Strong Limit Theorems in Noncommutative L2-Spaces (Lecture Notes in Mathematics, 1477) 9783540542148, 3540542140

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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1477

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen

1477

Ryszard Jajte

Strong Limit Theorems in Noncommutative L2-Spaces

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Author Ryszard Jajte Institute of Mathematics L6di University Banacha 22 90-238 Lodz, Poland

Mathematics Subject Classification (1980): 46L50, 46L55, 47A35, 60F15, 81C20

ISBN 3-540-54214-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54214-0 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

To

zny

"W'.:Lfe

'PREFACE

This book is a continuation of the volume "Strong limit theorems in non-commutative probability", Lecture Notes in Mathematics 1110 (1985). It is devoted mostly to one subject: the noncommutative versions of pointwise convergence theorems in L2-spaces in the context of von Neumann algebras. In the classical probability and ergodic theory the almost sure convergence theorems for sequences in L2 (over a probability space)

belong to the most important and deep results of these theories. Let us mention here the individual ergodic theorems, the results on the sure convergence of orthogonal series, powers of contractions, martingales and iterates of conditional expectations. The algebraic approach to quantum statistical mechanics suggests the systematic analysis of theorems just mentioned in the context of operator algebras. This is the main goal of this book. We consider a von algebra 1:>1 with a faithful normal state IjJ and take H = L2(M,IjJ) - the completion of

M under the norm

x

x e M.

Then we

introduce a suitable notion of almost sure convergence in H (generalizing the classical one) and prove a series of theorems which can (and should) be treated as the extentions of the well-known classical results (like individual ergodic theorems, Rademacher-Menshov theorem for orthoqonal series or theorem of Burkholder and Chow on the almost sure convergence of the iterates of two conditional expectations etc.). The classical pointwise convergence theorems for sequences in L2 are, as a rule, non-trivial extensions of much easier results concerning the convergence in L2-norm. The same situation is in the noncommutative case. Most of the noncommutative LZ-norm versions of the analogical classical results can be rather easily obtained by a natural modification of the calssical argument. Passing to the noncommutative almost sure versions needs as a rule new methods and techniques. Very often the algebraic approach makes much clearer the general idea which is behind the result concerning, say, real functions. At the same time the proofs provide some new tools in the theory of algebras. This is one of the reasons we decided to collect and prove in

VIII

a systematic way the results concerning the almost sure convergence in L2 over a von Neumann algebra. Only very few bibliographical indications have been made in the main text of the book. More complete information concerning the subject the reader will find in the "Notes and remarks" concluding the chapters. We hope that this book may be of some interest to probabilists and mathematical physicists concerned with applications of operator algebras to quantum statistical mechanics. The prerequisites for reading the book are a fundamental knowledge of functional analysis and probability. Many of the results presented in the book have been discussed and also obtained during the seminar on the noncommutative probability theory in L6dz University in the years 1985­1990. I would like to thank very much all my colleagues from this seminar for many interesting and fruitful discussions. I sincerely wish to thank Mrs Barbara Kaczmarska who took great care in the typing of the final version of the book.

L6dz, November 1990.

R. Jajte

Chapter 1. 1.1. 1.2. 1.3.

Chapter 2. 2.1.

2.2. 2.3.

2.4. 2.5. 2.6. Chapter 3. 3.1. 3.2. 3.3.

3.4. Chapter 4. 4.1. 4.2. 4.3. Chapter 5. 5.1. 5.2. ::i. 3.

ALMOST SURE CONVERGENCE IN NONCOMMUTATIVE L2-SPACES Preliminaries •.•.•..••••••••.•••••••••••••••.•••••.• Auxiliary results ••••.••.•••••..•••••....•••••..••••• Notes and remarks ••..•..•.••••.•.•.•....•••••••..•••. INDIVIDUAL ERGODIC THEOREMS IN ALGEBRA

L2

1 3 8

OVER A VON NEUMANN

Preliminaries .••••••.••••••••.•••••••••••••••.•.••••• Maximal ergodic lemmas .•••••••••••••••••••••••••••••• Individual ergodic theorems •••.••.•.•••.•••••.••...•• Ergodic theorems for one-parameter semigroups •.••.•.. Random ergodic theorem in L2(M,$) .•••.•.•.••.•...•.• Notes and remarks .••...••.•••...•.•....•....•..•....•

10 10 17

21 31 35

ASYMPTOTIC FORMULAE Preliminaries •••.•.•.•.•••••••.•••••••••••••••...•••. Asymptotic formula for the Cesaro averages of normal operators ••••.•••.•..••.••.••••••.••.•..••.••.....••• Ergodic Hilbert transform .•••••••...••.••..•••.•...•• Notes and remarks ..•.•.••••••...•.••....••.•..•••..•.

37 37

47 50

CONVERGENCE OF ITERATES OF CONTRACTIONS Preliminaries. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Main result ••.•.•••.•.•••••••.••••••..•.••.••••.••••• Notes and remarks....................................

52 52 62

CONVERGENCE OF ORTHOGONAL SERIES AND STRONG LAWS OF LARGE NUMBERS Preliminaries. • • • • • • • . • • • • • • • • • • • • • • • • • . • • • • • • . • • • • • • Rademacher-Menshov theorem and related topics .••••••• Notes and remarks....................................

64 64 84

x Chapter 6.

s .a . 6.2.

6.3.

Chapter 7.

CONVERGENCE OF CONDITIONAL EXPECTATIONS AND MARTINGLAES Preliminar ies .•..................•.........••....... Maximal inequalities and convergence theorems ...•••• Notes and remarks .•.•..•...•••...••..••...•••••••..•

85 85 89

MISCELLANEOUS RESULTS Preliminaries . Strong laws of large numbers .....•..........•.....•. Local asymptotic formula for unitary group in H •.• Notes and remarks .....•.............................

90 90

OPEN PROBLEMS...................................................

100

BIBLIOGRAPHY ..••..........•.. " ...•. " ......•.••.......•••. " .. .

103

INDEX.. .......•.. .. ..

112

7.l.

7.2. 7.3. 7.4.

.

. . ..•....

. .•. .. ....••. ..

97

99

Chapter 1 ALI!DST SURE ca.wERGElfCE IN l!fONCOIIJIIU'l'ATIVE

L -SPACES 2

1.1. Preliminaries Throughout the book we constantly use the terminology of operator algebras. In fact, only very little knowledge of this theory is needed for reading this volume. As we mentioned in the Introduction, this book is a continuation of [50J. All necessary (and sufficient for our purpose) information concerning the operator algebras has been collected in the Appendix to [50J. Let us begin with some notation. In the sequel

M

will denote a

a­finite von Neumann algebra with a faithful normal state M denotes the commutant of M. Proj M will stand for the set of all orthogonal projections in M. of M. For p E Proj M,

M+ will denote the cone of positive elements always pl = 1 ­ p. We shall write 1 for the

identity operator in

M*

In

the whole

book

denotes the predual of

M.

we shall discuss the problems concerning

H =

Hilbert space norm

M.

which is the completion

x

(GNS representation space for

of

M

M with

the

under the respect to

In the sequel we assume that M acts in a standard way, on the Hilbert space H = with a cyclic and separating vector Q such that set

=

(xQ,Q),

MQ = {xQ :

and the norm in For a S

E

X

E

for Q}

of

M by Hand

x

E

H.

M.

We shall identify

The norm in

H

M with the sub-

will be denoted by II II,

11 110>. p

E

Proj M we set

converges in norm in and

(with the usual convention Obviously, for all

s,n

inf 0 = +0». E

H,

we have

M}

2

and for

x

M

E

We adopt the following definition of the almost sure convergence in J. J _J _ fJ£fIIllTIOII.

A .oe.que.nc.e

p

M

in

¢ (1


0,

and

The following result is a noncommutative version -known theorem of Revesz (94).

1

(Le.

4>(ax)

us notice that these assumptions imply that all elements from

M.

r

,r,

•

preserving the state

normal maps. Indeed, let

k

i = l,2, ...

In the sequel, we shall consider several Schwarz maps in

< 56 1 / 2

(x s)

4>(x), aj's

for

a 1 .•• xeM).

are

ak Let

positive

be a bounded increasing net of positive

In particular, we have

o s Xs

S

x s' Then

a (x s) S

sup a(x s) and, by the positivity of a, sup a(x x s). Mores) S a(sup s s s over, since the state 4> is normal, we have $(sup a(x ) s s Thus

sup $(a(x s)) = sup $(X s ) s s 4>(sup Xs ) = $(a(sup x s)). s s

or $(a(sup Xs ) - sup a(x s)) s s

O.

14

Since

is faithful, we have

sup a(x s) = a(sup x s)' s s which means that

a

is normal.

Now, we shall prove the following maximal ergodic theorem. n.-l f.2.3. lHfORBI.

(xm) P

E

M

Em > 0,

M+,

C

Let lU put

"Oft

:E

\J=O m

= 1,2, .••

\J

ai

(i

TheYl thVte

= l,2, .•• ,k).

Let

exM:t6 a p/tOjee-UoYl

1>uch.that; we have

( 1)

aYld

sup

(2)

itOIt ¢orne cOiUtaYl:t6

depemU.Ylg only

A B k, k

OYl

ami ai£.

k,

m

aYld

•.. ,n k •

For k = 1 the result reduces to Theorem 2.2.1. Assume (2) holds for k - 1. Putting x s = 0 for s '# m, we obtain P.-'DO'.

(3 )

sup

that

lip s(k-1) m n k- 1

and

Applying Theorem 2.2.1 (for one kernel) with we find a projection (5)

p

E

M such that

(1 - Pm)PII", < 2Em

and - p)

s 2

Using the inequality

'"

:E

m=l

1

(l -

Pm)

Ci

= Cik ,

xm

= 1- Pm

and

15

for

z

=

(with

1 :ii v :ii k - 1),

we obtain

ps(k) ok + 2(1 _ p )S(k-1) m nk - 1

+ 2ps (k) (1 - p)phxm". nj{

Now, taking into account (5) it is enough to put

= 2Ak _ 1

A k

and

B

= 2B k

k

+ 4,

•

to obtain ( 2 ) , which completes the proof. Let us denote via the formula

0

(at)

be a Quantum dynam.ic.a1. ""em-igltoup

(n = 1,2, ••• ).

Then theJte

ex-v..u

for

T ;; 1,

.in

Let

M.

a pltodec.tion

p e M

w.c.h that - p) and T

lip

.r

at(xn)dtpll..,

0

Put 1

An

.r 0

(Xt(xn)dt

s

4TE

n

n = 1,2, •••

.

23

and sn = n

-1 n-1 :E:

k=O

Then we have, for 'r

T- 1

o

0.

k 1. T < (N + 1)

N

o.t(xn'dt = T- 1[A n + o. 1(An) + •.. + T

+

o.t(xn)dt]

«N + l'/N)sN+1(An'.

N

By the Goldstein's maximal lemma (for one kernell,

tion

p

E

there is a projec-

M such that

41(1 - p)

and N

1,2, ••• ,

n = 1,2, •..•

Moreover, 'I

1 IIPT-

at(xn,dtPII""

:;;

2I1PsN+1(An,plI""

:;;

o for

n

= 1,2, ••• ;

! .e,!. lJfBJIWI.

T;;: 1,

which ends the proof.

H,

:thVle

H

ex..Uu

:the

M

lP

such that

1,01t

evVlY

a. .6. in

H,

i :the (unique) ex..teMion 01, :the noJt.lllai. condUionai expec:ta:tion lE 01,

on:to:the von Neumann .6uba£gebJta

VuoI·

Then,

/U-

.u.mu

= lPl; Lt'ltVle

•

Let

(!JocJ.ated :to a quan:twn dyltOJllica£. .6em.tgltoup E

4En,

Take

C

H,

E

k > O.

Ma

01,

a

-invaJL.ian:t

e1emenU 01, M.

Then there is a projection

P

E

M

24

p)

¢(l -

and

T

5T£kl/ 2,




C8>

W = {w

tj will also be denoted by

is

(t l,t 2 , .•. );

a-invariant and

class of all isometries = ll(x)Q,

gebra

for

2.5.J.1Jf£OflBt

:that,

.601t

each

Then,

.601t

each

(T, :E, m) ,

In

The product measure

s

al of

x E M) .

F (M,

1-

H generated In the sequel

will by

of

11 E

(t , t

l

h

h

E E

2 ' ••• ),

H,

H,

the

that the al -

we assume

t ... f. ( t ) h

.6W1.WO 1'1

theJI.e ewu, an

h

E

H

PJlOO4..

.601t

be a map .ouch

.6uch that, trOIt p- a£mO.6t eveJI.Y

h ,

m-a£mO.6t eveJI.Y

t

E

dOM not depend

01'1

w and h

sentially bounded ultraweakly

B = L",,(T,:E ,m)

tET "" B is given by the formula

=

L",,(T,m,M)

m-measurable functions

IIf II = sup ess !If ( t) II •

J..-6

f. ( t) -

T.

Consider the weak*-tensor product algebra

v(f)

the

J..-6 weakly m- meMlVLab£e .

B can be identified with the von Neumann algebra the norm

to

we have that

h

-invaltiant

the (i.e.

f.: T'" F (M,

Let

=

M such for

M*, which is equivalent M acts in a separable Hilbert space H.

(Random eJl.90dic. theoltem).

p

Let us de-

stand

Xn(h,w)

on

the

M has the separable predual

assumption that

w =

T}.

E

of

(W,B,p)

dp = dt l dt 2 ••. the class of all normal*-endomorphisms a

•••

note by that

=

by

M.

C8>

of

es-

f : T ... M with

The tensor product state

v = ¢l

C8>

m

fEB. T

We split the proof of our theorem into a few steps. Let

Q

be a measure preserving transformation

of

T

and

let

32

: T

F(M,$)

for each

h

E

be a map such that H.

Folt eac.h

II n M

n

E

.thVte -u, a weaki.y

m -mea.w,wble map

H

.ouch

is a .oudably cho.oen ulVtaweki.y

m-

k-

( t ) (Qt ) .•.

k=l

"",

H,

h

T

I:: EO, n ::E

1

is weakly m-measurable,

Then we have the following lemma.

h

!tOIL eac.h

.that,

t

(Q t) h - h II

PI::: T

whVte

Proj M

0

PI::

meMwtable map .oat-U.ftying .the -tneQuaLU:y

>

J'

1 -

1::.

T

- =L ,m) 2(T,::E

H = L2(T,::E ,m,H). is generated by some endomorphism a(t)

Indeed, take tion, = 1

H

and

0

Q = u

1,

0

By the assump-

0

where

u

M.

of

is the isometric

tor in

L2(T,::E,m) generated by the transformation Q. Then, rather standard approximation we can show that the formula W(f)(t)

=

Put operaby

a

f E H,

defines some isometric operator on generated by the endomorphism

a

H. of

Moreover, L",,(T,::E ,m,M)

this =

operator is

L",,(T,::E ,m)

0

M,

given by the formula

Taking for

f

E

H

the function

fIt)

h,

we

have

that

the Cesaro

means = n

converge

-1

n

k=l

almost surely in

means that, for each

...

L:

d > 0,

-

H

k

t)h

to an element of

-

H,

say

there exists a projection

h.

This

33

say

T

Pc

-+

Proj M,

\!(P£) =

(* )

.r

such that

$(P£(t) )m(dt)

Ii.:

1

-+-

(t,),

-c

T

and

n - hll p £

IIS

n

as

0

-+

which ends the proof of our lemma. Now, we shall show that a.s. for

m-almost all

t

in

H,

T.

E

Indeed, let us notice that formula (*) implies the existence of a sequence

with

everywhere, say, for for every in

B =

t

L",(T,m)

CB>

Q

"',

m(To) = O. On the other hand, with entries there exists a matrix (x(s)) n,k T \ To

with

is a cyclic element in

B)

x(s) (t)P (t) II n,k £s "',M

o

k=l

-+

m-almost

such that

=

ilL

n

0

-+

such that

M

L

as

E

s = 1,2, .••

k=l (where

£n

for every

-+

.

s = 1,2, •••

and

and

t

E

T - T1

with

m(T1 ) = O.

Obviously, this implies that a.s. in for

m-almost all

t

E

H,

T.

Now, we are in a position to prove our theorem. T

the shift transformation in

Of course,

W,

T preserves the measure

Le.

Let us denote by

T(t 1,t2, ••• ) = (t 2,t3, ••• ). p = m CB> m CB> • • • • Put T] (W) =

34

= n(w)g(,w)

q(g)(w) and y(g)

4>(g(w) )p(dw).

f W

Obviously, the endomorphism

q: D

D preserves the state

be uniquely extended to the isometry L2(D,y)

L2(W,B,P)

-

h

h

H such that, for

E

a.s.

in

H

By the previous lemma, for every an element

q

H = L2(W,B,p) H, there exists p-almost all w E W, we have that

in

E

H.

- does not depend on w.

It remains to show that

y and can

h

To this end, it is

enough to prove the following proposition.

Let

h

H.

E

-

1-6

h-

h

const

M).

(E

Let us put

yhen

and, consequently,

The sequence where the

-n q z

t1(w), •.• , t n gence theorem

H-valued martingale, namely,

denotes the conditional expectation with

E(.IYn )

a-field

is an

Yn

of measurability of the first

= tn(w),

w

(13),

(qnz)

E

W.

By the Chatterji h(w),

for

n

respect

coordinates: martingales

p-almost

all

w.

to t1 =

converOn

the

35

other hand, the map

g

is isometric, so

-

hll

->

0

(the last convergence follows from rather simple estimations), gives

h(w) = z

p-almost everywhere and ends the proof.

which

•

2.6. Notes and remarks The maximal ergodic theorem 2.2.1 is due to M.S. Goldstein

[36]

(for one kernel). This result is almost always behind the individual ergodic theorems discussed in this chapter. Let us mention here that the first fundamental maximal ergodic theorem

for

operator algebras

was given by E.C. Lance [68] in 1976 and is still one of the most important results in the noncommutative ergodic theory. Goldstein's result goes back to function"

F.r.

Yeadon [107]

(similar to the function

the

who introduced g

appearing

in

"auxiliary

the sketch

of

the proof of Theorem 2.2.1). Yeadon discussed the case of semifinite trace

while

Goldstein considered an arbitrary state.

This case is

more difficult and needs a construction of an "auxiliary function" in such a way that the whole proof is some kind of interplay

g

between

the algebra and its commutant. Theorem 2.2.2 is a slight generalization of the results

in [48]

and [51]. The theorems 2.2.3, 2.2.4 and 2.3.4 are closely related to the previous results of D. Petz [87] and are taken from [57J. Theorem 2.3.1 was proved first for a unitary operator generated serving

by a

*-automorphisms of the algebra [48] and then extended to the

case of Schwarz map [51]. Theorems 2.3.4 and 2.4.3 are in the spirit

of

Dunford-Schwarz-

-Zygmund ergodic theorem for several kernels [113]. Theyare sions of the results of Conze and Dang-Ngoc [16)

and

of

L2-verPetz [87J

concerning the maps in von Neumann algebras. They are taken from [57]. Theorem 2.4.7 is in the spirit of Wiener's local ergodic theorem. A noncommutative analogue of the classical result of Wiener

context of von Neumann algebras) Theorem 2.4.7 is the

was proved version

of

by

S. Watanabe

the

Watanabe's

(in

the

[104]. result.

Theorem 2.5.1 is a noncommutative extension of the Kakutani [61]

and

36

Ryll-Nardzewski [93] random ergodic theorem. It is formulated family of isometries in the algebra

L2(M,¢)

for

generated by some endomorphisms

M (instead of measure preserving

transformations

sidered in the classical case). This theorem is an extension

of

a of

conthe

result of Dang-Ngoc [18] who proved such result for a family of endomorphisms of

M).

The general idea of the proof is similar

to

that

of Dang-Ngoc. It is worth noting that in one point our proof is quite different in comparison with that of Dang-Ngoc.

Namely,

in the con-

cluding part of the proof we follow the idea of S, G£adysz uses the Andersen-Jessen theorem [22]

while

Dang-Ngoc

[32]

who

follows

the

method indicated by C. Ryll-Nardzewski [93] (compare [50], Chapter 2).

Chapter 3 ASYMP'l'OTIC

FORIIULAE

3.1. Preliminaries In the previous chapter

we

proved

several

individual

theorems under the assumption that the contractions were induced by some kernels in tion in

M (as a Schwarz

H=

in

M.

Any kernel is a positive contrac-

map) and

so, using the clas-

sical terminology we can say that the kernel is

and

L1 contractThe positivity and

Lw

2. are just those properties of

ive. Moreover, it generates the contraction in contractivity of a kernel

ergodic

a

L

enable us to prove the individual ergodic theorems for extension to a contraction in

a

which

a

(or for its

H).

In this chapter we shall prove the results concerning some contractions in the algebra

H which are not necessarily generated by positive maps in M.

In the next section we assume that these contractions

are normal operators in

H

(so they are connected with the algebra

only via the general structure of the Hilbert space

M Of

H =

course, the lack of positivity will be then recompenseted by the existence of the spectral decomposition which will enable us to prove some asymptotic formulae. 3.2. Asymptotic formula for the Cesaro averages of normal operators It is well­known that, in general, the individual ergodic theorem does not hold for an arbitrary normal (even unitary) operator L

2 (Q, F , P )

where

(Q,F,p)

is a probability space.

It

a unitary operator

u

depends heavily (and only)

erties of the spectrum of

u

on

also well-

is

­known that the asymptotic behaviour of the Cesaro means

in

u

1.

n-1 L

n k=O

u

k

of

the local prop­

near the value one.

Recently, Gaposhkin [29] proved that if ure of a normal contraction operator

u

in

E

is the spectral meas­ then,

for

38

f

E

L2,

the ergodic averages of

converge almost surely to

f

u

(given by the mean ergodic theorem)

if

and only if

o < 11 -

E(z

zI

< 2 -n) f

-+

0

a.s.

The main goal of this section is to extend the Gaposhkin's result to the operator algebra context. We shall prove the following theorem. 3.2. J. TH£tJRBI.

u

Let

be a IlOJUllal coYLtltaction opeJtatOIt acting ..in

and let u =

.r

zE(dz)

C1

be i l i -6pectJuUI. lt .pltMentat..Lon wlih the -6pect1t.al meMUlLe

{ Iz I s

1}.

;'Olt

eveJty

and

a

=

Let

u

Then,

E(' )

!;

k ,

E

H,

(n

= 1,2, ••. l ,

we have the ;'ottow..ing Mymptot..Lc ;'oJUllUia

(1 )

atmoM

oUILe.ty M

wheJte

[k]

denotM the la.ll.gMt ..integeJt

m :> k. Put

and

where 1 n(l -

zn z)

F( • )

Then

m

wlih

39

Put aa

a k = F(z

= F( z

E

a

(1

E

1).

= 1,2, ... ),

For

(s

=

and

1,2, ••• ),

let us put for

for

Our proof will depend very heavily on the properties of us put (3 )

.r

gn

a

Ln ( z )

(dz ) •

(n

1,2, ••• ),

then we have CD

(4 )

:E

n=l

IIg nU2
1}

and it easily follows that

IL

2

n(z)1 2F(dZ)

Let

40 00

:E IIg nll2

n=l

C

2

(as a rule, the constant

:E a

k=l k is different in different formuale).

C

Let

us put ( 6)

°n,m

writing (7 )

g

2n+m

1,2, ... ,2 n-1),

(m

g n' 2

m in the form

m

=

n :E q=l

£

q

2n - q

o

with

or

1,

and applying the dyadic expension method well-known orthogonal series [3],

in

the theory of

we obtain the following representation

n

(8)

:E

k=l

.r

(9 )

Rn,q,j(Z)EE,(dZ),

a

with (10)

L

Rn,q,j (z)

q

2 +(j-l)2 n - q n

(z),

= 1,2, ••.. ,n.

ObviouslY, we have =

(11)

.r IRn , s. j ( Z ) I 2·F (dz ) •

a

Taking a suitable partition

of

the disc

a = {j z ]

write (12)

tl

j q

with mutually orthogonal vectors

t

E

H such that

$

1},

we

can

41

and

(13)




II (sn ­ s k)pll", '2

s

2 1 2 IIplsn _ s 2k 1 p nec /

To finish the proof it is enough to apply

s.e,s,

l1IfDRBI.

Let

ing the condition

'"

L:

n=1

(Bk) < "'.

to

for every £ > 0, we can find a prok 41(1 ­ p) < £ and such that

Proj M with

110 n ­ 011 .... 0 2

41

and

(B ) ,

and

('k)

{o n -

'"

L:

(109 n)2 lI!;nll2 n


0,

and a pO-6UJ.ve OpeJt.MOJt

and

and 4> (B ) m

s

(m

+

2m

1) 2 K:E

k=l

II

II2

l ' f,; 2' ••• , , m 2

+ 6 ,

B

m

theJt.e exL!.-t-6 E

M

-6uc.h -thM

be

a

a -6Y-6tem

71

K

+ 2

1

j=l

p( j).

Let us remark that using similar method one can prove the following theorem. 5.!.7. THEOIlfJI.

(9)

Let

1ft the COruiiilOll

log (n + 1) log (m + 1) I (E;n,E;m)

n,m=l

then

n

E;. j=l J wheAe

0

I
'6 and put

theJte ex.-Wt two

91

n

s(n) = 1. L f,;k. n k=l

(4 )

For

mr < n :il (m + l)r,

s(m r) + a(n,m),

m-rns(n) where

n

a(n,m) = L r m

( 5)

L

k=m r+1

For

k = 1,2, .•• ,

For

mr < n :il (m + l)r, d(n,m) =

we can write

f,;k·

x

let us fix

E

k

M such that

we set

..1...

mr

Then we have the following estimation

(7 )

:il

where

C(r)

c(r)m- r- 1

does not depend on

m or

n.

Putting (8 )

=

£!tl r+1

m

we have by (1) and (6) that ( 9)

L $(D m) = C(r) L m=l m=l

m- r- 1 ex>

(m+1)r L k=m r+1

s Co + Co m:1 m-2 < "',

$(

Ixk ) 12 )

92

Co > O.

for some

Id(n,m)1 2

Obviously, we 'have (10)

Dm,

mr < n

for

(m + 1)3.

Moreover,

"s(m r ) " 2 < -.

E

m=l

Indeed, by the assumption (2), ;;

m=l (since

>

r

"s (m

r)

"2

;; cm-2rmr(2-6) m=l

C E

m=l

m- r 6


(,

a •.6 • .l1t . H,

wlteJte

Pn(,.

For some partial solution, see [50J, p. 79. When

(P n)

are generated by conditional expectations

tive answer is given by Theorem 6.2.3.

The positivity of

the

posi-

Fa s

canbe

understood with respect to any fixed closed convex cone in H (compare Problem 4).

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algebra of von Neumann 1 almost orthogonal operators

41

conditional expectation 4, 23, 53, 85,88 conditional expectation of Accardi-Cecchini 91 contraction normal 38 positive definite 52 generated by a kernel 4, 10, 17, 52 convergence almost sure 2 almost uniform 2 'Clf conditional expectations 86 of martingales 34, 86 strong 2 uniform 3 dual. map 4 dynamical semigroup

22, 23

endomorphism 4, 31, 32 ergodic Hilbert transform filtration

47, 49

86,88

invariant state 13 vector 20 kernel

4, 10, 17

local ergodic theorem

10, 22, 28

map positive 4 Schwarz 4 4> -contractive 4 martingale 88 maximal ergodic theorem 11, 14, 58 mean ergodic theorem 18, 46, 57, 91 modular automorphism 19 conjugation 91 operator 19

operator modular 19 "orma1 38 unitary 37, 46, 47, 49 orthogonal projection 53, 86, 88 sequence 64, 68, 74, 78 series 64 positive cone 91,92 contraction map 4

91

£andom ergodic theorem

31

Schwarz map 4 semigroup of contractions 22, 23 quantum dynamical 22, 23 of Schwarz maps 21 weak * conditions 25 sequence orthogonal 64, 68, 74, 75, 78 quasi-orthogonal 69, 77, 84 of iterates 52 weakly stationary 62 shift transformation 33 spectral measure 37, 53 representation 38, 46, 53 spectrum 37 strong law of large numbers 64, 78 tensor product algebra 31 state 31 theorem of Akcoglu 91 Burkholder and Chow 53 Goldstein 11, 86 von Neumann 53 64, 68, 69 Stein 52, 92

113

'transformation weakly measurable 32 preserving measure 31 ultraweakly measurable

32

vector cyclic 1 separating

1