Operator Algebras in Dynamical Systems (Encyclopedia of Mathematics and its Applications 41)

This book is concerned with the theory of unbounded derivations in C*-algebras, a subject whose study was motivated by questions in quantum physics and statistical mechanics, and to which the author has made a considerable contribution. This is an active area of research, and one of the most ambitious aims of the theory is to develop quantum statistical mechanics within the framework of the C*-theory. The presentation, which is based on lectures given by the author in Newcastle upon Tyne and Copenhagen, concentrates on topics from quantum statistical mechanics and differentiations on manifolds. One of the goals is to formulate the absence theorem of phase transitions in its most general form within the C* setting. For the first time, the author constructs, within that global setting, derivations for a fairly wide class of interacting models, and presents a new axiomatic treatment for the construction of time evolutions and KMS states. The wealth of new insight offered here will make the book essential reading for graduate students and professionals working in operator algebras, mathematical physics and functional analysis.

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board R.S. Doran, J. Goldman, T.-Y.-Lam, E. Lutwak Volume 41

Operator algebras in dynamical systems

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ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Operator algebras in dynamical systems The theory of unbounded derivations in C*-algebras SHOICHIRO SAKAI Department of Mathematics College of Humanities and Sciences Nihon University, Tokyo, Japan

The right of the University of Cambridge to print and sell all manner of books was granted by Henry VIII in 1534. The University has printed and published continuously

since 1584.

CAMBRIDGE UNIVERSITY PRESS Cambridge New York

Port Chester

Melbourne

Sydney

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521400961 © Cambridge University Press 1991 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1991 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Sakai, Shoichiro Operator algebras in dynamical systems: the theory of unbounded derivations in C*-algebras/Shoichir6 Sakai. p. cm. — (Encyclopedia of mathematics and its applications; v. 41) Includes bibliographical references. ISBN 0 521 40096 1 1. C*-algebras. 2. Differentiable dynamical systems. 3. Harmonic analysis. 4. Operator theory. I. Title. II. Series: Encyclopedia of mathematics and its applications; v. 39. QA326.S26 1991 512'.55—dc20 90-48013 CIP ISBN 978-0-521-40096-1 hardback ISBN 978-0-521-06021-9 paperback

CONTENTS

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Preface Preliminaries Banach algebras, C*-algebras and W*-algebras Topologies on C*-algebras and W*-algebras Homomorphisms, *-isomorphisms and *-automorphisms Self-adjointness and positivity Positive linear functionals and states Commutative C*- and W*-algebras Concrete C*- and W*-algebras Representation theorems for C*- and W*-algebras Commutation theorem (von Neumann's double commutant theorem) Kaplansky's density theorem (Bounded approximations) Gelfand-Naimark-Segal representations Factorial and pure states Theorem (the Poisson kernel for the strip) Corollary (the analytic version) Theorem (the perturbation expansion theorem) Corollary (the complex version) Theorem (convergence on geometric vectors) Proposition (the restricted C*-system)

ix 1 1 2 2 2 2 3 3 5 5 5 5 6 7 7 7 9 11 14

Bounded derivations Introduction to derivations The commutation relation ab — ba=\ Continuity of everywhere-defined derivations Quantum field-theoretic observations (some unbounded derivations) Application to bounded derivations Uniformly continuous dynamical systems C*-dynamical systems and ground states

16 17 17 22 24 34 41 48

iii

Contents

3 3.1 3.2 3.3 3.4 3.5 3.6

Unbounded derivations Definition of derivations Closability of derivations The domain of closed *-derivations Generators Unbounded derivations in commutative C*-algebras Transformation groups and unbounded derivations

55 56 58 65 72 79 94

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8

C*-dynamicaI systems Approximately inner C*-dynamics Ground states KMS states Bounded perturbations UHF algebras and normal *-derivations Commutative normal *-derivations in UHF algebras Phase transitions Continuous quantum systems

101 101 107 114 130 155 \^ 175 133

References Index

207 217

PREFACE

Derivations appeared for the first time at a fairly early stage in the young field of C*-algebras, and their study continues to be one of the central branches in the field. During the past four decades, the study of derivations has made great strides. Their theory divides naturally into two major parts: bounded derivations and unbounded derivations. About thirty years ago, Kaplansky (in an excellent survey [97] on derivations) brought together two apparently unrelated results which stimulated research on continuous derivations. The first, related to quantum mechanics and due to Wielandt [190] in 1949, proved that the commutation relation ab — ba=l carinot be realized by bounded operators. The second, involving differentiation and due to Silov [180] in 1947, proved that if a Banach algebra A of continuous functions on the unit interval contains all infinitely differentiable functions, then A contains all n-times differentiable functions for some n. It is noteworthy that Kaplansky's observations three decades ago are still applicable to recent developments in the theory of unbounded derivations, although one has to replace quantum mechanics by all of quantum physics. Furthermore, the work of Silov continues to have a strong influence on the study of unbounded derivations in commutative C*-algebras. At an early stage, mathematicians devoted most of their efforts to the study of bounded derivations, rather than unbounded ones, even though the work of Silov and Wielandt had already suggested the importance of unbounded derivations. This, of course, is understandable because bounded derivations are more easily handled than unbounded ones. The study of bounded derivations has led to a beautiful mathematical theory that provides (among other things) essential tools for the study of unbounded derivations. In contrast to the study of bounded derivations, which is now approaching completion, the study of unbounded derivations is still in an early stage. Their study began much later and initially was motivated by the problem

x

Preface

of constructing the dynamics in statistical mechanics. It soon became apparent that the work of Silo v was also important in the study of unbounded drivations in commutative C*-algebras. The contents of the present book are based in large part on the author's lecture notes on the theory of unbounded derivations in C*-algebras given at the Universities of Copenhagen and Newcastle upon Tyne in 1977 [170]; also included are some of the extensive new developments discovered since 1977. Unbounded derivations is a fast growing field and the subjects it involves are quite diverse. Therefore I have made no attempt to give complete coverage of the theory. Rather, I have taken a somewhat personal stand on the selection of material. This selection is concentrated, for the most part, on the topics involving quantum statistical mechanics and differentiations on manifolds. There is a good possibility that the theory of quantum lattice systems in statistical mechanics may also be developed naturally within the context of unbounded derivations in C*-algebras (although its phase transition has not yet been established even for the three-dimensional Heisenberg ferromagnet). In fact, many theorems in the theory of quantum lattice systems have already been formulated for normal *-derivations in UHF algebras. One of the most ambitious programs in the theory of unbounded derivations is to develop quantum statistical mechanics within the C*-theory. Of particular importance is the generalization of phase transition theory to quantum lattice systems. The prospect of success is not necessarily gloomy, because the absence theorem for the phase transition theory (in one-dimensional quantum lattice systems with bounded surface energy) has already been successfully formulated in the quite general setting of UHF algebras ([5], [104], [168], [169]). This 'absence theorem' refers to a result where, in a restricted context, the temperature states can be shown to be unique. Hence the 'absence' of phase transitions. One of the goals of the present book will be to formulate the absence theorem for phase transitions in its most general form within the C*-setting. Compared with the theory of quantum lattice systems, the theory of continuous quantum systems is somewhat incomplete. Except for a few examples, time evolutions have never been constructed for interesting interaction models. As a matter of fact, even derivations have never been constructed globally. In the present book, we shall construct derivations globally for a fairly wide class of interaction models in the C*-setting. Also, an axiomatic treatment of the construction of time evolutions and KMS states will be attempted. However, it is not thoroughly discussed, and much is left to future research. A few words concerning the theory of unbounded derivations in commutative C*-algebras: I have not attempted to give complete coverage

Preface

xi

of recent important developments. (The reason for this is that the results are so numerous that it would be more suitable to publish another book concentrated on the subject.) Instead, I have included comments on matters related to transformation groups on locally euclidean spaces because I believe it may turn out to be one of the most important problems in the subject for future research. With respect to non-commutative manifolds, there is an excellent book by Bratteli [19], so I will not discuss them in the present work. References to theorems refer to the section number for example Theorem 5.2.2 refers to the theorem in Section 5.2.2. The author expresses his appreciation of the friendly care with which Professor P.E.T. Jorgensen has read the manuscript, of many valuable suggestions he made, and of his help in the proof reading. The author acknowledges also the invaluable assistance of Dr David Tranah and Ms Frances Fawkes, Cambridge University Press. Shoichiro Sakai Sendai, Japan

Preliminaries

In this chapter we shall state some fundamental facts used in this book. We shall also use elementary facts on C*-algebra and W*-algebras freely. The reader can find the proofs of them in the author's book [165] or other standard text books [41], [42], [139], [193], [194], [195], [196]. In addition we shall use some common notation without definition. The reader can find the appropriate definitions in [165].

1.1 Banach algebras, C*-algebras and W*-algebras Let s0 be a linear associative algebra over the complex numbers. The algebra $0 is called a normed algebra if there is associated to each element x a real number ||x||, called the norm of x, with the properties: (1) (2) (3) (4)

|| x || ^ 0, and || x || = 0 if and only if x = 0; ||x + 3 i < ||x||+ IWI; \\Xx\\ = \1\ ||x||, where X is a complex number;

If stf is complete with respect to the norm (i.e., if $4 is also a Banach space), then it is called a Banach algebra. A mapping xi—>x* of stf onto stf itself is called an involution if it satisfies the following conditions: (i) (x*)* = *; (2) (x + y)* = ** + }>*; (3) (xy)*=y_*x*; (4) (Ax)* = Ax*, X a complex number. An algebra with an involution is called a *-algebra. A Banach *-algebra is called a C*-algebra if it satisfies ||x*x|| = ||x|| 2 for xes/. It is easily seen that ||x*x|| = ||x|| 2 for xesrf implies ||x*|| = ||x|| (XGJ/).

2

Operator algebras in dynamical systems

A C*-algebra Jt is called a W*-algebra if it is a dual space as a Banach space (i.e. there is a Banach space Ji^ such that the dual (Jf^)*o{ Ji^ is Jf). We call Ji^ the pre-dual of Ji. In general, a dual Banach space is not necessarily the dual space of a unique Banach space. However the W*-algebra has a unique pre-dual space. The W*-algebras are also called von Neumann algebras and the present definition (W*-algebras) agrees with the familiar alternative definitions of von Neumann algebras. This Theorem is due to the author.

1.2 Topologies on C*-algebras and W*-algebras The topology defined by the norm || • || on a C*-algebra stf is called the uniform topology (or the norm topology). The weak *-topology o(Ji, Ji^) on M is called the weak topology or the cr-weak topology.

1.3 *-Homomorphisms, *-isomorphisms and ^-automorphisms Let ja/, $ be C*-algebras. A linear mapping of ,rf into J* is said to be a *-homomorphism if it satisfies: (1) (2) A *-homomorphism O of a C*-algebra $4 into $ is always normdecreasing (i.e. ||O(x)|| ^ ||x||). Moreover the image (yx) for x,yesi.

1.6 Commutative C*- and W*-algebras This section gives the spectral representation for commutative algebras (both C* and W*). Let Q be a locally compact Hausdorff space, and let C0(Q) be the algebra of all complex-valued continuous functions on Q vanishing at infinity. Define ||a|| = sup,eQ |a(t)\ and a*(t) = a(t) for aeC 0(Q). Then C0(Q) is a commutative C*-algebra. If K is a compact Hausdorff space, C(K) (the algebra of all complex-valued continuous functions on K) is a commutative C*-algebra with identity. The converse statement is also true: If si is a given commutative C*-algebra, then it is *-isomorphic to C0(Q), where Q is some locally compact Hausdorff space; if si is assumed to be commutative with identity, then si is *-isomorphic to C(K), where K is a compact Hausdorff space. Let (Q,/x) be a given measure space with /j(Q) < + oo, let L°°(Q,/x) be the C*-algebra of all essentially bounded ju-measurable functions on O and let L1 (Q, n) be the Banach space of all /i-integrable functions on Q. Then by the Random-Nikodym theorem, 1/(0,/*)* = L00(Q,iu). Hence L°°(Q,/i) is a commutative W*-algebra. More generally, let (F, v) be a localizable measure space (i.e. a direct sum of finite measure spaces) and let L°°(r,v) be the C*-algebra of all v-essentially bounded measurable functions on F, and let Lx(F,v) be the Banach space of all v-integrable functions on F. Then (L^F, v))* = L°°(F, v); hence L°°(r, v) is a commutative W*-algebra. Conversely, if M is a commutative W*-algebra, then it is *-isomorphic to 00 L (F, v), where (F, v) is a localizable measure space. Hence we also have a spectral representation for the commutative W*-algebras.

1.7 Concrete C*- and W*-algebras Let Jf be a complex Hilbert space and let B(Jf?) be the algebra of all bounded linear operators on Jf. We can define various topologies on B(34f). Although well known, they will be reviewed below. (1) The uniform topology This is given by the operator norm || a \ where ||a|| = sup, mK1 ||r^||. Take the adjoint operation a\->a* as the involution* (i.e. (a£,rj) = (?;,a*rj) (£,rjeJ4?)), where ( , ) is the scalar product (inner product) of 3ft). Then, B(3tf) is a C*-algebra and moreover any

4

Operator algebras in dynamical systems

uniformly closed self-adjoint subalgebra srf (i.e. aesrf implies a*estf) o is itself a C*-algebra. (2) The strong operator topology Let £eJf. The function ai—• || a£ || is then a semi-norm on B(jjf). The set of all such semi-norms {\\a£\\ | ^eJf} defines a Hausdorff locally convex topology on B(J^), called the strong operator topology. (^3^) The weak operator topology For each pair £, neJtf, the function a->|(a£,J7)| defines a semi-norm on B ( ^ ) . The set of all such semi-norms {\(ai,rj)\\^rjeJ^} defines a Hausdorff locally convex topology on £(Jf), called the weak operator topology. (4) The a-weak topology Let Tr be the trace function in B(Jf) and let T(Jf) be the set of all trace class operators on J-f. For aeT(J^\ define ||aHi = Tr(|a|), where \a\ = (a*a)1/2. Then T p f ) becomes a Banach space with the norm || -1| x. L e t / b e a bounded linear functional on B(J^) which is assumed to be continuous on bounded spheres of B(J^f) with respect to the weak operator topology. Then there is a unique trace class operator a on Jf such that/(x) = Tr(xa) for xeB(Jf) and | | / | | =Tr(|a|). Therefore T(Jf) can be identified with the Banach space of all bounded linear functionals of B(J^) which are continuous with respect to the weak operator topology on bounded spheres of #(Jf). Then it is known that Tpf)* = B(Jf). Hence B(JfP) is a W*-algebra and {y*x) in s/. Let J = {x\<j>(x*x) = 0, xesf}. Then J is a closed left ideal of si. Define a conjugate bilinear functional on the quotient linear space stjJ such that if xex^ yey^ then (x^y^) = (y*x) (here x 0 (resp.^) is the class containing x(resp. y)). The expression (x^, y^) does not then depend on a special choice of the representatives x, y. It will define a scalar product on sijJ under which sijJ will become a pre-Hilbert space. Let 2tf$ be the completion with respect to this scalar product. Then Jf ^ is a Hilbert space. Now we shall construct a ^representation of si on 2tf$ (i.e. a *-homomorphism of si into B(J^^)) via 0, denoted by {n^ Jf^}. Put n(a)x = (ax) a n d moreover [TfyOfl/Jfo] = ^ * (i.e. {TT 0,,#%} is cyclic), where [(•)] is the closure of (•) in 2/f #. If si has an identity 1 and 0 is a state on si, then £0 = 1^,. (2) From representation to state Conversely, if TT is a *-representation of J ^ into B(3tf) such that [7T(J/)^ 0 ] = ^ , then {TC, Jf} is unitarily equivalent to {^, ^ } , where is an extreme point in the convex set 9"^. If si has an identity, then 9^ is G(S/*, j/)-compact. The space y ^ with a (si*, si) is called the state space of si. In Chapter 4, we shall often use harmonic functions, and so we shall now state the fundamental facts on harmonic function, which we shall need in Chapter 4. Let C be the complex plane, and let D be the closed unit disk in C such that Z) = {z e C | | z | ^ l } . Let f$ be a non-zero real number and let 5^ be the closed strip of the complex plane such that Sp = {zeC|0 ^ Im(z) < 0 if p > 0 or p ^ Im(z) ^ 0 if j8 < 0}. Let D° (resp. Sfi°) be the interior of D(resp. Sfi) and let Dx = {zeD\z # - 1,1}. Consider a mapping co = (P/n)\ogi(l - z)/(l +z)ofD1in the z-complex plane onto Sp in the co-complex plane. Then it is a one-to-one bicontinuous mapping of D1 onto Sp and moreover it is a conformal mapping of D° onto S^0. Therefore, by Poisson's formula and the result from the Dirichlet problem, we have the following theorem (cf. [77]).

1.15 Perturbation expansion theorem (teR)

7

1.13 Theorem (Poisson Kernel for the strip) There exist two positive continuous functions K1(z,t),K2(z,t) that K 1 (z,*)dt+ J — oo

K2(z,t)dt=l

on Sp° x IR such

for zeSp°

J - oo

and for any two real-valued, bounded continuous functions fu f2 on U, P oo

u(z)=\

f* oo

f1{t)K1(z,t)dt+

\

J — oo

f2(t)K2(z9t)dt

for zeSfi°

J — oo

is harmonic on Sp° and moreover hm u(z) =J\(t)

and

z->t zeSn0

lim

u(z)=f2(t)(teU).

z^*t + i zeSjP

Furthermore ifu1 is a bounded continuous function on Sp which is harmonic on Sp°, and ux{t) =f1{t\ ut(t + ip) =f2(t) (teR), then u^z) = u(z)for zeSp. We shall not use the explicit form for the functions Kl9 and K2 but only the representation theorems stated here and their corollary:

1.14 Corollary (analytic version) Letf(z) be a bounded continuousfunction on Sp which is analytic on Sp°; then

f(t)K1{z,t)dt+ P f{t + iP)K2(z,t)dt

forzeS,0,

J — oo

Perturbation Theory We shall also need some fundamental results on bounded perturbations in Chapter 4. Here we mention some of them. Let A, B be bounded linear operators on a Banach space E. Let ut = exp(t^) and Tt = expt(A + B){teR) be exponential one-parameter groups.

1.15 Perturbation expansion theorem (teU) We have oo

Tt = ut+ X

r

utlBut2.tlBut3_t2.--utn_tn_lBut_tndt1dt2---dtn.

where the series is norm-convergent for all t.

8

Operator algebras in dynamical systems

Let Tt(0) = u, and

Proof t

\ ^

^

^

^

ti

^

t2

ti

t3

t2

tn tn

1

t tn

1

2

n-

Then ,0 and dot(x)Tfn(a) = T /n (da(x))^da(x)(a); hence da(x)|V(a) = da(x), where

14

Operator algebras in dynamical systems

da(x)| V(a) is the restriction of da(x) to V(a) and da(x)| V(a) is the closure of da(x)| V(a). Now let D(a) be the *subalgebra of stf generated by Via) (actually D(a) = V(a), see Dixmier-Malliavin [54]) and let C°°(a) be the *-subalgebra of all infinitely differentiable elements in s/ with respect to a - i.e., t\-+at(a) (fleC°°(a)) is infinitely differentiable with respect to G. Then D(a) a C°°(a). It is clear that C°°(a) c f]xe^{da(x)). We shall write da(x) = da(x)|C°°(a); then one can easily see that da([x,y]) = [da(x),da(y)] for x,ye^ and so the mapping x\-^da(x) (xe^) is a Lie homomorphism and Let J f be a Hilbert space. The W*-algebra of all bounded operators on Jf is denoted by B(Jf). A W*-algebra Jt in a Hilbert space J f means a weakly closed *-subalgebra of B(J4f). A factor is a W*-algebra with its center consisting of scalar multiples of the identity. By M\ we shall denote the commutant of M - i.e.

M' = {x'eB{tf)\x'a = ax' for all aeJi}. If M is a W*-algebra containing the identity operator in a Hilbert space Jf, then (e/#y (denoted by ^ " ) = M. In this case, ^ is also called a von Neumann algebra. Let M be a W*-algebra containing the identity operator 1 ^ in a Hilbert space J"f, and let <JT be the commutant of Ji. A linear operator T in ^ is said to be affiliated to Jl if for each unitary u'eJC', u'2(T) = 9{T) and u'*Tu' = T, where S(T) is the domain of T. If T is affiliated to Jl, we shall denote this by TnJi. Next we shall discuss the relationship between C*- and W*-dynamical systems.

1.18 Proposition (the restricted C*-system) Let \M, G,oc} be a W*-dynamical system with a locally compact group G; then there exists a C*-dynamical system {srf, Gya] such that stf is a a-weakly dense C*-subalgebra of' M with an identity and a is the restriction of a to stf. Proof Let v be a left invariant Haar measure on G and let LX(G, v) be the

group algebra of G. For aeJl, feL^G.v), put Tf(a) = \at(a)f{t)dv(t\ where the integral is defined by using the cr-weak topology on Jl\ then one can easily see that Tf(a)eJl. Moreover,

Lts(a )f(s)dv(s) Ls(a :)/(s)dv(s) Ls(a )f(r s)dv(s)l

l

Las(a(a)/(s)dv(s)

s)-f(s)\dv(s).

s

Unitary representations

Hence lim^ e \\oct(Tf(a))-Tf(a)\\

=0. For fl9f2eL\G,v),

+ «t(Tfl(ai))Tf2(a2)

15

al9a2eJ(9

xt(Tfl(ai))Tf2(a2) - Tfl(a)Tf2(a2) \\

^ MTfl{ai))\\ MTf2{a2))-Tf2{a2)\\ + ||a r (7> 1 (a 1 ))-T r i (a 1 )|| ||7> 2 (a 2 )||->0

(t-+e).

Let sf0 be the *-subalgebra of Ji generated by [Tf{a)\feLl{G,v\ aeJK}; then by the above consideration, || cct(x) — x || -• 0 (t -• e) for xestf0. Let stf be the uniform closure of stf0. For yesf, e>0, let xesrf0 be an element such that || y — x || < e; then

I *Ay) - y K I *tiy) - «rM I + I a t (x) - x || + || x - y \\. Hence lim r _ e||a f (y) — y|| ^2e. Since e is arbitrary, limf ^e || oct(y) — y || = 0. Therefore {J?/, G, a} is a C*-dynamical system, where & is the restriction of a to s/. It is easily seen that si is cr-weakly dense in M. D

Unitary representations Now let J( be a W*-algebra containing the identity operator 1 #> on a Hilbert space Jf, and let ti-^wf be a strongly continuous unitary representation of a topological group G on / such that utJ(u^ = Ji (teG). Then, put af(a) = Mrau,* (teG.asJi). It follows that each a, is a *-automorphism of M and that the mapping t\-^at is a homomorphism of G into k\x\(M\ For £e Jf7, let ^ ( a ) = (a^, f) (aG^#), where ( , ) is the inner product of jf. Then sup ||aKl

= sup ||aHl

^ sup \(au*Z,u*Z)-(au*Z^)\+

sup

Hence 11(^)^0^ — ^ | | - > 0 (ri-^^). Since all finite linear combinations of all vector states on M are dense in Ji^ || ( a ^ / —/1| -• 0 (t -> ^) f o r / e ^ ^ ; hence {^#, G, a} is a W*-dynamical system.

Bounded derivations

Introduction Derivations are defined by the familiar Leibnitz formula. As operators they may be bounded or unbounded. The bounded case will be discussed first. Derivations appear in various branches of mathematics and physics. They clearly have their origin in the concept of differentiation developed by Newton and Leibnitz. Lie algebras and their theory have a long history in mathematics. They may also be considered to be part of the theory of pure algebras. However, the study of derivations in operator algebras originated in quantum mechanics rather than in Newtonian mechanics. In late 1924, and early 1925, Heisenberg and Schrodinger both proposed explanations for the empirical quantization rules of Bohr and Sommerfeld. These explanations, which were originally known as matrix mechanics and wave mechanics respectively, are now known as quantum mechanics in the present theory of atomic structure. In 1931 Stone and von Neumann showed that these two formulations were equivalent. Heisenberg's formalism identified the coordinates of particle momentum and position with operators Pi and qj satisfying the canonical commutation relations: PiPj ~ PjPi = Wj ~ QjQi = 0;

ptqj - qjPi = - ife

where h is Planck's constant. Heisenberg's formalism was tentatively proposed in terms of matrix operators. However, a simple calculation with the commutation relations shows that they cannot be matrix operators. In the 1940s, it was of central interest for mathematicians as to whether or not the commutation relations could be realized by bounded linear operators on Banach spaces. Subsequently the studies of bounded linear operators on a Banach space have been extensively exercised on the commutators of bounded operators. Let £ be a Banach space and B(E) the algebra of all bounded linear 16

2.2 The commutation relation ab — ba=l

17

operators on E. For X, YeB(E\ put bx(Y) = [X, Y~\=XY- YX; then bx is a bounded derivation on B(E). Therefore, the theory of commutators stimulates the study of bounded derivations on Banach algebras. In this chapter we shall mainly discuss bounded derivations and, to some extent, unbounded derivations which arise from quantum field-theoretic observations.

2.1 Introduction to derivations Let s/ be a Banach algebra, b a linear mapping in s/. Then b is said to be a derivation in s/ if it satisfies the following conditions: (1) the domain S)(b) of b is a dense subalgebra in srf\ (2) b(ab) = b(a)b + ab(b) (a, be2>(&)). If Q)(b) = s/, then b is said to be a derivation on s/. If b is bounded, then b can be uniquely extended to a bounded derivation on jtf and so it may be considered to be defined on stf. Let stf be a *-Banach algebra, b a derivation in s/. Then b is said to be a *-derivation if it satisfies: (3) ae2>(5) implies a*e^(0 and ^(x n )-^j; implies j = 0. If b is closable, then b can be extended uniquely to the least closed derivation b, called the closure of b.

2.2 The commutation relation ab — ba = \ 2

Let L (IR) be the Hilbert space of all square integrable, complex-valued functions on the real line IR with respect to the Lebesgue measure, / a complex-valued measurable function on IR. For geL2(M), define Tfg = f-g; then Tf will define a closed linear operator in L2(U) with a dense domain. Let a = d/dt, b = T / o , where fo(t) = t. Then for geL2(U) with g'eL2{U)9

at = g{t) + tg\t)-tg\t) = Hence ab — ba = 1 on a dense subset of L2(IR); therefore the commutation relation is realized by linear operators in L2(U); however a and b are unbounded. If we take L 2 ([0,1]) as a Hilbert space, then b is bounded, but a is unbounded. However they cannot both be bounded; see later. Wintner [191] (1947) for Hilbert spaces and Wielandt [190] (1949) for general Banach spaces proved that this is impossible for bounded operators.

18

Operator algebras in dynamical systems

After these results, the theory of bounded linear operators on a Banach space has been extensively developed around the commutation relation. In the following, we shall discuss parts of these developments. 2.2.1 Theorem (Kleinecke [110], Sirokov [182]) Let si be a Banach algebra, 5 a bounded derivation on si. Suppose that S2(a) = 0; then S(a) is a generalized nilpotent - i.e., (|| ^(a)M || )1/w->0 (n^oo). Proof S(a)=V.3(a)1. formula,

Suppose that 3n{an) = n\S(a)n; then by Leibnitz's

Sn + 1(an + 1) = dn+1{ana) = Sn+I(an)a + ("

+ 1

jSn(an)S(a)

+ (n+l)n\d(a)n

+1

= S(n\S(a)'"3(a))a + (n + l)\S(a)n+1 =

(n+l)\d(a)n+1.

Hence dn(an) = n\S(a)n

(n = 1,2,3,...).

2.2.2 Corollary (Wielandt [190], Wintner [191]) Let si be a Banach algebra. Then there are no two elements a, b in si such that ab — ba= 1. Proof For xesi, let 5a(x) = ax — xa = [a, x]

{xesi)',

2

then 3a is a bounded derivation on si. 8 a{b) = Sa(l) = 0iiab — ba=l. Hence • by Theorem 2.2.1, 3a(b) is a generalized nilpotent, a contradiction. 2.2.3 Corollary (Singer-Wermer [181]) Let si be a commutative Banach algebra, 6 a bounded derivation on si; then 5(si) is contained in the radical of si. In particular, if si is semi-simple, then 3 = 0. Proof Let B(si) be the algebra of all bounded linear operators on si. For a,xesi, let Lax = ax; then LaeB(si). [5,Lfl](x) = (5L fl -L fl 5)(x) = 5(ax) - aS(x) = 8(a)x = Lmx. Hence Id, Lfl] = Lm and so \La, [La, 5]] = — [La, Lj(fl)] = 0, for a commutes with d(a). Hence by Theorem 2.2.1, LS(a) is a generalized nilpotent, and so (|| Ld(a)n || )1/M = (|| d(a)n || ) 1/n -> 0. D 2.2.4 Corollary (Silov [180]) Let C°°([0,1]) be the algebra of all infinitely differentiable functions on the unit interval [0,1]. Then there is no norm on C°°([0,1]) under which C°°([0,1]) becomes a Banach algebra.

2.2 The commutation relation ab — ba=\

19

Proof Suppose that C°°([0,1]) is a Banach algebra under a norm |||-|||. Consider the differential operator d/dt on C°°([0,1]). For each te[0,1], put Xt(f) = f(t) (/eC°°([0,1])); then AT, is a character and so it is continuous under the norm |||-|||. Hence | | / K III/III for /eC°°([0,1]), where ||-|| is the uniform norm. Suppose that |||/J||->0 and W/J, —0|||-»O, where g is an element of C°°([0,1]). Then \\f'H-g\\^0. Hence £ / # ) dt = fH(x) - fH(0) fog(t)dt. Hence ]**#(£) dt = 0 for xe[0,1] and so g = 0. By the closed graph theorem, the linear operator d/dt on C°°([0,1]) is bounded with respect to the norm |||-|||. By Corollary 2.2.3, d/dt (C°°([0,1])) is then contained in the radical of C°°([0,l]). Since ()te[O,1]J?t = 0, where J/t = {/eC°°([0, l])|/(t) = 0}, C°°([0,1]) is semi-simple and so d/dt = 0, a contradiction. • 2.2.5 Theorem (cf. Rosenblum [157]) Let srf be a C*-algebra, S a bounded derivation on s/. Suppose that 8(x) — 0 for a normal element x (i.e. x*x = xx*) of s/; then 5(x*) = 0. Proof d (exp(i/lx*)) = d (exp(Ux*) exp(iXx) exp( — iXx)) = (5(exp(Ux*)exp(iXx))exp( — iXx) + exp(Ux*) exp(iXx)^ (exp( — iXx)), n+1

n

AeC

n

Suppose that ^(x") = 0; then 5{x ) = S{x )x + x 5{x) = 0. By induction d(xn) = 0 (n = 1,2,3,...). Since 3(1) = 0 and 8 is bounded, 8 (exp( - iXx)) = 0. Therefore, S (exp(Ux*)) = 3 (exp(Ux*) exp iXx)) exp( - iXx). Moreover 8 (exp(Ux*)) exp( — Ux*)= (5(exp Ux* exp iXx) exp( — iXx) exp( — Ux*) = (5(expi(Ax* + Xx))exp - i(Xx + >lx*). Put f(X) = 8(cxpi(Xx* + Xx))exp - i(Xx + Ax*) = 5(exp(Ux*))exp(-Ux*); then f(X) is complex-analytic on the whole plane of the complex field. Moreover | | / W K H < 5 | | , for expi(/lx*+ Xx) and exp - i(Ax* + Xx) are unitaries. By Liouville's theorem, f(X) = a constant, and so f(X) = be a state on s/ such that \(j)(x)\ = \\x\\ then we

2.3 Continuity of everywhere-defined derivations

23

shall show that 0((/i 2 ) 1 / 2 = 0.

Hence (5{x)) = 0. Suppose that xn( = xn*)-+0 and 8(xn)-+y(^0). Let (j)n be a state on stf such that | n(y + xn)\ = || y + xn ||, and let 0 be an accumulation point of {(/>„} in the state space of «s/; then

\nj(y + ^ - ) - 4>ni{y)\ + I ^ O ) - 0o(y)l for some subsequence (n7) of (n) depending on y. Hence \(j)0{y)\ = llyll and so (j)0{d(y)) = 0. On the other hand, o = 4>nj(&{y + xnj)) = ci>nj(S(y) + s(xHJ)) - MKy)

+ y).

Hence (froiy) = 0, a contradiction. By the closed graph theorem, 3 is bounded.

• Theorem 2.3.1 can be extended to general semi-simple Banach algebras, though the proof is much more complicated and will be omitted here. 2.3.2 Theorem (Johnson-Sinclair [84]) Let srf be a semi-simple Banach algebra and let 5 be a derivation on si\ then 3 is continuous. 2.3.3 Notes and remarks The notion of derivations can be extended to a linear mapping of a subalgebra into a larger algebra as follows. Let J* be a Banach algebra and let UteB(J^) with non-negative spectrum (i.e. in the Stone representation Ut = exp(itH), H is a positive self-adjoint operator in Jf) such that OLt(a)= UtaUt* (aeJt) and H has a zero-eigenvalue vector £0 with ||£ 0 || = 1 and £ 0 is a cyclic vectorfor Ji. Then UteJi (teR) - namely, {Ut} is observable. Proof Since UtJiU* = Ji, it follows that UtJCU* = Ji'. Let x'eJi' and a,beJi. Then f{t) = (Utx'U*aU b£0) = {aUtx'U*£o, Ho) = {aUtx'£09 K o ) = (a exp(itif )x'{ 0 , Ko)Since H ^ 0, f(t) can be extended to a bounded analytic function / on the upper half-plane of the complex field as follows; f{t + is) = [a exp i(r + is)Hx'£0, b£0) = (aexp(itff)exp(-stf )x'f o, Ho)

(s > 0).

On the other hand, f(t) = (Utx'U*aU b£o) = (Ho, bUtx'*U*Z0) = K o , bUJ*^) =

= {U*b*at0, x'*S0)

(exp(-itH)b*ato,x'*Zo).

Therefore f(t) can be extended to a bounded analytic function / on the lower half-plane of the complex field. Since f(t) = f(t) = f(t) (teR), f(t) can be extended to a bounded analytic function on the whole complex plane

The general case

25

(Painleve's theorem) (cf. [77]). Hence by Liouville's theorem, f(t) = constant and so (Utx'U*a£0, b£0) = (x'a£0, b£0). Since M ^ o ] = JT, Utx'U* = x' and D so UteM" = J(.

Case 2 The general case (the case of positive energy without the assumption of the existence of a vacuum state) In this case, the theory becomes more complicated. We will use the idea of Arveson's spectral theory of one-parameter automorphism groups [6]. Let t\—>Ut be a strongly continuous one-parameter group of unitary operators on a Hilbert space, with the Stone representation Ut = exp(itH) = J^ oo exp(ity)dP(7). Let pt(a)=UtaU* (aeB(Jtv)); then t\-+pt is a a-weakly continuous one-parameter group of *-automorphisms on B(3tf). Let LX([R) be the group algebra of the real line IR consisting of all Lebesque integrable let f(y) = JX/(0exp(ity)dt 9 j8(/) = functions on U. For feL\U\ l-nfWPtdt (the integral is defined by using the a(B(jf), B(J^JJ-topology) Supp(/) is the support of / in the dual group U and U(f) = l^^f^U.dt. of U. The following fact is known: given a compact K a U and an open set WZDK, there is an heL^U) such that h(y)=l for yeX and h(y) = O for

yeU\W. 2.4.2 Lemma / / supp(/) c [>, oo), then j8(/)(a)P([A, ao))Jf aP([A + fi, ao))JtT for A, pen, aeB(JfT)9 where P([/, oo)) = J* dP(y) {leU). and supp(^)cz(A —g, oo) with £ > 0 ; then it Proof Suppose geL1^) is enough to show that U(h)P(f){a)U(g) = 0 for heL^U) with supp(/z)c= (— oo, X + ii — e). We may also assume that / , g, h have compact supports respectively. U(h)P(f)(a)U(g) = l \h(t)Utdt)[ = III

\f(s)UsaUfds)(

\g(r)Urdr

dtdsdrh(t)f(s)g(r)Ua+taU-a+r.

After the change of variables x = t9 y = s + t, z = r — s (note the Jacobian (J(x, y9 z)/J(t9 s, r)) = 1), and an application of Fubini's theorem, one finds

dtdsdrh(t)f(s)g(r)Ut+sU.s+r = {([dxdydzh(x)f(y =

- x)g(z + y- x)UyaUz

\\h*(f-gz)(y)UyaUzdydz,

26

Operator algebras in dynamical systems

where gz{x) = g(z + x).

h*(f'gz) = HNz) supp(/*$ z )cz supp(/) + supp(#z) = supp(/) + supp(#) cz In + k - e, oo). The last set is disjoint from supp(/i), and so h*(f-gz) = O. Hence O U(h)P(f){a)U{g) = 0.

One-parameter groups 2.4.3 Theorem (Borchers [16]) Let t\-^cct be a G-weakly continuous oneparameter group of*-automorphisms of a W*-algebra M containing the identity operator in a Hilbert space ffl. Then the following two conditions are equivalent: (1)

There is a strongly continuous one-parameter unitary group with non-negative spectrum (namely, Ut = exp(tiif), H^O) such that ott{a)=UtaU* (aeJ/, ten). (2) There is a strongly continuous one-parameter unitary group t^>VteJt with non-negative spectrum such that cct(a)= VtaVt* (aeJi, teW). Proof (2)=>(1) is trivial. We shall show (1)=>(2). Let H = J ^ y d P ( y ) be the spectral decomposition of H; then P([t, oo)) = 1 when t ^ 0. For teU9 let el9 e2 be projections in M such that el9 e2 ^P([t, oo)) and let e± v e2 be the supremum of ex and e2 in Jtv, where Mv is the set of all projections in Ji. Since (1 - P([f, o o ) ) ) ^ ^ + e23tf) = 0 , ^ ^ ^ ^ P([r, oo)). Therefore there is the largest projection Pt in Ji such that Pt ^ P([r, oo)). Since UsPtU* < P ( [ t , oo)) and U8PtU*eJf9 UsPtU* = Pt for seU. Clearly Ptl ^ Pt2 if tx ^ t2. For each teR, define qt = As0 strongly as t -• oo. Moreover qt = 1 for t ^ 0. Thus there is a unique projectionvalued measure Px on R such that Pi([>, oo)) = qt(teU). Let Kf = J^ oo exp(ir7)dP 1 (y); then VteM and t->K t has non-negative P^Piy) spectrum. Since VtP^)XJ* = Px(y) for teU and yeR, P(y)P1(iu) = for y, /XGR. Let K = \™ydPl(y); then P x ([t, oo))^P([t, oo)) (teR) and the commutativity of {P(y)\ and {PI(JU)} implies H — K^O, where H — Kis the closure of i / - K . Now let Wt=UtV.t; then ^ = expk(/f - X). Let 1^ = J*^ exp(i/iOdP2(/i) and yt(a) = WtaW* (aeJt, teR). Let F, be the largest projection in Jt such that Ft ^ P 2 ([t, oo)); then UsFtU* ^ t/ s P 2 ([t, oo))L/s* = P 2 ([*,oo)) and so UsFtU* = Ft(seR). Similarly 7sFfKs* = F f . Since K,H^K + tFt. Let X + tF, = J ^ y dP3(y) be the spectral decomposition of K + ti7,; then, if t ^ 0 , P 3 ([t, oo))^P([t, oo)) and P^fr, oo))

Actions by 1R"

27

P3([£, oo)). By the maximality of P^t, oo)), P^lt, oo)) = P3([r, oo)) (f > 0); hence X = X + t¥t (t > 0), and so Ft = 0 (t > 0). By Lemma 2.4.2, for feL^U) with supp(/) c [r, oo) (t > 0), y(/)(fl)JT = y(/)(a)P2([0, oo))Jf c P2([t, oo)pf for aG^#. Let e be the orthogonal projection of 3tf onto ly(f)(a)^f}. Since y(f)(a)eJ(9 eeJi and so e^P 2 ([f, oo)) (t>0). Hence e = 0. Therefore y(f)(a) = 0 for a e J and feL^R) with supp(/) c [*, oo) (t > 0).

Since supp(/) = - supp/, y(gf)(a) = 0 for aeJi and gel}(R) with supp(^) cz (-oo,0)u(0, oo). For geL^R) with J#(r)dt = O, ^(0) = 0; any such g can be approximated by h with Iz, zero in neighborhood of 0 and so y(g)(a) = ig(t)yt(a)dt = 0, $Rg(t)dt = 0 (ZeJtJ and so is the inner product of Un. Now suppose that there exists a closed convex subset A in Un such that no straight line is contained in A and P(A) = 1 ^ , where 1 ^ is the identity operator on Jf. We shall call this condition the spectrum condition. Take an element co^A and let Ao = A — co. Since Ao contains no straight line, there is a linearly independent family {/i,/ 2 ,•••»/„} of linear functional such that A o g {xeR"|all fj{x) ^ 0}. Let fs(x) = (tpx) (xeUn); then =

exp(i(foj,x-co})dP(x) JJA

=

I

exp(i0 for XGA 0 , wAfj.exp( — i < ^ , c o > ) = exp(i2ifJ) for where Hj ^ 0. 2.4.4 Proposition Under the above spectrum condition, let Jibe a W*'-algebra containing the identity operator on a Hilbert space Jf such that utMu? = Ji

28

Operator algebras in dynamical systems

"); then there is a strongly continuous unitary representation ofUn such that VtaV* = utau* {aeJt, teUn).

t\-^Vt(eJi)

Proof By the above consideration, MAfJexp( — i(faj9 co>) = exp(Ui/y) with Hj^O. Let Hj = S™aDydP{y) be the spectral decomposition of //,. Then by the consideration in the proof of Theorem 2.4.3, there is the largest projection qXJ in Ji such that qXJ ^ Pj{[K oo)). Since usqkju* ^ P,([/l, oo)) for seUn and usqXjus*eJf, usqk jU* = qXj for seR". By the considerations in the proof of Theorem 2.4.3 there is a projection-valued measure Qj on U such that Qj&k, oo))= A M < ^ M J in . # . Let 7^, = i™aDexp(U.y)dQJ) = J r exp(i< t, x + co>) dQ(x) = J r exp(i< t, x >) dQ(x — co).

D Now for w, = J R nexp(ioit be a a-weakly continuous representation of Un by *-automorphisms on a factor Ji containing the identity operator in a Hilbert space Jf. Then the following two conditions are equivalent: There is a strongly continuous unitary representation t\-^uteB(J^) of Un satisfying the spectrum condition 'Sp(u)czA' such that (xt(a) = utau* (aeJe,teUn). (2) There is a strongly continuous unitary representation t\-^Vt{eJi) of W1 such that Sp(V) g Sp(u) c A and a,(a) = VtaV* (aeJt, teUn).

(1)

To prove Theorem 2.4.5, we shall provide a lemma. 2.4.6 Lemma Let t\—>Wt be a strongly continuous unitary representation of Un into the unitary group of B(Jf); then xoeUn belongs to Sp(w) if and only if there exists a sequence {£„} of elements in Jf such that \\£n\\ = l and 2 ^ 0 ( t t ^ o o ) for each teU\

Actions by R"

29

Proof Let Wt = Jexp(i)dP(x). Suppose that xoeSp(W). Take a decreasing sequence {Kn} of compact neighborhoods of x0 such that f ) ^ ^ ^ { x o } . Let Pn = \Kndp(x); then Pn * 0. Take an element iH in PHJtT with || {„ || = 1; then

i)— 112 —>0

(n->oo).

Conversely suppose that \\(Wt - exp(i)l)£ B || -+0(n^> oo) for each re(R" with | | ^ | | = 1. F o r ^

= fdllPM^H2 f |exp(i