kingman's subadditive ergodic theorems for *­automorphisms, a random ergodic theorem and a non­commutative local ergodic theorem for quantum dynamical semigroups. Chapter 3 is devoted to the theory of martingales in von Neumann algebras. Conditional expectations in von Neumann algebras and martingale convergence theorems are important in particular in the theory of measurement in quantum mechanics. The non­commutative martingale convergence theorems of N. Dang­Ngoc and M. S. Goldstein will be proved. Chapter 4 deals with the strong laws of large numbers in the context of von Neumann algebras. Among others, Batty's results will be presented. Chapter 1 has a preparatory character. In it we shall discuss some properties of the almost uniform convergence in von

IV

Neumann algebras. These notes do not cover, of course, all the results concerning the almost uniform convergence in von Neumann algebras. In particular, we do not discuss the ergodic theorems for weights, only for states. This book is written mainly for a reader familiar with the theory of probability but may be of some interest for those mathematicians and physicists who are interested in some techniques of operator algebras and their applications. As we mentioned, our main goal is to present some ideas which lead us from the classical results well-known in the probability theory to their non-commutative versions, and, consequently to the applications in quantum field theory. In this sense our book is "homogeneous". Most of the results presented here have been obtained recently (Lance 1976-78, Kummerer 1978, Dang-Ngoc 1978, 1982, Yeadon 1975-1980, Watanabe 1979, Goldstein 1981, and others). Some theorems will be proved for states and some of them only for traces. Since a state, in general, is not subadditive on the lattice of projections, the techniques for non-tracial states are, as a rule, quite different from those used for traces, and are also much more difficult. It is worth noting here that very often the arguments needed for traces are similar to the classical ones; but in some cases, a new approach is necessary. The prerequisites for reading this book are the fundamentals of functional analysis and probability. The elements of the theory of von Neumann algebras are collected in the Appendix. We refer also to the Appendix for the terminology and notations used in the book. The Appendix is almost self-contained and can also be read separately, before studying the main chapters. These notes were prepared during my stay at the University of Tennessee in Knoxville and at the Center for Stochastic Processes at the University of North Carolina at Chapel Hill. I am deeply indebted to all my colleagues from both these ingtitutions for the warm hospitality they extended to me. Many thanks are due to Ms. Cindi Blair and Mrs. Margaret Ravitch for their patient and skillful typing of the first draft of the manuscript. I sincerely wish to thank Mrs. Ruth Bahr who took great care in the typing of the final version.

Chapel Hi 11, October 1984

R. Jajte

CONTENTS INTRODUCTION . . . CHAPTER 1. ALMOST UNIFORM CONVERGENCE IN VON NEUMANN ALGEBRAS 1.1 Preliminaries... . . . . . . . . . . . . . 1.2 Various kinds of 'almost sure' convergence in von Neumann algebras. . . . 1.3 Non-commutative version of Egoroff's theorem 1.4 Notes and remarks . . . . . . . . . . CHAPTER 2. ERGODIC THEOREMS 2.1 Preliminaries.. . . . . . . . . . . . . . . 2.2 Almost uniform ergodic theorems. . . . . . . 2.3 Non-commutative subadditive ergodic theorems. 2.4 Random ergodic theorem. . . . . . .... 2.5 Local ergodic theorem for quantum dynamical semigroups. . . . 2.6 Notes and remarks

. 1

4 5 8 9 .11 .12 .32 .47 .52 .55

CHAPTER 3. 3.1 3.2 3.3

CONVERGENCE OF CONDITIONAL EXPECTATIONS AND MARTINGALES IN VON NEUMANN ALGEBRAS .57 Conditional expectations Almost uniform convergence of conditional expecta.64 tions and martingales .80 Notes and remarks

CHAPTER 4. 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

STRONG LAWS OF LARGE NUMBERS IN VON NEUMANN ALGEBRAS Preliminaries Independence Almost complete convergence in von Neumann algebras Strong limit theorem for orthogonal sequences A non-commutative extension of Glivenko-Cantelli 's theorem Generalized Kolmogorov's inequality for traces and some of its consequences. Strong laws of large numbers for traces Convergence rate in the strong law of large numbers Notes and remarks

.81 .81 .82 .83 .87 .90 .91 100 105

VI

APPENDIX , 1. Generalities, commutant, bicommutant theorem, A-invariant subspaces . Positive linear functionals, GNS-representation, pure 2. states and irreducible representation . 3. Partial isometries, polar decomposition, equivalence of projectors . 4. Locally convex topologies on B(H) 5. Hilbert-Schmidt and trace-class operators, predual of a von Neumann algebra . 6. Normal positive linear functionals, traces, normal positive linear mappings, Kadison's inequality, singular positive linear functionals, metrization of the strong topology in the unit ball of a von Neumann algebra. 7. Measure topology, non-commutative Lp-spaces The algebra LoJr 0 and q E Proj A, there is a projection rEA such that r < q, ¢(q-r) < E and [Ix n r l l < E for l a r oe enough.

Proof:

Indeed, let 0 < En O. By (i) we can find a sequence (r n) c Proj A with ¢(r.L) < E and a sequence of positive integers m(n) such that n n Ilxmrnll < En for m > m(n). Let q Proj A be ql ve n , Then, by the normality of w, 0 and we can fix no such that E < E no and q) < E". Putting r = q[0,E 2), we have r < q, ¢(q-r)




m(n o)'

0 and assume

7

that (i) holds.

By (*), we find a sequence (Pn) nE 1 = Pl':::' P2 .:::. ... ,

m(n).

Put P = inf Pk· Then /, m(n o)' This means that xm ->- a almost uniformly. 0 By an easy modification of the above proof we can show the implication (i) ->- (iv). Namely, for a given a f P t Proj A and E > a we find a sequence of projections P = Pl .:::. P2 .:::. ... with IlxmPnl1 < E for m > m(n) and 0, there exist pro j e c t i on s Pn inA wi t h Pn t 1 and I I xnPn I I < E for all n . Then ¢(p m) -> l-E for n > m which means that (i) holds so that (iii) implies (i ).

It remains to prove the implication (iv) ->- (iii). Let E > 0, Ek < Ek+l ->- E and a < ok ->- O. To show that (iii) holds it is enough to find an increasing sequence (qk) c Proj A and an increasing sequence of positive integers such that < ok and I !xnqkl I < E k for n.:::. n k (then we can put P1 = ... = P = 0, Pn1+l = ... = P 01' nl n2= etc. Thus it is enough to show that i f E < E', > a and !!xnP!1 < E for n > f, where P t Proj A, then there exist q E Proj A and L'> f such that q.:::. p, < Ii and Ilxnqll < E' for n > f'. Let (P t, t eT) be a maximal family of mutually orthogonal projections in A such that Pt and I IxnPtl I + a as n ->- oofor every t E T. This family is at most countable (because there is a faithful normal state ¢ on A; c omp . Appendix). Since ¢ is normal and faithful, from (iv) it follows that there exists a seuuence (P k) of mutually orthogonal projections in A such that 'f Pk and IlxnPkl1 ->- a as n ->for k=l ,2, . . . . k= 1 N Taking N large enough we obtain - L Pk) < and, consequently, N k=l < Ii for q = P + L Pk' Ilxnqll < E' for n sufficiently k=l large. The proof is completed. 0

a


- (i), (iii) ->- (i) and (iv) ->- (iii) is the same as in Theorem 1.2.1, so it remains to prove the impl ications (i) ->- (iv) and (iv) -s- (ii). Let (i) hold, and let 0 f P E Proj A. Put Ek = 2 - k - 1 ¢(p ) > O. We find qk E Proj A such that n I k) . as n

-r

00,

Putq=P,,!\qk' k=l + Moreover
0 there exists a projection pEA such that ¢(l-p) < E and IIp(xn-x)pll -+ 0 as n -+ "". We omit the formulation of natural versions of the conditions (i), (iii) and (iv). Theorem 1.2.1 and Theorem 1.2.2 hold also for the bilateral version of the conditions (i)-(iv). We leave the proof to the reader.

1.2.4 DEFINITION.

A sequence (x in A (in A i f n) to be bilaterally almost uniformly convergent to x

1.3. Non-commutative version of Egoroff's theorem. We start with the following proposition 1.3.1. PROPOSITION

A be a von Neumann algebra acting in a Hilbert (xi) in A converges strongly to X o then, for every E > 0, there exists a sequence (Pi) c Proj A such that Pi -+ 1 strongly and II(xi-xO)Pi l l < E for i=l,2, ...

space

H.

Let

If a sequence

We may assume II \ II :5.- 1 and "o = O. Put Yi = xixi. Then, for every hE H, we may have IIYihl1 = Ilxixihll :5.- Ilxill Ilxihll :5.- Ilxihll,

Proof.

9

(Yi) converges strongly to zero. Put Pi= ei([O,[ 2 J), where denotes the spectra 1 measure of y.. Then 1

1

y.1 = f u e 1. ( d u ) ->

f

[2

o

[[2,lJ

[

e . ( d u) 1

>

[2 (

which impl ies t ha t ( Pi) converg es strongly to 1 .

1 - P l'

),

t-1oreover, we have

IIxiPil12 = IIPixixiPl1 < IlxixiPi II = II YiP i II < which ends the proof.

£: 2 ,

0

1.3.2 THEOREM (Non-commutative Egoroff's theorem). Let A be a von Neumann algebra with a faithful normal state ¢. Let (x n ) be a sequence in A converaent to x in the strong operator topol09Y. Then, for every projection pEA and any [ > 0, there exists a projection q p in A and a subsequence (x ) of (x ) such that ¢(p-q) < [ and 1 I (x -x)ql I nk

-;-

0 as k -;-

nk

n

00

l·le may assume that P there exists a sequence (P n) and Pn -;- 1 strongly. Choose n > "r Put ql = P Then nl'

Proof.

= 1 and x = O. By Proposition 1.3.1, of projections in A such that IlxnDnlln l such that

< clZ

( 1)

Putting 02 = On

' I'le have q2

2

But Ilxnq2112=

n

2

n

and

< 1/2

2

q1' q,(Q2- ql) < [/2

=

(1)

and llYn

2

= \

2.

1

q211 n 2' Consequently Ilx m q2 11 < 1/2 2. By induction we obtain a decreasing sequence (qn) of projections in A and a sequence of indices n l < "z -

Putting q = inf q, we qet ¢(l-q) < [ and k

0, which completes the proof.

0

1.4 Notes and remarks. The' pointwise' convergence in von Neumann algebras discussed in Chapter 1 was introduced first by I. Segal [117J and has been used systematically in the so called 'noncommutative

10

theory of integration'. This theory has been much developed by Segal [117] and Dixmier [30] independently for semi-finite traces. Now there exists the theory of 'integration' not only for traces but also for states and weights. Some information and indications to the literature can be found in the Appendix. We note here that the relationships between many kinds of convergence in von Neumann algebras have been discussed by Segal [117], Ogasawara and Yoshinaga [94], Padmanabhan [95], Lance [82], Stinesprinq [121], Batty [11] and, very recently, by Petz [103] and Paszkiewicz [98]. In particular, D. Petz [103] introduced the notion of the quasi-uniform convergence and A. Paszkiewicz [98] discussed the relationships between some kinds of 'pointwise' convergence for unbounded sequences of operators. He proved,among other things, that, for bounded sequences, the almostuniform convergence coincides with the quasi-uniform convergence. Some parts of the proof of Theorem 1.2.1 are taken from [98]. In f981 the author uses very successfully the Halmos' theorem" on two subspaces" [55]. The non-commutative version of Egoroff's theorem is due to Saito [114]. Closely related to the problems discussed in this chapter is the paper by Radin [107] in which the author introduced the notion of "¢-almost every state on a C*-algebra A, where ¢ is a trace on A" (bicommutant). He proves that any *-automorphism of A" is implemented by some point transformation in the state space of A, defined ¢-almost everywhere. A more general situation (when ¢ is an arbitrary state) is considered by Luczak [142J.

CHAPTER 2 ERGODIC THEOREMS 2 . 1. Pre 1 i min a r i e s

Let us assume that the "kinematical structure" of a physical system is given by a von Neumann algebra A with a faithful normal state ¢ (see INTRODUCTION). Then the dynamics of the system is described by a continuous one­parameter group (at) of *­automorphisms of A. As a rule, this group satisfies some conditions related to ¢ (for example, ¢ is at­(sub)invariant). In comparison with the classical situation, where the "kinematical structure" is described by a differentiable manifold U, the group (at) corresponds to the group (Bt) of diffeomorphisms of U (see for example [85]). If there exists a measure ]1 on the manifold U which is Bt­invariant (for compact manifolds such a measure always exists), then by the pointwise ergodic theorem of Birkhoff, the limit of the ergodic averages n -1

lim n­ l I BtF n­>­oo t=O exists almost surely for all ]1­integrable functions (observables) on U. Using the language of the von Neumann algebras theory, the Birkhoff theorem (for bounded functions) in its simplest form can be formulated in the following way. Let A = ILoo(rI, B, p) be a von Neumann algebra of all bounded measurable functions over a probability space (rI, B, p). If a is a *­automorphism of A given by a measure preserving automorphism of rI, then the limit of the ergodic averages akf exists almost uniformly for all f E A, where

As we shall see, this formulation is suitable for the generalization to the non­commutative von Neumann algebra context. An irreversible dynamical system is described by a quantum dynamical semigroup, i.e. by a continuous one­parameter semigroup (at) of normal positive maps of a von Neumann algebra A (see for example [41]). That is why it is reasonable to consider the ergodic theorems not only for groups of *­automorphisms but also in a more general situation: for normal positive linear maps of a von Neumann algebra. The information about the asymp­ . , . } -1 \.n­l k. tOtlC behav10r of the ergod1c averages sn(a = n Lk=O a lS of great

12

importance. In this chapter we shall prove several theorems on the almost uniform convergence of the ergodic means sn(a), where a is a normal positive map of A into itself. Z.Z.

Almost uniform ergodic theorems

In this section we shall prove several theorems concerning the almost uniform convergence of the ergodic averaqes. We start with a non-commutative analogue of the mean ergodic theorem. Let us remark that there exist many versions of the mean (statistical) erqodic theorem in von Neumann algebras which have important applications in mathematical physics. We formulate here one of them in its simplest form (for a more qeneral situation see for ex. [71J). Z.Z.l THEOREM (Mean ergodic theorem). Let A be a von Neumann algebra with a faithful normal state ¢; in A with o l

¢(ax)


- x in ll..z(As'CP). We have sn(x j ) ->- x j in IL by Theorem Z.Z.l.

ppoof.

For

II s n

o sn(x)

(x k

=

> 0, we find k

- x k liz < o x , we obtain

o /Z.

and no' such that Ilx-x k liz < Z

Z

0

Of course, here Ilxllz = cp(x).

/Z and Since

0

Ilx-x k liz < Iis n (x)-sn (x k )ll z + Iis n (x k )-x k liz < o 0 00 000 for k large enough. o Taking into account the fact that all xk's are a-invariant, we obtain the formula we are looking for. 0 A

Z.Z.5.

PROPOSITION.

elements a

l,a Z,a 3,a 4

For each x from A+

element y from A such that x-x iah e r e

X

Ppoof.

A and

> 0 one can find foup 2 (i = l,Z,3,4) and an (a, -a + i(a - a + (y - ay), z) 3 4)
0 there exist elements a a l, Z yEA such that (* )

A+ with

As and
O we obtain the equality (*) with l Z Z Z 1cp(3 } < . 0 i

Z.Z.6. PROPOSITION.

the assumptions about a as in the theorem,

there exists a linear positive map a': (i ) (i i )

( iii) Proof.

A'

->-

A'

such that

0.'1 < 1

(a' (y)£,,1;) < (y£"t;) fop 0 .2 YEA' (a(x)Y£,,1;) (xc ' (y)£"t;) fop all X

E

A, YEA'.

It is a simple consequence of lemma (A.6) (in Appendix).

In

15

fact,it is enough to put (for a fixed 0 YEA'), = = = Then by (A.6) there exists Z E A' such that 2cU. 2 = (xlzI Put IzI = a'(y). The standard reasoning completes the proof. D 2.2.7. PROPOSITION.

A and

from

Let (a.).

let E. > 0 (i 1

1

E

1E

I).

I be a family of positive operators Assume that q is a projector from A'

qA'q implies (a.xCU < E.(XCU . 1 1 there exists some p E Proj A such that ¢(p) .:::.

such that 0 for i

E

x

E

(i E I). Then and IlpaiPI

I.

Ei

It is enough to choose p as the projection onto the subspace X = [rq; rEA']. In fact, we then have pEA (because A'X = X). Moreover, for rEA', we have (pa.prqCrqU = = 1 1 1

Proof.

=

< 1 = 1 Thus pa.p < E.l over 1 ­ 1 E A1. On the subspace paiP = O. Consequently, pa i P < Eil and Ilpa.pll < E. follows. Evidently, ¢(p) = _> D 1 1 1

­

PROPOSITION. Let 0 < X E A and 0 < rEA', such that for all 0 y r, y ( A'. If q = support r, then for each 0 y E qA'q, the inequality holds. 2.2.8.

The proof is standard. If 0 y cr for some c > 0, (y E A'), then c­ly r and also Let 0 yE qA'q and qn = = the value of the spectral measure of r on the interval (l/n,oo). Then q n­ < nr and q n -+ q (strongly). Also y ­< Ilyll q, and q ny q n­ 0 be given and let < E for all i=1,2,3,4, with ¢(p) .:::. 1 - 8E and such that lis n (a.)pll 1 00

and n=1,2, . . . .

This is the key point of the proof and it differs from Goldstein's ([47J, Theorem 1.2) only in some details. Let

Proof.

For y E L, we put

16

-

where a

alE, and let g(y) =

there is a matrix 1 -

4

y = (Yik)

N

I I Y' and let 0 i =1 k= 1 1 k


1

N

- L

n=1

En- 1 ¢(x n)

(*)

and IIPNsk(xn)PNII",< En

for k,n

L

Yn,k

1 , Z, ... , N,

Put

( **)

N

E

L Y < l} n , k= 1 nk

and g(y) where gn(Yn,l ,Yn,Z"" ,Yn,N) -1

-

with x n = En Xn ' Since L is weakly compact, there is an element Y which the maximum value of 9 is attained. Put Y

N

= 1 -

and let 0

N

L Y

n,k=l nk Y

-

Yij = Yij

YN, Y E A'; let us fix 1 < n < N, and put for (i,j) t (n,k)

Ynk = Ynk + y. Then the matrix (Yi j) belongs to L. g((y i j)) g((Yij))' hence

Consequently, we have < hence (Z)

Let qN = support YN· Then by Proposition Z.Z.8, for 0 Y E qNA'qN' we also have (Z). By Proposition Z.Z.7, there is a projection PN E A such that ¢(PN) and IlpNsk(xn)PN11oo < £n

for k, n

1,Z, ... , N.

Let us put a'(Y1z)' Y12 = a ' ( 13), .. ·,Y 1,N-l = a'G1N). Yl,N = 0 Y21 = aIG ZZ)' Y 22 = a'G Z3),oo.,Y 2,N_l = a'(YZ,N), .Y2,N = O.

Yn

YNl = a ' (YN,2)' YN,2 = a'(Y N,3)"" ,YN,N-l = a'(Y NN), YN,N = 0, where a ' is defined in Proposition Z.Z.6. From the properties of a ' it follows that the matrix (Yik) just defined belongs to L. Consequently,

20 (3 )

g((Yik)) On the other hand, we have - (Yn1E:,E:)] +

= + (xna'(Yn2)CE:) -

+

+ +

+ ((x n + a(x n))a'(Yn3 CE:) - 3(Y n3E:,E:) + [(xnYn,NE:,E:) + + (x

+ a(x

n

N

+ ...

n)

+ 9n(Ynl'Yn2'···'Yn,N) +

I k=l N

I k(YnkE:,E:) . k=l By the properties of a' we have N

I k(YnkCE:) k=l Consequently, gn(Ynl N

-

I (YnkCU. k=l From (3) and (4) we obtain N

N

N

Since I k=l

n

N

I I n=l k=l

I(YnkE:,E:) n,k=l

(5 )

k < 1, from (5) we have -

N

I

n,

k= 1

(Ynk C U


1 - I n=l so that PN satisfies the conditions (*) and (**). {N such that PN a}

Q=

16

xe (dA)

E

There is a net

converges weakly to some positive operator a

A.

We have

21

for k,n
1-2

co

I

1

s- ¢(x ) n=1 n n

IlpxnPl1 2s n for all n=1,2, ... Putting in theorem 2.2.12 Xl = x and x n = 0 for n > 1, we obtain 2.2.14. COROLLARY. Let a satisfy the assumptions of Theorem 2.2.12. Then, for each S > 0 and x E A+, there is a projection PEA such that lips n (x)pll < 2s and ¢(p.L) -< 2 .'t..Ld. E

and

From Corollary 2.2.14, one can deduce the following maximal ergodic theorem of Lance. 2.2.15. THEOREM [3D]. If a satisfies the assumptions of theorem 2.2.12 then, for each x E A+ io i t.h. Ilxll < 1, there exists a positive operator c E A such that Ilcll < 4, ¢(c)-< 3¢(x)1/2 and s (x) < c n for aZZ n=l, 2, . . . . Proof.

Let us remark that for z s A+ and p s Proj A, q=l-p we have

z < 2pzp + 2qzq. pzp + qzq Indeed, from (p-q)z(p-q) > 0 it follows that pzq + qzp and, consequently, z = (p+q)z(p+q) < 2[pzp +qzq]. Put s = ¢(x)1/2 By Corollary 2.2.14, there exists a projection p s A such that sn(x) < 2ps (x)p + 2(1 - p) s (x)(l - p) n

< 4sp + 211xll (1 - p),

n

22

where = ¢(x)1/2 and ¢(pl) < 2 c = 4 p + 211xll(1 - pl. Th;n Ilcil proof is completed. 0

=
nt'

nt

22t, and let N be a positive integer such

We can write N nt'

We have Y

) + ... + apnt-l(y

nt

=

pn

+ r, where p,

t

= s

nt

n -2

n

+(n

t

n p-l (Y (Yl) + " . + a t ntP+l

t-2)a

1

+

l)

n (Yl)+"'+ a

(Y

l),

Moreover,

=

)

+ ... + (n , - l)a t + a

nt

r-positive

((n

(Y

n -1

+ (a t

l)

- l)a

t

t(p+l)-2

nt p

(Y

(Y

+

l)

+

l)

(Yl))'

Consequently, sN ( Y ) - Y

S N( Y1)

A

-l[ N Y +

n

(Y

=

t t

­1 ) N-l( )) N ( Y1 + a ( Y1 + . , . + a Y1 + «{y - l }«

n

t

) + ... + a

n t-2

1

pnt-l

((n

ntP+l +(n

where

n (Yl)+ .. · + a

t-2)a

+ (Yl

+ a(Yl) + .. , + a

+

+ aN-l(y

n

t-2

l))]

n

-2

n

-2

a t - l)a t and

t

(Yl) ) (Yl))'

(Y

l))

(Y

n t)

- l )o

- n ntp

t(p+l)-2

+ (a

ntp

=

t(Yl+2a(Yl)-t

(Yl) +

(Yl)) + (Yl)+

24

y t, p, r

(ntP+r)-

(ex

ntp

(Yl)+ex

ntP+l

(Yl)+···+

-1 ntp n t ((n t - 1) ex (Yl) + ntP+l nt(p+l )-2 +(nt-r)ex (yl)+···+ex (Yl))· + ex

ntP+r-l

1

(Yl) -(ntp + r)-

1

Since ex is a contraction in IL 2, we have 11

-1 8 211 2 :5- 2N t n t

Moreover, Nt

and

Bt,s' Gt , p , r , s

(

2(n

tP+r)-ln t·

2 2t n

opera tors Tt , s , t. The re exist As enjoying the following properties:

N implies P




2 Gt,p,r,s s=o the convergence of the series being taken in the norm

topology in

IL 2.

In a similar way we can write A

_

sN (z) - z = N

2

s=o

Z

t,s

1

1

pn t spn (zn ) + NtN- 0t + A t,p,r t t

, 0t

with Zt,s , Dt , s ' L t , p, r , s

I I Zt ,51 12 < :5- 2-

sl

2- s I I Zt

2 ot,s

s=o

E

A t , p,r

2 L t,p,r,s s=o

As and such that

I I 2 ' 1lOt, s I I 2

:5- 2 - s i lOt

1I 2

'

I I Lt , P, r , s I I 2:5-

[At,p,r l 2 . 1

Put - 2-(t+s) = 2-(t+s) . (t,s , (t,p,r,s By Theorem 2.2.12 (Goldstein's maximal ergodic theorem), there exists a projection p ( A such that

25

I


0 be given. Fix t such that l
t

E2 - 6 .

\ e-(t+s) ( 2 ) ¢Tt,s 1 By the

(7)

.

inequality, we have

uniformly convergent (i.e. in norm algebra A. On the other hand, sk(Yn )p

11'11,.,)

s =0

sk(T

=I

sk(T t s)p , s=D ' where the last equality is in the norm I 1'11 2 , t

L

)p is t's to some element from the

From this estimation it follows that the series

Consequently, we have

26

I [2

where lOt f\P

E

A and II St P II

Yt,p,r P

E

E

A

At , p, r P

00

lOt ,

A and IISt,p,rPlloo

E

s k (z n ) P t °t P

- s - t).1:2 For the same reason, we have

s=O

A and I Is k (z ) p II nt


0

and 0 s a

E

a 'l '

A+,

L

j=l

x.

J

< l }.

Define a function 9 on K putting

= T(ax 1) + T((a + o.(a))x 2) + ... + n+ T((a + ... + a 1 ( a ) ) x

n

- E(T(x ) + ZT(X Z) + ... + 1

By the weak compactness of K, there exists a point in K, say , ... ,x ) such that g(x , ... ,x (x max g. Let x E A, 1,x Z n 1 n) K

o




1i m r-+ oo

so Xsk(a)X

11\(a)1/2 En(r)hI1

(E nr ( )sk(a)E nr ( )h, h) < lim -

( )h, h) nr

=

2

=

dXh, h) ,

for each k.




f

(1 - A) Q(dA) > 1/2 Q[0,1/2], [0,1/2]

so 1 - q

=

Q[O, 1/2)

2(1 - X),


_00 t

Of course, as in 1° should be treated as the (unique) extension of as to In the case of equality in 1°, we have n-l k 1 x = L u xl ' n k=O 00

so then {x

n}

is a sequence of ergodic averages.

In one of the next

34

sections we shall prove the following subadditive ergodic theorem in the von Neumann algebra set up. 2.3.3. THEOREM [64].

If {x

n} is a subadditive sequence in A, then and bilaterally almost uniformly to an

n-lx n converges in

A

a -invariant element

x

E

ll, 1 •

To prove a similar result for the continuous parameter t, we need more restrictions on x. Let us remark that, as it was noticed by Kingman [69], some additional assumptions concerning x t are necessary even in the classical case (when A is a commutative algebra over a probability space (Q,F,y)). Let edA{!xl} denote the spectral measure of lxi, that is, [x ]

f; edA{lxl}.

A subadditive map x is said to be separable

there exists a sequence 0

sk

if

1 such that, for all A>O, we have (11 )

as N + 00 (in the strong operator topology).

For A > 0 let us put

and (13) In one of the next sections the following theorem will be proved. 2.3.4 THEOREM. bet x : [0,00) + map such that the (semi) group weakly continuous.

(A,¢) be a separable subadditive (et

,t

E

T) of

is

If

fOO ¢(p(A))dA


_00

a in (i) should be treated as the (unique) extension of the

automorphism a to IL l '

We shall prove the followinq theorem which is the non-commutative extension of a recent result of Derrienic [28J. Compare also [64J, [65J. 2.3.6 THEOREM. Then n

_1

x

n

Let (x ) be a quasi-subadditive sequence in D...l(s)(A,¢). n converges in ILl and bilaterally almost uniformly to an

a-invariant element x of ILl' and ¢(x)

= y.

Before startinq the proof of Theorem 2.3.6 we shall formulate a few lemmas. We shall follow the basic ideas of Kingman r691 and Derrienic [28J. However, in our context, we cannot perform some operations, for example, to pass to the lim inf or lim sup with the sequences of operators under considerations (as in [69J and [28J with sequences of real functions). To avoid such procedures we start with the following lemma.

36

2.3.7 LEMMA. n.. (A,¢)

is)

of

o

Let, fop n,m = 1,2,., " bn,c and an(m) be the elements n satisfying the inequalities c 2::. 0, and m


O.

Choose a subsequence (m

in such a way tha t ¢(c

ms)

s)

n

0 bilatepally

of positive integers m

< E2 with I Em < E/2. ms s =1 s

Putting E =

II e s=l lO,E m ] (c m), we ha ve ¢ (1 - E) < E/2. Let (P m) be a sequence of s s E 0 projections from A such that ¢(l - Pm) < ------=iT m and IIPman(m)Pmll

2

as n

-+

Put P

00.

A Pm

m=l In particular, Ilpan(m)pll

Then, fo r each m, Ilpan(m)pll

0 as n

00

E lip, we have Putting Q Em fo r n > Nm ¢(l-Q) < E and 1\ QbnO II -< I I Qc m I \ +IIQan(m)O\ I < 2Em , for s s n > max(ms,N s)' which means that b 0 bilaterally almost m uniformly. 0 The main idea of the next lemmas is to reduce the proof of our theorem to the case of non-negative quasi subadditive sequences (un) q, (u ) satisfying the condition inf n_ = O. More exactly, we shall show
0 i=O

fo r n

1,2, ...

with some x, w E n.. 2 (A, ¢) and such that ¢ (x n) ¢(x) = inf n n n-l Then we apply to the sums I a i the Yeadon's theorem and, finally, x i=O after some transformations, we use the previous lemma. In the further considerations we neglect some parts of linear functionals on A. We use the Takesaki's decomposition of a functional fEA* into its normal and 'singular' parts: f=fn+f s' where fnEA* and f s is singular in the sense of Takesaki (see references below). The decomposition is, of course, the analogue of the classical result of Hewitt and Yosida (see for example [36J, p. 163.).

37

2.3.8 LEMMA.

Let Nand S be two bounded linear functionals on A.

Assume that N is normal (i.e.

N

E

A*) and S is positive and singular

in the sense of Takesaki ([126J, p.

127, see also Appendix).

Then

N + S > 0 implies N > O.

Let p be an arbitrary orthogonal projection from A. By Takesaki's theorem (see Appendix (A.41), p. 126), a positive linear functional S on A is singular if and only if, for every non-zero projection q E A, there exists a non-zero projection rEA such that r q and S(r) = O. It follows that there is a sequence (Pk) of mutually orthogonal projections in A such that p = Pk and S(Pk)=

Proof.

= O. Indeed, it is enough to take in A a maximal family of mutually orthogonal non-zero projections (p a ) such that pa -< p and S(p a ) = O. Such family is at most countable (since A is semifinite; see (A.40)). Put On =

... ).

Then by the normality of N, we have that

N(p) = lim N(O ) = lim [N(O ) + S(O )J > O. n n n n nthe spectral theorem we obtain N(a) ends the proof. 0

>

0 for all 0

2.3.9 LEMMA. Ccomp , [28J,[69J). Theorem 2.3.6.

(h

Put

1 m

Ym = iii

- ax k_ l),

Since p is arbitrary, by

n


1.

Then there exists a sequence (zn)'

0 < zn

E

III (A,¢) such that,

for

every m > n, we have

n-l k 2a y k=O m Moreover,

Proof.

< x

sup n

nl m-l

n

2

+ a (-

l lv, III

m k=l


z(l»y-C. k

o (1).

s

41

Let us take x = x 2k for k = 1,2, ... The sequence (x k) is obviously k quasi-subadditive with respect to a 2 (and with the same constant C = sup llhklll). We can now repeat the same reasoning as for (x k). In k

particular, we put y'

m

= L

m

and obtain the formula analogue to (22) z'

where

V

=

V

o is

of A*,

o+

0'

s

(29 )

-

a weak* limit point of

and

z' - the corresponding element

the corresponding singular functionals in Takesaki's

decompositions of suitable a' and 8'. obtain 2y z'(l) 2y - C Let us notice now that

Since n - 1 ep(x

2n

)

-+

2y, we (30 )

(31 ) The sequences 1

m

m

1

m

m

(32 ) L (x - ax 2 k_2) k=l 2 k- l are bounded in ILl-norm (since x r - aXr- 1 -< xl + o h r- 1; compare the proof of Lemma 2.3.9.). Taking the suitable weak*-limit points v l and v of these sequences, we obtain

L (x - ax 2 k_l) and k=l 2 k

2

v0 =

1 + v 2) and v

0=

v1 + av2.

Denote by zi the norma 1 parts of vi (i =1,2). Takesaki's decomposition gives imediately 1

z = 2 (z , + z2) and

z(l)

The uni queness of

z'

Consequently, we have z(l) = gives y

( 33)

(34)

(zl(l) +

z'(l), and (30)

C y - 2'

The repetition of the procedure just described will give us

(35)

42

-

2n

-

forn=1,2 ...

which means that y = z(l) and ends the proof of lemma 2.3.10.

0

2.3.11 PROOF of Theorem 2.3.6. Let (x n ) satisfy the conditions (i) and (ii) of Definition 2.3.5. By lemma 2.3.10 there exist in A two elements and w such that n-l u = x + an (w) - I a i x .::. 0, wit h cjJ (x) = y ( 36) n n i =0 It is easily seen that (u ) is a nonnegative quasi-subadditive sequence n in satisfying the condition . cjJ(u n) cjJ(x n) lnf ----= inf - y = O. n n n n n-l a i Moreover, by Yeadon's theorem 2.2.17, the averages n -1 Li=O x

x

converge bilaterally almost uniformly and in so it is clear now that it suffices to prove our theorem for the quasi-subadditive sequences (x ) which n are nonnegative and satisfy the condition ¢(x ) inf __n_ = O. ( 37) n

By Yeadon's theorem 2.2.17, the averages 1 k-l

k

im

a

(38)

xm

converge bilaterally almost uniformly to some am-invariant element X eo such that ¢(x m) = ¢(x m)· Putting A

A

a (m) n

where k

1 k- 1

k

a

im xm - x 1 mk m m + k a zm'

k(m) is defined by the equality n

( 39)

mk + r (0




0

and

m =

1,2, ...

(44)

We shall prove that then (45)

holds.

We need a few lemmas.

2.3.13. LEMMA.

For a *-automorphism a of A and Ql,Q2' ... QN

E

Proj A

45

we have that

V

l

0,

U is ultraweakly continuous, i.e. the function is continuous in t on [0,00) for each E A* and x

We shall prove the following theorem of S. Watanabe [133J.

E

A.

53

2.5.1. THEOREM [133], [191 .(A non-commutative local ergodic theorem). Let A be a von Neumann algebra and a = (at)t>O a dynamical semi-group on

A.

Then,

for each x

E

A,

the integrals

converge almost uniformly to x

as T -+ O.

Without any loss of generality we can assume that A acts (in a standard way) in its GNS-representation space Hcp = IL 2 (A,cp), with the cyclic and separating vector S. Put

Proof.

for 0 For each x

E


0 we found a projection pEA such that as n -+

and

< E,

which ends the proof.

00

D

77

3.Z.l7 Let F, w) be a probability space and let (F n) be an increasing sequence of subalgebras of F. Let us treat the conditional expectattions lP n = IEn(oIF n) as the operators acting in F, w). Then, by the martingale convergence theorem, the sequence IE(fIF n) converges almost surely, for every f E ll.Z' The ll.Z­version of the martingale theorem is true also for some projections lP n which are not necessarily conditional expectations. Namely, E. Stein [lZO] proved that, for every increasing sequence lP n of positive orthogonal projections in F, w) the sequence lPnf converges almost everywhere for each f E ll.Z' In the next section we shall show that a similar result can be proven in the von Neumann algebra context for traces. Let A be a finite von Neumann algebra with a faithful normal tracial state ¢. We adopt the following definition. 3.2.18

DEFINITION

(Comp. [34].

A sequence an:

ll.Z(A,¢)

ll.Z(A,¢)

is said to satisfy the condition of Duncan i f there is a positive constant C such that (58 )

for all sequences (Pl'PZ"") of mutually orthogonal projections from and all positive integers n.

A

We start with the following maximal lemma for positive operators in ll.Z(A,¢) . 3.Z.l9

LEMMA. Let an: ll.Z(A,¢) ll.Z(A,¢) be a sequence of positive operators (i .e. anx O, there is a projection qEA such that for k=l ,Z, ...

(59)

and (60)

First assume that x Ak,k' then

+ 0) -1jJ(A + 0) = 1jJN(A - 0) O,k O,k k,k - 0) = 0, we get by virtue of (19), (20), and (21) that for

Putting, for convenience, 1jJN(A 1jJ(A

k,k every real

A amd each k=1,2, ... ,

there is an

i

between 0 and k-l

such

that 1jJN(A

i,k

+ 0) -1jJ(A i,k + 0) - 11k S1jJN(A - 0) -1jJ(A - 0) s1jJN(Ai+l,k - 0) -1jJ(Ai+l,k - 0) + 11k;

thus,

for an arbitrary projection p from p[1jJN(A.

1,

k+ l) -1jJ(A.

1,

A,

we have

k+O)]p-l/k P

S p[1jJN(A - 0) -1jJ(A - O)]p S P[1jJN(Ai+l ,k - 0) -1jJ(A i+ l ,k - O)]p + 11k p and,

as

a consequence of the above inequality, we obtain that

IIp[1jJN(A i,k + 8) -1jJ(A i ,k + 8)]pll + 11k, 8=±0 which gives sup -CO-

k=l Pr o o j ,

Let

n e[O 'X n )

I

n=l

0':

C

¢(x

n)

0

(35 )

a.u.

Then

96

so we have that formula (30) holds. Moreover, (31) is satisfied (by assumption) and ) = 0 (because E, are symmetrically distributed). n n Repeating the part of the proof of Theorem 4.7.4 (after formula (31), we obtain (35), which was to be proved. D In the case of nonsymmetrically distributed E,n we need more restrictions on {x k}. 4.7.6 THEOREM. Let {E,k} be successively indepehdent and identically distributed, and let ¢ and {x n} be as described in Theorem 4.7.4. If, additionally, T(E,1) = 0 and

xk x n

k

Co n

for k

n,

(36)

then (34) implies (35). Proof.

In the same way as in the proof of Theorem 4.7.5, we show that a.u.

and thus, it is enough to prove that

We have the following estimation

J

00

1

n

1

-

­­ I I T(E,k) I Xn k=l

n

­­ I Xn k=l

x

AT(edA{IE,l I}) k

because, writing the polar decomposition E,k ­

= uklE,k ­ Ilukll T(jE,k ­ lE,k ­

Consequently,

where

and

we have IT(E,k ­ = T(IE,k ­

= lE,k l e(xk,oo) {1E,kl}·

and

97

S1 ->- 0 because

n

I

m"'O

(m+l) l(e[x,x

m m+l

I

)

m