Tomita-Takesaki Theory in Algebras of Unbounded Operators (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1699

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Atsushi Inoue

Tomita-Takesaki Theory in Algebras of Unbounded Operators

Springer

Author Atsushi Inoue Department of Applied Mathematics Fukuoka University Fukuoka 814-0180, Japan e-maih sm010888 @ ssat.fukuoka-u.ac.jp

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme Inoue, Atsushi: Tomitak-Takesaki theory in algebras o f unbounded operators / Atsushi lnoue. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; Hong K o n g ; L o n d o n ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo : Springer, 1998 (Lecture notes in mathematics ; 1699) ISBN 3-540-65194-2

Mathematics Subject Classification (1991): 47D40, 47D25, 46L60, 46N50 ISSN 0075-8434 ISBN 3-540-65194-2 Springer-Verlag Berlin Heidelberg New York 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free tbr general use. Typesetting: Camera-ready TEX output by the author SPIN: 10650140 41/3143-543210 - Printed on acid-free paper

Preface

Main part of this note is a summary of my studies during several years of the Tomita-Takesaki theory in O*-algebra. In 1995 I began this work for the preparation of the seminars during the summer of 1996 at the Mathematics Institute of Leipzig university. I wish to thank Professor K. D. Kiirsten and K. Sehmfidgen for their warm hospitality, for their interest in this work and for their encouragement. Further, I wish to thank them for many helpful discussions and suggestions when they visited the Department of our university in 1996, 1997. I also aeknowlege Professor J. P. Antoine (Louvain Catholique University), Van Daele (Leuven Katholieke University), W. Karwowski (Wroclaw University), G. Epifanio and C. Trapani (Palermo University) and A. Arai and M. Kishimoto (Hokkaido University) for giving me opportunities of the seminars and lectures about this work for their colleagues and graduate students of their Mathematical Departments and for many invaluable discussions and many helpful suggestions. I should like to thank Professor H. Kurose and Dr. Ogi for their encouragement and many helpful conversations. It remains for me to express my gratitude to M. Takakura for typing this mannuseript in TeX. This work was supported in part by Japan Society for the Promotion Science and Japan Private School Promotion Foundation. August 1998

Atsushi Inoue

Contents

Introduction .................................................. 1.

Fundamentals of O*-algebras

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1.1 O* -al g eb r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 W e a k c o m m u t a n t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 I n v a r i a n t subspaces for O * - a lg e b r a s . . . . . . . . . . . . . . . . . . . . . . . 1.4 9 I n d u c e d e x t e n s i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 I n t e g r a b i l i t y of c o m m u t a t i v e O * - a lg e b r as . . . . . . . . . . . . . . . . . . 1.6 Topologies of O * - a l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 U n b o u n d e d g e n e r a l i z a t i o n s of yon N e u m a n n algebras . . . . . . . 1.8 * - r e p r e s e n t a t i o n s of , - a l g e b r a s . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 T r a c e functionals on O * - a lg e b r a s . . . . . . . . . . . . . . . . . . . . . . . . .

8

12 15 19 20 23 26 27

29

2.

S t a n d a r d s y s t e m s and m o d u l a r s y s t e m s . . . . . . . . . . . . . . . . . . . 2.1 Cyclic generalized v e c to r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 S t a n d a r d s y s t e m s a n d s t a n d a r d g e n e r al i zed v ect o r s . . . . . . . . . 2.3 M o d u l a r g e n e r a li z e d v e c to r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 G e n e r a l i z e d C o n n e s cocycle t h e o r e m . . . . . . . . . . . . . . . . . . . . . . 2.6 G e n e r a l i z e d P e d e r s e n a n d Takesaki R a d o n - N i k o d y m t h e o r e m 2.7 G e n e r a l i z e d s t a n d a r d s y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 50 57 60 87 95 106

3.

Standard weights on O*-algebras ......................... 3.1 W ei g h t s a n d quasi-weights on O*-algebras . . . . . . . . . . . . . . . . . 3.2 T h e r e g u l a r i t y of quasi-weights a n d weights . . . . . . . . . . . . . . . . 3.3 S t a n d a r d weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 G e n e r a l i z e d C o n n e s cocycle t h e o r e m for weights . . . . . . . . . . . . 3.5 R a d o n - N i k o d y m t h e o r e m for weights . . . . . . . . . . . . . . . . . . . . . . 3.6 Standard weights by vectors in Hilbert spaces ..............

111 114 119 125 129 143 162

Physical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 171 186 196

4.

4.1 4.2 4.3

Quantum moment problem I ............................. Q u a n t u m m o m e n t p r o b l e m II . . . . . . . . . . . . . . . . : ........... Unbounded CCR-algebras ...............................

VIII

Contents

4.4 4.5

S t a n d a r d s y s t e m s in the B C S - B o g o l u b o v model . . . . . . . . . . . . S t a n d a r d s y s t e m s in the W i g h t m a n q u a n t u m field theory . . . .

209 212

References ....................................................

225

Index .........................................................

239

Introduction

This note is devoi~ed to a study of the O*-approach of the Tomita-Takesaki theory (T-T theory) for von Neumann algebras. As well-known, the T - T theory plays an important rule for the structure of von Neumann algebras and the physical applications. An extension of the T - T theory to *-algebras of closable operators called O*-algebras has been given for the Wightman quantum field theory and for quantum mechanics, and we have tried to develop systematically the T - T theory in O*-algebras for studies of the structure of O*-algebras and the physical applications for several years (Inoue [4, 5, 9, 10, 14, 17 -19], Antoine-Inoue-Ogi-T~apani [1], Inoue-Karwowski [1], InoueKarwowski-Ogi [1], Inoue-Kiirsten [1]). The main purpose of this note is to summarize and develop these studies. Let 79 be a dense subspace in a Hilbert space ~ and denote by s the set of all linear operators X from 79 to 79 such that 79(X*) D 79 and X'79 C 79. Then s is a *-algebra under the usual operations and the involution X --~ X t - X* [79. A *-subalgebra of /?(79) is an O*-algebra on 79 in 7-/. O*-algebras were considered for the first time in 1962, independently by Borehers [1] and Uhlmann [1], in the Wightman formulation of quantum field theory. A systematic study was undertaken only at the begining of 1970, first by Powers [1] and Lassner [1], then by many mathematicians, from the pure mathematical situations (operator theory, topological ,-algebras, representations of Lie algebras etc.) and the physical applications (Wightman quantum field theory, unbounded CCR-algebras, quantum groups etc.). Powers defined and studied the notions of closedness, selfadjointness and integrability of O*-algebras in analogy with the notions of closedness and self-adjointness of a single operator, respectively, and investigated the weak commutant which plays a fundamental role in the general theory. The weak c o m m u t a n t 3/Vw of an O*-algebra M on 79 in 7-/is defined by 3d~ = {C E I 3 ( ~ ) ; ( C X ~ I r l ) = ( C ~ [ X t r I ) , V X E Ad,V~,r/E 79}, and it is weakly closed *-invariant subspace of the *-algebra B(7-/) of all bounded linear operators on 7-/, however is not an algebra in general. The self-adjointness of A/I implies Adw79 ~ c 79 and it implies that A&w is a yon Neumann algebra. A survey of the theory of O*-algebras may be found in the recent monograph of Schmfidgen [21]. In Chapter I the general theory of O*-algebras is introduced.

2

Introduction

In Chapter II the notion of standard generalized vectors which makes possible to develop the T-T theory in O*-algebras is defined and studied. We started such a study in case of O*-algebras with cyclic vectors. But, an O*-algebra is not spatially isomorphic with a direct sum of O*-algebras possessing cyclic vectors in general, in other words only a very special subfamily of O*-algebras has representations with cyclic vectors. On the other hand, the concept of cyclic vectors proved to be very useful for studies of O*-algebras. These facts suggest that a generalization of the notion of cyclic vectors would provide a useful tool for investigations of wider class of O*algebras. Here we pursue this idea. \%% explain how to define the notion of generalized vectors. Let M be an O*-algebra on ~9 in 7-{ and (0 E }f. If (0 { iP then every operator in M doesn't act on (0. How do we treat with (0? Three ways are considered: (i) Rigged Hilbert space : ~P[t~] C ~ C IPf[t~]. As usual, (0 is regarded as an element of the topological conjugate dual ~gf[t~] of the locally convex space Z)[t~] equipped with the graph topology t~ on Z). (ii) Generalized vectors: A generalized vector ~ for M is a linear map of a left ideal Z)(A) of M into 7:) such that )~(AX) = A)~(X) for each A E A4 and X E ~P(,~). The generalized vector A(0 for Ad by ~0 is defined by Z)(l~o ) = {X E ~d;~0 E ~D(Xt*) and xt*~0 E D} and A~o(X ) = Xt*~0 for X E ~D(A~o). This is the reason we call such a map ), generalized vector. (iii) Quasi-weights. A quasi-weight ~ is a map of the algebraic positive cone "P(cJI~,) = { E XkXk't"Xk E ~ , } of a left ideal 9Iv of A// into R + such k

that ~a(A + B) = ~(A) + p ( B ) and ,(,XA) = X,(A) for each A, B E P(ff[~) and ~ _> 0. The quasi-weight cv~o by ~0 is defined by r = D(A~o ) and

link ~01l for k

k

P(9l~eo ). Let Ad be a closed k

O*-algebra on 7:) in ~ such that Ad'~D c D and ~ be a generalized vector for Ad satisfying (S)1 1((TP(1)f N ~D(1)) 2) is total in 7-{. The commutant k c of k t 3 is defined by D(;~c) = {K E Adw; ~c E D s.t. KA(X) = N~K,VX ~ Z~(A)} and AC(K) = ~K for K E 7P(kc), and it is a generalized vector for the yon Neumann algebra Ad~. Suppose that (S)2 Ac((TP(Ac) * A ~D()~c))2) is total in 7Y. Then the commutant Ace of ,~c is similarly defined and it is a generalized vector for the von Neumann algebra (Adw).'' The maps A(X) --~ A ( x t ) , X E 7?(A) t ~ 7P(A) and ACC(A) ~ ACC(A*), A E 7P(Acc) * N 7P(Acc) are closable conjugate linear maps whose closures are denoted by Sx and S~cc, re/ 1 1 / 2 and S~cc = J~ccAl~c be polar decompospectively. Let S), = "oxz_~x sitions of Sx and Sxcc, respectively. By the Tomita fundamental theorem, -it ~ t v 4cc(JM~)'J~cc = M ~ and A ~itc c ~[ ,~A~ ~ V A,-~cc = (JMw) , t E I~. But, we it don't know how {Axcc}teR acts on the O*-algebra M in general, and so we define a generalized vector which has the best condition. A generalized vector A for Ad is standard if (S)1, (S)2 and the following conditions (S)3 it vt (S)5 hold: (S)3 A),cc:D C 59, vt E ~ . (8)4 ~A. , xitc c J v l z~- -~. aAN-cict ~- M , E ]t~. it --it (S)5 A~cc(I)(s ~/)(s = ~(s162~ / ) ( s vt E ~ . Suppose s is a

Introduction

3

standard generalized vector for A/[. Then S'~ -- S~cc,t E ~ -* a t ( X ) it --it A~xA~ (X 9 A/I) is a one-parameter group of *-automorphisms of M and the quasi-weight ~ generated by A satisfies the KMS-condition with respect to (a~}t~ ~. To apply the unbounded T - T theory to more examples we weaken the above conditions (S)3 ~ (S)5 and define the notion of modular generalized vectors. A generalized vector A for A4 is modular if it satisfies the conditions (S)1, (S)2 and the following condition (M): There exists a dense subspace s of 79[tM] such that (M)I A/[s c s and (M)2 it A i c c E C ~ for all t 9 ]~. ~Ve need the notion of generalized von Neumann algebras which is an unbounded generalization of von Neumann algebras. An O*-algebra J~4 is a generalized yon Neumann algebra if A/[~79 c 7:) ! ! ! and M = (~4w) c K {X 9 s C X C , vC 9 AAw}. The other unbounded generalizations of von Neumann algebras were considered by Dixon [2], Araki• [1] and Schmiidgen [19]. Suppose A is a modular generalized vector for ~4. Then there exists the largest subdomain 79M of 79 satifying the conditions (M)I and (M)2, and A can be extended to a standard generalized vector As for the generalized von Neumann algebra ( M ,~r~M~'Jwc. The standardness and the modularity of generalized vectors in the following special cases are investigated: A. Standard systems with vectors. B. Standard tracial generalized vectors. C. Standard systems for semifinite O*-algebras. D. Standard generalized vectors in the Hilbert space of all Hilbert-Schmidt operators. E. Standard systems constructed by von Neumann algebras with standard generalized vectors. The Connes cocycle theorem and the PedersenTakesaki Radon-Nikodym theorem for yon Neumann algebras are generalized to standard generalized vectors for generalized von Neumann algebras. In Chapter Ill the notion of standard weights is defined and studied. Weights on O*-algebras (that is, linear functionals that take positive, but not finite valued) have naturally appeared in the studies of the unbounded T-T theory and the quantum physics. The algebraic positive cone P(Ad) and the operational positive cone M+ are defined and the corresponding weights are defined. The GNS-construetion of a weight ~ is important for studies of O*-algebras like for positive linear functionals on O*-algebras and for weights on von Neumann algebras. In the bounded case r _- {X c 7k4;~(XtX) < oo} is a left ideal of M, but it is not necessarily a left ideal of 7k4 in the unbounded case. For example, the condition 9~(I) < co does not necessarily imply ~(XtX) < co for all X 9 A/L So, using the left ideal r {X 9 A/I;~((AX)t(AX)) < oo,VA 9 M} of ,&4, the GNS-representation 7r~ and the vector representation A~ are constructed. We give two important examples of weights. For any ~ 9 79 the positive linear functional w~ on fl'[ is defined by w~(X) = (X~I~),X 9 M, but if ~ 9 7-/\ 79 then the definition of the above w~ is impossible. So, we need to study the quasi-weight w~ defined by ~ as stated at the begining of this section. Another important example is a regular (quasi-)weight. A (quasi-)weight ~ is regular if ~ = sup f~ for some

4

Introduction

net. {f~} of positive linear functionals on 3/1. An important class in regular (quasi-)weights which is possible to develop the T - T theory in O*-algebras is defined and studied. A faithful (quasi-)weight ~ on P ( M ) is standard if the generalized vector A~ defined by A~,(rrv(X)) = k ~ ( X ) , X E ~r is standard. Suppose ~ is standard, then the modular automorphism group {cr~}tciR of .r n r is defined and ~a is a { o / } - K M S (quasi-)weight. To generalize the Connes cocycle theorem for (quasi-)weights on O*-algebras, some difficult problems arise. Let g) and ~b be standard (quasi-)weights on P ( M ) . In case of von Nemnann algebras rr~ and rc~ are unitarily equivalent, but in case of O*-algebras they are not necessarily unitarily equivalent. So, the unitary equivalence of rr~o and rcw is first characterized, and in this case the Connes cocycle theorem for standard (quasi-)weights on generalized yon Neumann algebras is generalized. The Radon-Nikodym theorem for (quasi-)weights on O*-aIgebras is also studied. In Chapter IV the T - T theory in O*-algebras studied in Chapter II, III is applied to quantum statistical mechanics and the Wightman quantum field theory. The quantum moment problem for states on an O*-algebra is first studied. Many important examples of states f in quantum physics are trace functionals, that is, they are of the form f ( X ) = t r T X with a certain trace operator T. It is important to consider the quantum moment problem (QMP): Under what conditions is every strongly positive linear functional on an O*-aigebra a trace functional? This was studied by Sherman [1], Woronowicz [1,2], Lassner-Timmermann [1] and Schmiidgen [2,4,21] etc. Main results of Schmiidgen are here introduced. QMP is also closely related to unbounded T - T theory. In fact, if f is a trace functional on M , then f ( X t X ) = t r ( X f 2 ) * ( X ~ ) , X C Jtd for some positive Hilbert-Schmidt operator ~, and so rrI is unitarily equivalent to a subrepresentation of the *-representation rr of M on the Hilbert space 7-g | ~ of all Hilbert-Schmidt operators on "/-gdefined by rr(A)XQ = A X f 2 for A, X E 3.4. As stated in special eases D in Chapter II, such a representation rr is useful for the T - T theory in O*-algebras. Hence, as QMP for weights, it is important to consider when a weight ~ on Ad+ is represented as ~ ( X t X ) = tr(Xt*f2)*Xt*f2, X E ~Ii~ for some positive self-adjoint operator f2. This problem was recently considered by Inoue-Kiirsten [1] and is here introduced. Standard generalized vectors in unbounded CCR-algebras are studied. Let .4 be the canonical algebra of one degree of freedom, that is, a ,-algebra generated by identity I and hermitian elements p and q satisfying the Heisenberg commutation relation: [p, q] = - i i . The Schrgdinger representation ~r0 d of .4 is defined by (Tr0(p)~)(t) = - i ~ ( t ) and (Tr0(q)~)(t) = t~(t), ~ E 8 ( ~ ) . Von Neumann [1], Dixmier [1] and Powers [1] considered when a self-adjoint representation 7r of .4 is unitarily equivalent to a direct sum ~,~ioTr~ (3 of *representations 7c~ which are unitarily equivalent to the Schr6dinger representation 7r0. Such a representation is called a YVeyl representation of the

Introduction

5

cardinal I0. The Powers results are here introduced. Furthermore, it is shown that a Weyl representation of countable cardinal is unitarily equivalent to the self-adjoint representation 7r| of A defined by ~D(Tr| = 8(JR) | L2(I~) { r E L2(]~) | L2(R); T L 2 ( R ) c 8 ( R ) } and 7r| = 7co(a)T for a C .4 and o(3

T E ~(7r|

and f2~ -- X ~

TM

e-~3./2 j~ e | ~

(~ > 0) is a standard vector for

r~=0

7r| where {fn}~=0,1,... is an ONB in L2(N) consisting of the normalized Hermite functions. Let M be an O*-algebra on 8(]~) generated by 7r0(A) and fo | f00 and 7r a self-adjoint representation of 3/l on 8(1~) | L2(N) defined by 7r(X)T = X T for X E M and T C 8(]1{) | L2(N). Then the positive o(3

self-adjoint operator g?{~} -~ ~ n

c~f~ | f ~ (c~ > 0, rz = 0, 1 , . . . ) defines a 0

modular generalized vector hs?(o~) for the self-adjoint O*-algebra 7c(2t4), and ) ~ ( ~ ~ is standard if and only if c~n = e ~ , n E N U {0} for some ~ E N. Standard generalized vectors and modular generalized vectors in an interacting Boson model and the BCS-Bogoluvov model are given. Standard generalized vectors in the Wightman quantum field theory are studied. The general theory of quantum fields has been developed along two main lines: One is based on the Wightman axioms and makes use of unbounded field operators, and the other is the theory of local nets of bounded observables initiated by Haag-Kastter [1] and Araki [1]. The passage from a Wightman field to a local net of yon Neumann algebras is here characterized by the existence of standard systems from the right wedge-region in Minkowski space.

1. F u n d a m e n t a l s of O*-algebras

In this chapter we state about the basic definitions and properties of O*algebras without the proofs except for Section 1.9. In Section 1.1 the notion of closedness (self-adjointness, integrability) of an O*-algebra is defined and studied in analogy with the notion of a closed (self-adjoint) operator. In Section 1.2 the relation between the self-adjointness of an O*-algebra M and the weak commutant 3d'w are investigated. In Section 1.3 invariant subspaces for O*-algebras are studied. For a closed O*-algebra M on 79 in 7-/, there are some pathologies between a subspace [Y)~ o f / 9 which is Adinvariant (i.e. AJgJ[ c 9J[) and the projection P ~ onto the closure 9)I of 9)I with respect, to the Hilbert space norm. For any Ad-invariant subspace ~ of :D the closure Adgn of the O*-algebra Ad[gN - {X[gJI; X E 3//} is a closed O*-algebra on the closure ~ t ~ of 92R with respect to the graph topology tM in 9)I. But, the projection P ~ of 7-/onto 9JI does not belong to M ~ in general. When P ~ E Ad'w, another closed O*-algebra 3/Ip~ on P N / ? in 9J~ can be defined, and 3/Ip~ is an extension of 2td~ (denote by Ad~ -< Adp_~), but Adgn r A//_p~,~ in general. In this section the relation between Mgn and Adp~, and the self-adjointness of these two O*-algebras are investigated in detail. The different notions of cyclic vectors and strongly cyclic vectors for O*-algebras are defined and investigated. In Section 1.4 induced extensions of O*-algebras are introduced. Let 3,t be a closed O*-algebra on /9 in ~ . C ~D. Then Ad~ is a yon Neumann algebra and X is affiliSuppose 3,~w~D ' ated with the von Neumann algebra (A&w)' for each X ~ .M, so that it is possible to make use of the yon Neumann algebra (M'w)' for studies of O*algebras. Thus the condition 2td'w2? C ~Dis useful for studies of O*-algebras. Even if M ~ is a von Neumann algebra, the condition AdwT? ' C 79 does not necessarily hold. In this section it is shown that if .A,l~ is a yon Neumann algebra then there exists a closed O*-algebra .hf on a dense subspace g in 7-t such that 3,4 -4 Ar, Af'w = M',v and A/~wg c g. In Section 1.5 the relation between a commutative O*-algebra M and the yon Neumann algebra (M~w)' is investigated. The commutativity of an O*-algebra M doesn't necessarily ! ! imply the commutativity of the von Neumann algebra (3dw) . There exists a commutative self-adjoint O*-algebra 79(A, B) generated by commutative essentially self-adjoint operators A and B such that ('P(A, B)'w)' is a purely infinite von Neumann algebra. It is shown that Ad is integrable if and only if

8

1. Fundamentals of O*-algebras

(3/l")' is commutative. In Section 1.6 several topologies on O*-algebras are introduced. The (quasi-)uniform topology,/)-topology and X-topology which are generalizations of the operator-norm topology in case of bounded ease are introduced. They are different in general in case of unbounded case. Further, the other topologies called weak, a-weak, strong, strong*, a-strong and a-strong* are introduced. The relations among these topologies are investigated. In Section 1.7 the notions of extended W*-algebras and generalized von Neumann algebras which are unbounded generalizations of yon Neumann algebras are introduced. In Section 1.8 the basic properties of ,-representations of *-algebras are noted. In Section 1.9 strongly positive linear functionals on O*-algebras are studied. Many important examples of states in q u a n t u m physics are trace functionals, that is, they are of the form f ( X ) = t r T X with some positive trace class operator T. Here trace functionals are studied for the preparation of quantum moment problem studied in Chapter IV.

1.10*-algebras Let 2) be a dense subspace in a Hilbert space ~ with inner product ( I )- By 12(2)) (resp. 12c(2))) we denote the set of all linear operators (resp. closable linear operators) from 2) to 2), and put s

= {X E s

2)(X*) D 2) and X*2) c 2)}.

Then s c s c s and s is an algebra with the usual operations X + Y, c~X and X Y , and s is a *-algebra with the involution X --+ X* ~ X* [2). We remark that if there exists an element X of 12'(29) which is closed, then 29 = 7-/and hence/2*(29) equals the algebra/3(7-/) of all bounded linear operators on T/ (Lemma 2.2 in Lassner [1]). D e f i n i t i o n 1.1.1. A subalgebra of s contained in s is said to be an O-algebra on 2) in 7-/, and a ,-subalgebra of s is said to be an O*-algebra on 79 in 7-/. We first define the notion of a closed O-algebra in analogy with the notion of a closed operator. D e f i n i t i o n 1.1.2. Let 3A1 and Ad2 be O-algebras on 791 and 792 in 7-/, respectively. We say that flA2 is an extension of 2M1 if 791 c 792 and there exists a bijection t of Adl onto fl/I2 such that t ( X ) I791 = X for each X E fl41, and denoted by Adl -< 3A2. Let A/I be an O-algebra on 2) in 7-{. We define a natural graph topology on 2). This topology is a locally convex topology defined by a family {]l IIx;X

1.1 0-aloebras *

rs

9

3/l} of seminorms: II~llx= I1~11§ IIX~ll, ~ c z~, and it is called the induced (or 9raph) topology on 19 and denoted by tz4. D e f i n i t i o n 1.1.3. An O-algebra 34 on I9 is closed if the locally convex

space 19[tz4] is complete. Let 2Vl be an O-algebra on i/) in 7Y. We denote by ~(Ad) the completion of the locally convex space 19[t~]. Then it is clear that

~(.M) c ~(M) -

n D(x). XCA4

For the closure of an O-algebra we have the following T h e o r e m 1.1.4. (1) Let 34 be an O-algebra on 19 in ~ . We put 2 = XF~(M),

XCM,

= {2; x ~ M } Then y ~ is a closed O-algebra on O ( M ) in ~ which is the smallest closed extension of M . (2) Let Ad be an O*-algebra on 19 in ~ . Then ~(Ad) = / ) ( A 4 ) and Y~ is a closed O*-algebra on O(A4) in ~ which is the smallest closed extension of 34. P r o o f . See Lemma 2.5 in Powers [1]. The above M is called the closure of M . E x a m p l e 1.1.5. Let ~ = L2(]~) and 19 the space C ~ ( R ) of all infinitely differentialbe functions with compact supports. We define O*-algebras M1 and Ad2 on i/) as follows:

M1 = { E

Z ~k~ t k ~

;~k,~ ~ r

,~,.~ c N u (0)},

k=0 /=0

M2={}-~A(t) ~

;AeC~(R),n~Nu{0}}.

k=0

Then ~ ( A d l ) equals the Schwartz space 8(Ii{) of all infinitely differentiable rapidly decreasing functions. In fact, M1 is the O*-algebra of the SchrSdinger representation of the *-algebra generated by identity 1 and two self-adjoint elements p, q satisfying the Heisenberg commutation relation: p q - qp = - i l . This will often appear in this note. The O*-algebra A42 on 19 is closed.

10

1. Fundamentals of O*-algebras

\Ve next define the notion of self-adjointness of O*-algebras. Let 3/l be an O*-algebra o n / 9 in ~ . We put v*(M)= M* =

N v(x*), XE34 - x**fv*(M);x

c M},

A XCM

M** =

c M}.

-

T h e n we have the following P r o p o s i t i o n 1.1.6. Let M be an O*-algebra on /9 in ~ . T h e n Ad* is a closed O-algebra on ~D*(Ad) and 3d** is a closed O*-algebra on ~D**(M) such t h a t M -< 3/l -< M** -< M * . These O-algebras 3/l, M , 3/l** and M * d o n ' t coincide in general. P r o o f . See L e m m a 4.1 in Powers [1] and Proposition 8.1.12 in Schmfidgen [21]. D e f i n i t i o n 1.1.7. An O*-algebra M o n / 9 in ~ is said to be algebraically self-adjoint (resp. essentially self-adjoint , self-adjoint ) if A/l** = M * (resp. 3// = r M = 34*). A closed O*-algebra A4 is said to be integrable (or standard) if X* = X t for each X C M . For integrable O*-algebras we have the following P r o p o s i t i o n 1.1.8. Let Ad be a closed O*-algebra o n / 9 in 7-/. T h e following statements are equivalent: (i) A4 is integrable. (ii) X is a self-adjoint o p e r a t o r for each X c M h -- { X E M ; X = X t } . If this is true, then 3,l is self-adjoint and M - {X; X E M } is a , - a l g e b r a of closed operators in ~ equipped with the strong s u m X ' I - Y - X + Y = X + Y, the strong scalar multiplication c~.X = a X , a r 0 = a X , the strong 0 a=0 p r o d u c t X . Y =- X Y = X Y and the involution X ~ X* = X t . P r o o f . See T h e o r e m 7.1 in Powers [1] and T h e o r e m 2.3 in Inoue [1]. E x a m p l e 1.1.9. Let p be a regular Borel measure on L2(IR n, ~). We put

19 = { f E ~ ; / I p ( z ) f ( z ) 1 2 d t , ( z )

~n and ~ =

< oo for all polynomials p},

1.10-al~,ebras *

cr

11

and for any polynomial p define an operator 7r(p) on 77 by

@(p)/)(x) = p ( x ) f ( x ) , f 9 Then M - {~r(p); p is a polynomial} is an integrable O*-algebra on 77. E x a m p l e 1.1.10. We define an essentially self-adjoint operator H1 and symmetric operators nil1 in L2[0, 1] by

{

771 =

{f 9 C~176 1];f(n)(o)

H~

- i ~dF 771

~

= f(n)(1), n = 0 , 1 , 2 , ' ' "}

n771 = { f 9 771; f(k)(0) = f(k)(1) = 0, k = n , n + 1 , - . . } .d

tnH1

-~[~771,n 9 Nu{o}.

Let Adl and had1 be the closed O*-algebras ~(H1) and 7)(~H1) on 771 and n771 generated by H1 and nil1, respectively. Then we have 0J~l

---~ o.A-41 1MI~;~= . . . ~:;~=n.A/[l~.~_ . . 9 =;g=n -<~ . / ~ *1* = n .A/I*1 ---- - / ~ 1 ~-- JVtl::~=O./V~l,?~ AA*'-~ A.4* C 1~

and hence 3/ll is self-adjoint and nfldl is algebraically self-adjoint but not self-adjoint for each n c i%t, and 0Ad 1 is not even algebraically self-adjoint. E x a m p l e 1.1.11. The O*-algebra 2t4 1 in Example 1.1.5 is essentially selfadjoint. In fact, it is well-known that N =

t2 - ~

[8(]R) is an essentially oo

self-adjoint operator in Ad~-i and 8 ( R ) = 77~(N) - A 77(Nk)' which implies k=l

v'(M1) - N v(x*) c N k 1

(M1)

k=l

Hence .Ad 1 is essentially self-adjoint. The O*-algebra J ~ 2 in Example 1.1.5 is not self-adjoint. In fact, suppose_vM2 is self-adjoint. Then AJ2 -< A41 and Adl is self-adjoint, it follows that AJ1 = 2t42, which contradicts Hence A,12 is not self-adjoint.

Cg'(IR)~S(N).

In analogy with the notion of a closed (symmetric, self-adjoint) operator, we have defined the notion of a closed O-algebra (O*-algebra, self-adjoint O*-algebra). We here arrange those relations on the following diagram:

12

1. Fundamentals of O*-algebras Single o p e r a t o r T

O p e r a t o r algebra M

T : closable T : symmetric

3// : O-algebra M : O*-algebra

TcT* T : closed

,

XcX*,

Vxt =XEM M : closed

% D ( T ) is complete w.r.t. the g r a p h n o r m II lit

,

Z)(T) = "p(T)

29 is complete w.r.t. the g r a p h t o p o l o g y t3a

,

D:

N

D(x)

XEAd

T : self-adjoint

M : self-adjoint

% Z)(T*)

:

Z)(T) X6A4

T* = T

M : integrable X*=X,

Vxt=xEM.

1.2 W e a k c o m m u t a n t In this section we define a weak c o m m u t a n t of an O*-algebra which plays an i m p o r t a n t rule for the studies of O*-algebras. D e f i n i t i o n 1.2.1. Let M be an O*-algebra on ~D in ~ . T h e weak comm u t a n t A d " of M is defined by

34" = {C E B(~);

Proposition

(CX{b)

= (C(IXtr~) for all X E J~ and ~,r] E D } .

1.2.2. Ad'w is a weak-operator **

in B(7-g) such t h a t M ' w = ( M ) w

I

closed *-invariant subspace

/

= Mw-

P r o o f . See Proposition 7.2.9 in Schmfidgen [21] and L e m m a 2.3 in InoueTakesue [1]. In general, A,I'w is not an algebra as Proposition 1.2.4 shows later. So, we define the strong c o m m u t a n t s A,I~ and A,/' s of Ad as follows:

~ : = {c ~ M " ; c':D 9 D}, M " s = {C E M'w ; C D C D and C*:D c 29}.

1.2 Weak commutant

13

It is easy to show that Ad~ is a subalgebra of B ( ~ ) contained in M~ but, M ' is not *-invariant in general. If M is closed, then M ' 5 is weak-operator closed and Ad~ is a yon Neumann Mgebra on 7~. We have the following P r o p o s i t i o n 1.2.3. Let M be a closed O*-algebra on 79 in ~ . Consider the following statements: (i) M is self-adjoint. (ii) Ad'w -- Ad;. (ii)' X is affiliated with the von Neumann algebra (M'w)' for each X C M .

(iii) M'w = ( M ) 5 (iii)' t**(X) is affiliated with (M'~)' for each X E 2t4. (iv) M~w is a yon Neumann algebra. Then the following implications hold:

(i)~

(ii) (iii) ~ z ~; ~(iv). (ii)' (iii)'

Any converse implications don't hold in general. P r o o f . See Lemma 4.5 and Lemma 4.6 in Powers [1]. We investigate the relations of the statements in Proposition 1.2.3 in c ~ e of the O*-algebra generated by one symmetric operator. L e m m a 1.2.4. Let 79 be a dense subspace in a Hilbert space ~ and ~(A) the O*-algebra on 79 generated by A t = A in s Let UA be the partial isometry on 7-/ defined by the Cayley transformation (A - iI)(A + iI) -1 of the closed symmetric operator A. Then U~ ~ ~ ( A ) ~ for each n E N. P r o o f . See Lemma 3.2 in Powers [1] and Example 8.5.1 in Schm/idgen [21]. P r o p o s i t i o n 1.2.5. Let 79 be a dense subspace in a Hilbert space 7-/and "P(A) an O*-algebra on 79 generated by A t = A E s Then the following statements are equivalent: (i) A is essentially self-adjoint. (ii) 7~(A)~ is a von Neumann algebra. (iii) 79(A) is algebraically self-adjoint. P r o o f . See Lemma 3.2 in Powers [1] and Theorem 2.4 in Inoue-Takesue

[1]. P r o p o s i t i o n 1.2.6. Let, D and A be in Proposition 1.2.4. Suppose 7)(A) is closed. Then the following statements are equivalent: (i) 7)(A) is integrable.

14

1. Fundamentals of O*-algebras (ii) 7)(A) is self-adjoint. (iii) V(A)~ = V(A)'. o~

(iv) A is essentially self-adjoint and D =

N

n=l (v) A ~ is essentially self-adjoint for n = 1, 2,.-- . P r o o f . See Theorem 2.1 in Inoue-Takesue [1]. E x a m a p l e 1.2.7. Let ~ ) I , H 1 ; J ~ 1 and , ( D l , n H l , n . A 4 1 ( n E N U {0}) be in Example 1.1.10. Then the following results hold: (1) Let T be a closed operator in L2[0, 1] defined by 29(T) = {f C C[0, 1]; f ( t ) - f(O) =

g(s)ds for some g c/;2[0, 1]},

Tf = -ig,

/0'

f C D(T)

( f ( t ) - f(O) =

g(s)ds).

Then H1 and ntTT1 (~7. E N) are essentially self-adjoint operators in L2[0, 1] such that n/-/1 = [-I1, and 0/-/1 is a symmetric operator in L2[0, 1] such that 29(oH1) = {f E 29(T);f(1) = f(0) = 0} and oH1 c T. (2) M1 = P(H1) and r~2M1 = 7)(nil1), and so ~(H1) is self-adjoint, 7)(~H1) is algebraically self-adjoint but not self-adjoint for each n E N, and 7~(oH1) is not even algebraically self-adjoint. (3) 29, =

N k=l

0~)1 = f l I ) ( ~ 1 k) C 29"(5~ k-1

= ecru[0, 1],

o~

= ~)1, ?2 E ~ . k=l R e m a r k 1.2.8. (1) P ( A ) " is not a v o n Neumann algebra if A is not essentially self-adjoint. (2) The algebraic self-adjointness of ~(A) does not necessarily imply the self-adjointness of P(A) (see 7~(,~H1),n c ]~ in Example 1.2.7), and so the implication (iii) => (ii) in Proposition 1.2.3 does not hold in general. In particular, even if Ad" is a yon Neumann algebra, A/['w ~ Ad' in general.

-N

(x)

(3) The condition 2) =

n

29(A'*) does not always imply that A is

n:l

essentially self-adjoint (see T~(0H1) in Example 1.2.7). The essential selfOQ

adjointness of A does not necessarily imply the condition 2P = n D(A'~)" If n=l so, then /)(A) is always self-adjoint by Proposition 1.2.6, which contradicts the above (2).

1.3 Invariant subspaces for O*-algebras

15

(4) The essential self-adjointness of A does not necessarily imply that of A n for n = 1, 2 , . . . (Proposition 1.2.6 and the above (2)). (5) The implication (ii) ~ (i) in Proposition 1.2.3 does not necessarily hold as the following example shows: Let n c 1%t. Let M be a closed O*algebra on ,tiP1 generated by nil1 and {~ | ~; ~, r]C ,~Z)I}, where (~ | ~)~ = (r E ,gD1. As seen in Example 1.2.7, ~H~ is essentially self-adjoint and P(~H1) is not self-adjoint, and so A4 is not self-adjoint. It is clear'that M"

= M'

=

C~.

1.3 I n v a r i a n t s u b s p a c e s for O * - a l g e b r a s m

Let Ad be a closed O*-algebra on :D in 7/. We

denote by ]C the closure of a

subset K] of ~ with respect to the Hilbert space norm and denote by ~t~ the closure of a subset g)l of T) with respect to the graph topolog:7 t2r V~;e first define an induction of M. For any projection E in .A4(v we put

XEE~ = ENd,

X C M , ~ E T);

ME = {xz; x ~ M}.

Then M E is an O*-algebra on E:D in ET-{ and is called an induction of JM. For inductions of O*-algebras we have the following P r o p o s i t i o n 1.3.1. Let A4 be a closed 0*-algebra on :D in ?-/ and E a projection in A4~w. The following statements hold: (1) If M~w is avon Neumann algebra, then (ME)~w is a v o n Neumann alge! bra which coincides with the reduction (Mw)E of the von Neumann algebra M'w.

(2) E 9 c 9*(ME) C E g * ( M ) (3) Suppose E:D c T). Thcn the following statcmcnts hold: (3)1 M E is closed. (3)2 :D*(A4E) = E:D*(M). (3)3 A4E is self-adjoint if and only if E:D = E:D*(M). P r o o f . See Proposition 3.1 in Ikeda-Inoue [1]. Let ~}I be any ~/-invariant subspace of Z). The closure M ~

of the O*-

algebra A d V ~ - {X[g)I;X E ~I} is a closed O*-algebra on ~ t ~ in ~ . The following questions for the projection PN and the O*-algebra 3 / l ~ arise: Q1. Does the projection P N belong to M~w? Q2. Suppose P N E M'w. Does the closed O*-algebras Mp,~ and A d ~ coincide? The following example shows that Q1 is not affirmative in general.

16

1. Fundamentals of O*-algebras E x a m p l e 1.3.2. Let ~ = L2(]~) and d~ = { f c C~(I~); d ~ f E 7-/for n = 0, 1,-.-}, A = -i~

d

Then M ~ :P(A) is a self-adjoint O*-algebra o n / 9 in 7-{. We put = { f E C~(It{); f(t) = 0 for each t E ~ \ [0, 1]}. Then 9J[ is a M-invariant subspace of ~ and P N is given by f(t), t C

( P N f ) (t) = O,

[0, 1]

t ~ [0, 1]

for all f e 7-/. By the simple calculation we have P ~ ~ M~w . We next have the following results for the relations between .A/IN and Adp~: T h e o r e m 1.3.3. Let Ad be a closed O*-algebra on /9 in ~ and ~Y[ a Ad-invariant subspace of/9. Consider the following statements: (i) 9J~ is essentially self-adjoint , that is, Ad~ is self-adjoint. (ii) P~/9* ( M ) = ~ t ~ . (iii) P ~ E dcl~ and A/IG~ = d~4~. (iv) P ~ / 9 c ~D.

(v)

e M'w.

Then the following implications hold: (i) =~ (iii)

(iv) =~ (v).

(ii) In particular, if 2M is self-adjoint, then the above statements (i), (ii) and (iii) are equivalent, and the above statements (iv) and (v) are equivalent. Any converse implications don't hold in general. P r o o f . See Theorem 4.7 in Powers [1] and Theorem 3.3 in Ikeda-Inoue

[1] E x a m p l e 1.3.4. Let ~D be a dense subspace in a Hilbert space 7-/ and any non-zero element o f / ) . Then s = D, and so Pc*(vjg = I and s

= L;t(/9)c,(z~)4 = s

This shows that the implication (iii)

(i) in Theorem 1.3.3 does not. hold in general.

1.3 Invariant subspaces for O*-algebras

17

E x a m p l e 1.3.5. Let 79 = { f 9 C~[0,2];f(~)(0) = f(r0(1) = f(~)(2),n = 0 , 1 , ' ' " }, .d A = -~

f29.

Then 224 = 7~(A) is a closed O*-algebra on 29 in L2[0, 2]. We put 92R = { f e C ~176 [0, 2]; f(,0 (t) = f(n)(t) = 0 for each t C [1, 2] and n = 0, 1 , . . . }. The following statements hold: (1) ~J[ is a M-invariant subspace of 2) and

(P~f)(t) = [~It)'

0
for each f c L2[0, 2]. Hence we have P ~ E Ad'w but P~29 g~ 29. This shows that the implication (v) ~ (iv) in Theorem 1.3.3 does not hold in general. (2) M ~ r M p ~ . This follows since Mm~ and Adp~ are unitarily equivalent to the non-selfadjoint O*-algebra 0 M 1 and the self-adjiont O*-atgebra M1 in Example 1.1.10, respectively. %Ve define the notions of cyclic vectors for O*-algebras. D e f i n i t i o n 1.3.6. Let JM be a closed O*-algebra on 29 in 7~. A vector ~0 in 79 is said to be ultra-cyclic (resp. stron9ly cyclic , cyclic ) if Ad~0 = 2) (resp..A4~O t~ = D, M~o = ~ ) . As seen later in Example 1.3.11, two notions of cyclic vectors and strongly cyclic vectors are different. It is shown later that the two notions coincides in case of integrable commutative O*-algebras with the metrizable graph topology (see Theorem 1.5.5). P r o p o s i t i o n 1.3.7". Let J~ be a closed O*-algebra on 29 in 7-{. Consider the following statements: (i) Every non-zero vector in 29 is strongly cyclic. (ii) Suppose ~ is any Ad-invariant subspace of 29. Then 9)I = {0} or ~ t . M z 29. (iii) Every non-zero vector in 7) is cyclic. ' * 7) C D } --- C I . (iv) M ' s = {O E Ads;O (v) Supose 9JI is any essentially self-adjoint M-invariant subspace of D. T h e n 9)I = {0} or ~ t , ~

= 29.

Then the following implications hold:

(i) ~ (ii)

(iii) => (iv) =:> (v).

18

1. Fundamentals of O*-algebras

In particulaL if 34 is self-adjoint, then the statements (iv) and (v) are equivalent to A4'w = C I . P r o o f . See Theorem 5 in Gudder-Scruggs[1]. We give some concrete examples for (strongly) cyclic vectors: E x a m p l e 1.3.8. Let 79 be a dense subspace in a Hilbert space 7-/. Every non-zero vector ~ in 79 is ultra-cyclic for s E x a m p l e 1.3.9. Let 341 be in Example 1.1.5 and 34 = 341, that is, M is a self-adjoint O*-algebra on 8(1~) in L2(R) generated by the momentum operator P and the position operator Q on 8 ( ~ ) defined as follows: (Pf)(t) = -if'(t)

and ( Q f ) ( t ) = t f ( t ) ,

f E 8(]~).

Then the following statements hold: (1) 3 4 " = C I . In fact, take an arbitrary C E 34'w- Since C commutes the spectral projections of P and Q, it follows that C commutes with the unitary operators U(s) - e i~P and V(s) = e i~Q for all s E R. Here, ( U ( s ) f ) ( t ) = ei~tf(t) and (g(s)f)(t) = f ( s + t) for f e 8(IR) and s E N. It is well known that {U(s), V(s); s C ]t{}' = C I (see von Neumann [1]), which implies C = c~I for some a E C. (2) A vector ~0(t) - e - ~ in 8(I~) is strongly cyclic for 34. (3) Let ~ be any non-zero vector in C~(I~). Then, O ~ P ~ 7 ~ I , and so is not cyclic for 34. By (1) we have PM7 ~ 34"" This shows that the implication (iv) => (iii) in Proposition 1.3.7 does not hold in general. E x a m p l e 1.3.10. Let .Ad2 be a closed O*-algebra on C ~ ( I ~ ) given in Example 1.1.5. Since Ad2 c Adl, it follows from Example 1.3.9, (1) that (Ad2)'w = C I , which implies that every vector in C~(II{) is not cyclic for JbI2. Next example shows the difference between cyclic vectors and strong cyclic vectors. E x a m p l e 1.3.11. Let fl/[ be a closed O%algebra on 79 in 7-/, ~r a dense subspace of t / contained in 79 and A / a closed O*-algebra on I9 generated by 3,/ and {(~ | 7) [l); ~, r] c g}. Then every non-zero element of S is cyclic for A/, but A / h a s no strongly cyclic vector. The concrete example of a cyclic but not strongly cyclic vector for an O*-algebra has been given in Example 8.3.18 in Schmiidgen [21].

1.4 Induced extensions

19

1.4 I n d u c e d e x t e n s i o n s Let Jk4 be a closed O*-algebra on :D in ~ . Let C be a ,-invariant subset of M~w containing I. We put = linear span of CT), k

k

k

= {2; x c M} Then it is e ~ i l y shown that M is an O-algebra o n ~ such that d~4 -~ M -~ e c ( M ) -~ M * , where e c ( M ) is the closure of ~4. We call this O-algebra ec(Ad) the induced extension of M determined by C. The induced extension ec(Yk4) is not an O*-algebra in general. We consider when e c ( M ) is an 0 " ; algebra. P r o p o s i t i o n 1.4.1. Let M be a closed O*-algebra on 2? in 7-/ and C a 9-invariant subset of ~#w containing I. The induced extension e c ( M ) is an O*-algebra if and only if C2 c M~w . In particular, e M - ( M ) is an O*-algebra if and only if M ~ is a yon Neumann algebra. P r o o f . See Proposition 2.1 in I n o u e - K u r o s e - 0 t a [1]. We next investigate whether the domain D ( e c ( A d ) ) is C-invariant. T h e o r e m 1.4.2. Let Jk4 be a closed O*-algebra on 2) in ?Y. Suppose C is a *-invariant subset of ;~4~w such that I E C and C2 c A/Vw. Then the following statements are equivalent: (i) D ( e c ( . A 4 ) ) i s C-invariant. (ii) C" c M~w and e c ( M ) = e c 2 ( M ) . . . . . ec,,(M). If this is true, then C" c ec(M)~w . In particular, if M'w is a yon Neumann algebra, then e ~ a - ( M ) is a closed O*-algebra such t h a t M -< e ~ - ( M ) , e ~ - ( M ) ' w = M " and M ~ w : D ( e ~ , ( M ) ) C : D ( e ~ , ( M ) ) . P r o o f i See Theorem 3.1 and Corollary 3.2 in I n o u e - K u r o s e - 0 t a [1]. E x a m p l e 1.4.3. L e t / P be a dense subspace in 7-I and A = A t E s We denote by jk4 the closure of the O*-algebra P ( A ) . Let U - UA be the partim isometry defined by the Cayley transformation of A. By L e m m a 1.2.4, we have U s c ~4~w for each n E N. We have that e U , u , v . } ( ~ 4 ) is an O*-algebra if and only if A is essentially self-adjoint. In fact, suppose eU,u,u. } (jk4) is an O*-algebra. By Proposition 1.4.1 we have U*U = UU* = I, and hence A is self-adjoint. The converse follows from L e m m a 1.2.4 and Proposition 1.4.1.

20

1. Fundamentals of O*-a,lgebras

1.5 Integrability

of commutative

O*-algebras

For the integrability of commutative O*-algebras we have the following T h e o r e m 1.5.1. Let. A4 be a closed commutative O*-algebra on 59 in 7-t. The following statements are equivalent: (i) 33 is integrable. (ii) 33 is self-adjoint and (33~w)' is a commutative yon Neumann algebra. (iii) 33'w = 3"l' and (33'w)' is a commutative von Neumann algebra. (iv) The yon Neumann algebra (33~s)' is commutative. (v) There is a commutative yon Neumann algebra .4 such that. X is affiliated with .4 for each X E 33. (vi) There is a commutative von Neumann algebra .4 such that X is affiliated with "4 for each X E 33h. P r o o f . See Theorem 7.1 in Powers [1] and Theorem 9.1.7 in Schm/idgen [21]. For the integrable extension of commutative O*-algebr~ we have the following P r o p o s i t i o n 1.5.2. Let Ad be a commutative O*-algebra on 59 in 7-t. The following statements are equivalent: (i) There exists an integrable O*-algebra 2M1 acting in the same Hilbert space 7-/as M such that 2t4 -~ M 1. (ii) There exists a yon Neumann algebra C on 7-t contained in M~w such that C' is commutative. If this is true, then the induced extension ec(Ad) determined by C can be taken for 33 1P r o o f . See Proposition 9.1.12 in Schm/idgen [21]. P r o p o s i t i o n 1,5.3. Let Ad be a closed commutative O*-algebra on 59 in ~ . Suppose there is a subset A / o f Adh such that Af generates 3/1, and B1 and B-2 are strongly commuting self-adjoint operators for each B1, B2 E Af. Then 2td* is an integrable commutative O*-algebra, and 59"(3,t) = A N 59(~n). BEArnEN

Further, we have that Ad* = ez4-(Ad) and (3,4*); = A {N}'. BEAT

P r o o f . See Theorem 9.1.3 in Schmiidgen [21]. T h e o r e m 1.5.4. Let ~4 be a closed commutative O*-algebra on 59 in 7-/ and A / a subset of 2t4h such that Af generates Ad. The following statements are equivalent:

1.5 Integrability of commutative O*-algebras

21

(i) M is integrable. (ii) Ad is self-adjoint and (B1 + iB2)* = B1 - iB2 for each B1, B2 E A/. (iii) Every element B of A/is essentially self-adjoint,/) = A A 79(~n) BEN'nEN

and (B1 + iB2)* = t31 - iB2 for each B1,/?2 ~ J~. (iv) B1 and B2 are strongly commuting self-adjoint operators for each B1, B2 E A / a n d Mw79' C l). P r o o f . This follows from Proposition 1.5.3 and Corollary 9.1.14 in Schmiidgen [21]. For cyclic vectors for an integrable commutative O*-algebra we have the following T h e o r e m 1.5.5. Let M be an integrable commutative O*-algebra on 79 in ~ such that the graph topology tz4 is rnetrizable. Then the following statements are equivalent: (i) M has a strongly cyclic vector. (ii) Ad has a cyclic vector. (iii) The von Neumann algebra (A4'w)' has a cyclic vector. P r o o f . See Theorem 9.2.13 in Schmiidgen [21]. The main assertion in Theorem 1.5.5 (the implication (iii) ~ (i)) is no longer true in general if the graph topology is not metrizable as seen in next example. E x a m p l e 1.5.6. We denote by 79 the set of all functions f in L2(]~) with compact support, and define M g f = g f for 9 E C(]~) and f c 79. Then Mc(R) =_ {A/l/; f E C(]R)} is an integrable O*-algebra on li) in L2(]R) which has no cyclic vector, and ((Adc(R))~)' : ML~(]~). The vector (0(t) ~e

_t 2

, t E I~ is cyclic for the commutative yon Neumann algebra ML~(R ).

For the polynomial algebra 7)(A, B) generated by commutative symmetric operators A and B in Et (79) we have the following C o r o l l a r y 1.5.7. The following statements are equivalent: (i) 79(A, B) is integrable. (ii) ~(A, B) is self-adjoint and there exists a normal operator C which is an extension of A + i B . (x?

(iii) A and B are essentially self-adjoint, l) = [~ (79(~n) N 9 n=l

there exists a normal operator C which is an extension of A + i B .

and

22

1. Fundamentals of O*-algebras

(iv) A and B are strongly commuting self-adjoint operators and P(A, B) is a closed O*-algebra satisfying P(A, B)',,,59 c 59. We next study self-adjoint extensions of P(A, B). Suppose that A and B are essentially self-adjoint. We put 59oo(A,B) = {~ 9

N

59(~,~m)N59(~m~);~m~

=~m~

n,rnE]N

for all n, m 9 1%1},

Aoo= AF59oo(A,B),

B ~ = BF59oo(A,B).

Then it is easily shown that Aoo, Boo 9 s B)), and A C Aoo c A and B c Boo C B. For the polynomial algebra T)(Aoo, Boo) we have the following P r o p o s i t i o n 1.5.8. Let A and B be commuting essentially self-adjoint operators in s (59). The following statements hold: (1) P(Aoo, B~) = P(A, B)*, and so they are self-adjoint O*-algebras on Doo(A, B) and 79(A, B)'w is a v o n Neumann algebra. (2) Suppose 70(A, B) is self-adjoint. Then 59 = 59oo(A, B) and "P(A, B) = 7)(Aoo, B ~ ) . (3)Suppose A and B are strongly commuting. Then 59*(P(A,B)) = oo

59oo(A, t3) = [~ (D(A '~) C? 5 9 ( ~ ) ) , P(A, B)* = T)(doo, Bo~) = e~(A,B)~ n:l

(P(A, B)) and they are integrable. P r o o f . See Proposition 9.3.13 in Schmfidgen [21]. We showed in Proposition 1.5.8, (1) that if A and B are essentially selfadjoint, then P(A, B)* is self-adjoint. But the converse of this result doesn't necessarily hold ~ seen in next example. E x a m p l e 1.5.9. Let A be an essentially self-adjoint operator in 7-/such that A 2 and A+A 2 are both not essentially self-adjoint. Since A is self-adjoint, 7)(A) * is integrable by Proposition 1.5.3. Further, since P(A 2, A + A 2) = P(A), it follows that 7)(A 2, A + A2) * is integrable though the operators A 2 and A + A 2 are both not self-adjoint. We finally introduce the Schmiidgen construction of non-integrable selfadjoint commutative O*-algebras. T h e o r e m 1.5.10. Suppose ,4 is a properly infinite von Neumann algebra on a separable Hilbert space 7-/. Then there exists a self-adjoint O*-algebra 7)(A, B) on 59 in T/generated by commuting essentially self-adjoint operators A and B in s such that (P(A, B)'w)' = A, and A '~ and B ~ are essentially

1.6 Topologies of O*-algebras

23

self-adjoint operators for each n E 1~. P r o o f . See Theorem 9.4.1 in Schmiidgen [21].

1.6 Topologies

of O*-algebras

In this section we introduce several topologies of an O*-algebra. Let Ad be a closed O*-algebra on 7:) in 7-{. A. W e a k , s t r o n g a n d strong* t o p o l o g i e s Let s

be the set of all linear operators X from 79 to ~ such that

79(X*) D D. Then s (79, 7-{) is a *-preserving vector space equipped with the usual operators X + Y, a X and the involution X t =- X* [79. For each ~, rl E 79 we put pc,(x)

=

I(X~l~J)l,

p~(X) = IIx~ll, p~(X) =p~(x)+p~(Xb,

x ~ ct(z),~).

The topology on s defined by the family {p~,n(-); (, r/ ~ 79} (resp. {p~(-);~ ~ D}, {p~(.);( E 7)}) of the seminorms is called the weak (resp. strong, strong* ) topology on s 7-l) and denoted by rw (resp. r~,r*). It is shown that the locally convex space s 7-/)[Z~*] is complete. For each subset A / o f s 7-/) we denoted by ~ w (resp. ~ , ~ f f ) the closure of Af in ffl(D, 7-/)[rw] (resp. s s (73, 7-/)[r*]). The induced topology of the weak (resp. strong, strong*) topology on AJ is called the weak (resp. strong, strong* ) topology on Ad. It is easily shown that M[Tw] and M[T~*] are locally convex *-algebras. B. a-weak, a-strong and a-strong* topologies

We put oo

D ~ ( M ) = { { ~ } c 79; )-~(IK~II 2 + IIXCnll 2) < 0o for all X C A/I}, O(3

n=l 0o

ptr

= [ ~ IIx4~N2]a/2, n=l

p~o}(x)

=p{~o}(x)+p{~.}(x*),

x ~M

for {~n}, {tin} E 79~(M). The locally convex topology defined by the family {p(~},{~}(-) ;{{n}, {rln} E 79~ (resp. {p{{~}(.); {{n} C D~

24

1. Fundamentals of O*-algebras

{p~}(-); {~,~} 9 D~ is called the (,M)-a-weak (resp. (JA)-a-strong, (JM)-a-strong*) topology on M and denoted by 7"?,w z4 (reap. 7"b~, z4 7"~ ). It is easily seen that Ad[7"/w M ] and M[7"~~] are locally convex *-algebras. In particular, the (s (D))-a-weak ((/2* (~D))-a-strong, (s ('1)))-a-strong*) topology on 3d is simply called the a-weak (resp a-strong, a-strong*) topology and denoted by 7"ow(resp. 7",~, 7"*s). It is easily shown that M[7"~aw],AA[7"~w],Jr4 [7.~M] and A417"~*~]are locally convex *-algebras, and the following relations among these topologies hold: Tw -~ Ts -~

7.s

.,<

A

A

A A J. Ad 3A T* A/I 7-~w -< rSs -< ~s , where the symbols r2 >- 71 and ~ mean that the topology r2 is finer than

7"2

the topology 7"1.

C. U n i f o r m and quasi-uniform t o p o l o g i e s A subset ~ of 7P is said to be M-bounded if it is a bounded subset of the locally convex space ~[tz4]. Let 9J[ be a M-bounded subset of ~D. We put IlXll~ =

sup I(X~l~)l,

x 9 M

(,nEg~ and

Py,~(X)

=

sup IlgX~ll,

4EgN

x 9 M

for Y 6 M I , where M I is an O*-algebra on D obtained by adjoining the identity operator I. The locally convex topology defined by the family {]l" I]~; ~ is M-bounded } (resp. {Py,~(-); Y 6 2Me, ~J~ is M-bounded})is called the uniform (resp. quasi-uniform) topology on M . Since 3.4 is closed, it follows from Corollary 2.3.11 in Schmfidgen [21] that 9)I is a bounded subset of D[t~] iff 9Jr is a bounded subset of D[tc,(z~)], and so the uniform (resp. quasi-uniform) topology on M equals the topology induced by the uniform (resp. quasi-uniform) topology on s Hence we denote by r~ (resp. rq~) the uniform (resp. quasi-uniform) topology on M . T h e o r e m 1.6.1. M[%] is a locally convex *-algebra and A417.q~] is a locally convex algebra such that 7.u -< 7.qu. The topologies % and 7.q~ equal if and only if the locally convex *-algebra Ad[7.~] has the .jointly continous multiplication.

Proof. See Theorem 3.1 and Theorem 3.2 in Lassner [1].

1.6 Topologies of O*-algebras

25

Further studies of the uniform topology and the quasi-uniform topology have appeared in Inoue-Kuriyama-Ota [1], Lassner [1] and Schmiidgen [21]. D. p - t o p o l o g y a n d , k - t o p o l o g y For each A E (3,tl)+ and put.

I(x~l~)l

p A ( X ) = sup ~ez~ (ACIC) '

X E 3,t,

where A/0 = cxz for A > 0. This defines the normed space

9.1a = { x 9 M ; p A ( X ) < ~ } with the norm PA -- PA [9~A. We note that

U

9.In = 2~4, further, the

Ae(AA1)+

relation 0 < A _< B implies that the injection td,B : (9-,[A : P A ) ~ ('Q[B : f i B ) is a norm-decreasing map. The inductive limit topology for the normed spaces {(9,1A : PA); A 9 ( M I ) + } is called the p-topology on 3,'1 and denoted by %. Theorem

1.6.2. Ad[%] is a bornological locally convex *-algebra.

P r o o f . See Theorem 1 in Jurzak [1]. For each A 9

MI

we

put

AA(X

sup

) =

tlx~ll

X 9 M,

where A/O = oo for A ) O. This defines the normed space

~3a = { x 9 M;AA(X) < ~ } with the norm AA -- AA [ ~ A , and the spaces ~ A constitute a direct set. The inductive limit topology for the normed spaces { ( ~ A ; AA); A 9 M I } is called the A-topology on Ad and denoted by rx. It is clear that % -< rx, but M[r~] is not even a locally convex algebra, in general. Further studies of the p-topolog:7 and A-topology have appeared in Arnal-Jurzak [1], Inoue-Kuriyama-Ota [1] and Jurzak [1]. We here arrange the relations among all topologies defined above: Theorem

1.6.3. The following diagram among the topologies holds: Tqu ~- Tu ~"- T w

3.

A

T,k ~ Tp

A

"~ T s

-'~ Tq,u

3.

T a w -~ Tas ~

3. TA .

26

r 1. Fundamentals of O * -aloebras

1.7 U n b o u n d e d g e n e r a l i z a t i o n s of v o n N e u m a n n

algebras In this section we introduce extended EW*-algebras and generalized yon Neumann algebras which are unbouded generalizations of von Neumann algebras.

A. E x t e n d e d W*-algebras An O*-algebra M on /9 in 1-/ is said to be symmetric if (I + X t X ) -1 exists and lies in Mb for all X 9 M , where M b is the set of all bounded linear operators in M . T h e o r e m 1.7.1. A closed symmetric O*-algebra is integrable. P r o o f . See Theorem 2.3 in Inoue [1]. D e f i n i t i o n 1.7.2. A closed O*-algebra Ad on /9 in 7-I is said to be an extended W*-algebra (simply, an EW*-algebra) if it is symmetric and 3db -{A; A 9 Adb} is a yon Neumann algebra on 7-/.

B. Generalized von N e u m a n n algebras To define another unbounded generalization of von Neumann algebras, we first introduce unbounded commutants of an O*-Mgebra. Let dM be a closed O*-algebra on /9 in 7-/. We define unbounded commutants and unbounded bicommutants of M as follows: M : -- ( s e s

.hd'c = {S C s Ad~c = {X e s M's = {x 9 s

(s'X~lw) : ( S ~ l X t . ) for all X C .A4 and ~,~/C/9}, S X = X S for all X E AJ}, (CX~Ir/) _ (C~lxtr/) for all C 9 M'~ and ~,~7 9 = sx

for all S 9 M ; }

Then we have the following Proposition

1.7.3. (1) Ad~ is a subspace of s

and (AA~)b =

I

Mw[/9. (2) A/l'c is an O*-algebra on 79, which equals A/I~ A s (3) Ad" c is a closed O*-algebra on /9 containing M-~ • s /I ! I (Mw3w = Mw. (4) d~4~c~cis a closed O*-algebra o n / 9 containing ~-~w N s (5) Suppose dye'/9 c / 9 . Then AA~,r = {X c s

is affiliated with (Adw)' '} D M'c'c D ~ w

P r o o f . See Lemma 4.1 and Theorem 4.2 in Inoue [9].

and

As

1.8 *-representations of ,-algebras

27

Further studies of u n b o u n d e d c o m m u t a n t s of O*-algebras are t r e a t e d in Epifanio-Trapani [1], Inoue [9], I n o u e - U e d a - Y a m a u c h i [1], K a s p a r e k - V a n Daele [1], M a t h o t [1] and Schmfidgen [17]. D e f i n i t i o n 1.7.4. A closed O*-algebra 54 on i/:) in 7/ is said to be a generalized yon N e u m a n n algebra if Adw79' C 79 and 54wc" = .Ad. P r o p o s i t i o n 1.7.5. Let 34 be a closed O*-algebra on 7:) in 7-/such t h a t ! A/lw79 c 7P. T h e following s t a t e m e n t s are equivalent: (i) Ad is a generalized von N e u m a n n algebra. (ii) A4 = {X c / : t ( / ) ) ; X is affiliated with the yon N e u m a n n algebra (A/t'w)' }. (iii) 34 = (A4'~)'[79 ~ N/:t(79). P r o o f . See Proposition 2.6 in I n o u e - U e d a - Y a m a u c h i [1].

1.8

,-representations

of ,-algebras

In this section we define *-representations of *-algebras and note their basic properties. D e f i n i t i o n 1.8.1. Let .4 be an algebra. A representation rr of A on a Hilbert space 7-/ is a h o m o m o r p h i s m of A onto an O-algebra on a dense subspace 79(rr) of 7/ satisfying rr(1) = 1 whenever M has an identity 1. A representation 77 of a *-algebra .4 is said to be hermitian or a *-representation if ~(x*) = 7r(x) t for all x E .4. D e f i n i t i o n 1.8.2. Let 7h and 7r2 be representations of .4 on a Hilbert space 7/. If 7rl(x) C 7r2(x) for each x E A, t h a t is, T)(771) C 79(7r2) and Trl(x)~ = 772(X)~ for all x C .4 and ~ E 79(771), then 7r2 is said to be an extension of 777 and denoted by 771 C 772. D e f i n i t i o n 1.8.3. Let 7r be a representation of .4 on a Hilbert space 7/. We denote by t~ the g r a p h t o p o l o g y t~(A) on 79(7r) with respect to the O - a l g e b r a 7r(-4). If 79(Tr)[t~] is complete, then 77 is said to be closed. Let 7r be a representation of an algebra -4. We denote by ~(Tr) the completion of 79(r)[t~]. T h e n we have

xE~4

Proposition

1.8.4. Let ~r be a representation of -4. We put

28

1. Fundamentals of O*-algebras : = re(x)

9 A

T h e n 5 is a closed representation of A which is the smallest closed extension of re and ~ is a closed representation of .4 satisfying 7r C ~ C ~. If 7r is a , - r e p r e s e n t a t i o n of a *-algebra A, then ~ = ~. P r o o f . See L e m m a 2.6 in Powers [1]. Hereafter let .4 be a *-algebra. For a *-representation re of .4 we put

"~(re*) = A z)(re(x)*) xEA re*(x) = ~(X*)* [V(~*),

x~A

v(re**) = ['-'l z)(~*(x)*) xEA. T h e n we have the following

P r o p o s i t i o n 1.8.5. re* is a closed representation of A and re** is a closed *-representation of ,4 satisfying re C 5 c 7r** C 7r*. These representations re, ~, re** and re* d o n ' t coincide in general. P r o o f . See L e m m a 4.1 in Powers [1} and Proposition 8.1.12 in Schmiidgen

[21]. Definition 1.8.6. A , - r e p r e s e n t a t i o n re of .4 is said to be self-adjoint (resp. essentially self-adjoint , algebraically self-adjoint ) if re = re* (resp. ~r = rr*,re* = re**). We r e m a r k t h a t a . - r e p r e s e n t a t i o n re of A is closed (resp. self-adjoint, essentially self-adjoint, algebraically self-adjoint) iff the O*-algebra re(A) is closed (resp. self-adjoint, essentially self-adjoint, algebraically self-adjoint). Let re1 and re2 be representations of A on Hilbert spaces ~ 1 a n d 7-/2, respectively. We define an intertwin 9 space lI(rel, re2) of re1 and rr2 which is an i m p o r t a n t tool in representation theory by

I[(rel, 71"2) = {K E ~(~-~1, ~t'~2); KTr1(32) = re2(z)K, Vx e .A}. T h e n we have the following

P r o p o s i t i o n 1.8.7. Let re, rel and re2 be , - r e p r e n t a t i o n s of A. T h e n the following s t a t e m e n t s hold: (1) lI(re, re) = re(A)' and lI(re, re*) = r e ( A ) ' .

1.9 ~iYace functionals on O*-Mgebras

(2) n(~i,~2) c n ( ~ , ~ )

29

c ~(~1"*,~2"*),

(3) ~ ( ~ 1 , ~ 2 ) * C ~ ( ~ 2 " , ~ 1 " ) , ~(~1,~2")* C ~(~2,~1").

P r o o f . See Proposition 8.2.2, 8.2.3 in Schmiidgen [21]. D e f i n i t i o n 1.8.8. Let 711 and 71-2 be *-representations of .4 on Hilbert spaces 7-ll and 7-/2, respectively. If there exists an i s o m e t r y U of 7Y1 onto 7-12 such t h a t U:D(~rl) = :D(Tr2) and nl(x)~ = V*n2(x)U~ for all x E .4 and E ~P(7~1), then nl and 7r2 are said to be unitary equivalent and denoted by 71"1 ~ 71"2 .

For the u n i t a r y equivalence of two , - r e p r e s e n t a t i o n s we have the following T h e o r e m 1.8.9. Let nl and ~2 be closed , - r e p r e s e n t a t i o n s of a , - a l g e b r a .4 with identity in 7-/1 and 7-/2, respectively such t h a t 7r~(A)'~D(Tri) c :D(Tr~) (i = 1, 2). Suppose

(i) ~(~1, ~2") = n(~l, ~2) and ~(~2, ~1") = ~(~2, ~1); (ii) lI(nl, 7r2)7-/1 is total in 7t2 and 1I(7r2, 7rl)?/2 is total in 7-/1; (iii) one of the following s t a t e m e n t s for yon N e u m a n n algebras (71"1 (.A)/w) ' a n d [712~[.A JwJ ~' "~' hold: (iii)l (7rl (.4)~w) ' and (7r2(A)~w)' are s t a n d a r d yon N e u m a n n algebras, in particular, they are von N e u m a n n algebras with cyclic and s e p a r a t i n g vectors. (iii)2 7rl (A)'w and n2(.A)~ are properly infinite and of coutable type. (iii)3 7-ll and 7-12 are separable, and (Trl(A)~w) ' and (n2(A)~w)' are von N e u m a n n algebras of type III. T h e n ~1 and 7r2 are unitarily equivalent. P r o o f . See T h e o r e m 3.2 and Corollary 3.6 in Ikeda-Inoue-Takakura[1].

1.9 Trace

functionals

on

O*-algebras

Let A be a *-algebra. A linear functional f on A is said to be positive if f(x*x) >_ 0 for all x E .4. For a positive linear functional f on A we can construct the G e l f a n d - N a i m a r k - S e g a l representation as follows: T h e o r e m 1.9.1. Let f be a positive linear functional on a *-algebra A. T h e n there exists a closed *-representation 7rf of A on a Hilbert space 7-/f

30

1. Fundamentals of O*-algebras

and a linear map I f of `4 into the domain T)(lrf) of rrf such that I f ( A ) is dense in T)(~rf) with respect to t~s, and (Af(x)llf(y)) = f(y*x) and Af(xy) = ~rf(x)Af(y) for all x, y 9 `4. The pair (Trf,)`f) is uniquely determined by f up to unitary equivalence. We call the triple (Trf, Af, ~ f ) the GNS-construction for f . P r o o f . We give simply the proof. From the same proof of the Schwartz inequality we have

If(y*x)l 2 < f(y*y)f(x*x),

x,y E ..4.

And so, A/f - {x C .4; f(x*x) = 0} is a left ideal of .4 and the quotient space )`f(.4) = {)`f(x) - x + AYf; x c .4} is a pre-Hilbert space with inner product

()`f(x)lIf(y)) = f(y*x),

x , y 9 .4.

We denote by T/f the Hilbert space obtained by the completion of the preHilbert space )`f (.4). ~Ve can define a .-representation 7r~ of .4 by

~ ( ~ ) ) ` I ( y ) : )`f(xy),

~ , y 9 .4

and denote by 7rf the closure of ~r}. Then )`f is a linear map of .4 into :D(lrf) satisfying )'1(.4) is dense in Z)(Trf)[t~s] and I f ( x y ) = 7rf(x)If(y) for all x, y 9 .4. Let (Try, A~) be a pair of a closed *-representation 7r~ of .4 on a Hilbert space T/~ and a linear map A~ of .4 into T)(Tr~) satisfying )`~(A) is dense in :D(~r~)[t~], and (A~(x)lI~(y)) = f(y*x) and A'f(xy) = 7r'f(x)A~(y) for all x, y E `4. Here we put

u)`s(x):)`~(x),

z cA.

Then U is an isometry of )`f(A) onto i~(`4), and so it can be extended to ! an isometry of T/f onto ~ f . The extension is also denoted by U. Then it is easily shown that U:D(Trf) : :D(zr}) and rl(X ) = U*TC~(x)U for all x E `4. This completes the proof. We consider positive linear functionals on O*-algebras. Let A/I be a closed O*-algebra on :D in 7-/with the identity operator I. We define two positive cones ~(Ad) and A/I+ as follows: t

.

M},

k

M+ : {x c M;x

>_ 0}.

A linear functional f on M is said to be positive (resp. strongly positive ) if f ( X ) >_ 0 for each X E ~~ (resp. M + ) . It is clear that every strongly positive linear functional on M is positive, but the converse does not hold in general as seen next:

1.9 Trace functionals on O*-algebras

31

E x a m p l e 1.9.2. Let A4 be the O*-algebra generated by the position and momentum operators Q and P on the Schwartz space S ( ~ ) and put A = ~ ( Q + iP). It can be checked that (AtA - I ) ( A t A - 2I) r P(A4). Hence there exists a positive linear functional f on M such that

f((AtA

-

I ) ( A t A - 21)) < O.

(1.9.1)

Let {~n} be the orthonormal basis in L2(R) consisting of the eigenvectors of the number operator N = AA t. Since oo

( ( A t A - I ) ( A t A - 2I)~l~ ) = E ( n -

1 ) ( n - 2)l(~l~n)l 2 > 0

n=O

for each ~ E S(]~), we have (AtA - I ) ( A t A - 2I) E All+, which implies by (1.9.1) that f is not strongly positive. Many important examples of states in quantum physics are trace functionals, that is, they are of the form f(X) = tr TX with a certain positive trace class operator T. In this section we define and study trace functionals in detail for the preparation of considering quantum moment problem in Chapter IV : Given an O*-algebra Ad, under what conditions is every strongly positive linear functional on .A4 a trace functional? Let ~I(T/) be the set of all trace class operators on 7-{. Every operator oo

T in GI(?-/) can be represented as T

Etn~n|

~n,

where {tn} C C,

n:l

c~

E

=

Itnl < cx~ and {~n}neN' and {~n}neN' are orthonormal sets in 7-/ with

n=l

N ' = {n E N; t,~ # 0}. Further, the trace norm I/(T) - trlT I equals E

It~l"

n

In case T* = T we can have in addition that tn E ~ and ~n = ~n for all n E 1~' (see KSthe [1], w Further, we put G~ = ~n -- 0 for all n E N \ N ' . oo

If the preceding conditions are fulfilled, then we call the sum E

tn~n | ~ a

~,=1

canonical representation of T. We define some subsets of GI(T/) as follows: GI(JM) = {T E B(TY);TT-/c ~P,T*~ c T~ and X T , X T * E GI(TI) for all X E A4}, GI(Ad)+ = {T E G I ( ~ 4 ) ; T > 0}, 1G(A4) = {T E B(7-t);TX and T * X are closable and T X , T * X E G I ( ~ ) for all X E .A4}, 1~(./~)+

= {T E l~(./~);V

We have the following

~ 0}.

32

1. F u n d a m e n t a l s

of O * - a l g e b r a s

L e m m a 1.9.3. The following statements hold:

(1) (~il(fl/[)

C

(2) 6 1 ( M )

= { r (5 N ~ ) ; T ~

I(~(..A/[).

c ~ , r * ~ c l) and X T Y (5 G 1(?-g) for all X, Y (5 A4 } and it is a *-subalgebra of/3(~) satisfying M G I ( M ) = G ~ ( M ) . (3) 1 G ( M ) = {T (5/3(7{); T ~ c I ) * ( M ) , T * ~ C ~ * ( M ) and X*T,X*T* E G I ( ~ ) for all X (5 A4}, and it is a *-subalgebra of/3(7-/) satisfying 1 G ( M ) M = 1G(M). CX3

IIX~IIIIY~II < oo for all X , Y (5 M , then we

Let {~,~}, {r/n} (5 D. If ~ rt,--1

say that the series L

('~ | ~ converges absolutely with respect to M .

n=l oo

L e m m a 1.9.4. (1) Let { ~ } and {7]~} in D. Suppose the series ~

~ON

~=1 oo

converges absolutely w.r.t. AA. Then T -

~-~,~,~ | ~

E GI(A4) and tr

n=l

X T Y r = ~--~,(X~n[Yrh~) for all X , Y E A4. oo

(2) Let T (5 ~ l ( J t 4 ) and T = ~

t n ~ | ~ a canonical representation of

oo

T. Then ~-'(tn~,0 | ~ converges absolutely w . r . t . M . n=l

P r o o f . (1) It follows from Kgthe [1], w T ~

s

5 that

(~ | ~ (5 C i l ( ~ ) and tr T = ~ ( ~ l r / ~ ) .

n:l

(1.9.2)

n:l oo

Take arbitrary X E .A.4 and x (5 7-/. Since ~ n=l

X, Y E Ad, it follows that n

{(X},~k | ~ > } k

c z~,

1 n

&m(X}

:

rx,

k=l 7q~

O0

k 1

k=l

[[X~,~llllYrb~l[ < 0o for all

1.9 Trace functionals on O*-algebras

33

oo

which implies by the closedness of Ad that T T / c D and X T = E n

X~ | 1

Similarly, we have T * ~ C D. Hence it. follows from (1.9.2) that X T E ~1(~) oo

and tr X T = E ( X ~ , d r b J , which means that T E ~ l ( 3 A ) . n

1

(2) Since X t X T , YtYT* E G I ( ~ ) for all X , Y E A/l, it. follows that

I(XVX(t,~n)Kn)l

ItnlllX4~ll 2 = ~ hEN

nEN hen
and

ItnlllYwnll2 = ~ hEN

I(Y*Y(<w,JIn.)[

nE~]

= ~

I(YWT*~/nlV~)I

hEN which implies

~-~lt,,lllXS~llllYw,~ll <_ (~lt.lII n

X

8,112) 1/2 (~-~lt,.lllYw.lI 2) 1/2

n

n

This completes the proof. L e m m a 1.9.5. (1) Every element T of GI(Ad) can be written as T = (T1 - 1 3 ) + i ( T 3 T4) with 7~ E GI(M)+(i = 1 , . . . ,4). (2) Suppose :D[tM] is sequentially complete. Then ~ 1( M ) = ~ 1(s (:D)), and so G I ( M ) [ T ) is a two-sided ideal of L:t(/9). P r o o f . (1) By the *-invariance of G I ( M ) we may assume that, T is hermitian. Let T = E t,~,~ | ~ be a canonical representation of T. We put n

N+

= {n c N; t~ > 0},

T+ -~ E

tn(n | ~n.

hEN+ Since E(tn~n)| converges absolutely w.r.t. A//, it follows that E (tn~n)| n nEi%l+ ~n also converges absolutely w.r.t. A,t, which implies by Lemma 1.9.4 that

34

cr 1. Fundamentals of 0 * -aloebras

T+ E ~I(.A/~)+. Since T T_

E

T+ - T _> 0 and T,T+ E ~I(.A/~), we have

~I(.M)+.

(2) Take a r b i t r a r y T E ~1(./~) and A , B E s

Let T

~_jtn~n |

=

?l

be a canonical representation of T. By the closed graph t h e o r e m we have tM = tot(v). Hence we have

IIA~ll _< IlX~ll and IIB~ll _< IIr~ll,

~ c 79

for some X, Y E A4, which implies t h a t ~-~(tnG~) | ~nn converges absolutely n

w.r.t. s proof.

By L e m m a 1.9.4 we have T E G l ( s

This completes the

L e m m a 1.9.6. T h e following s t a t e m e n t s are equivalent: (i) A4 is self-adjoint. (ii) I~I(Ad ) = 1 G ( M ) . P r o o f . (i) ~ (ii) This follows from L e m m a 1.9.3, (2). (ii) =* (i) Take an a r b i t r a r y ~ E 79"(M). T h e n we have ~ @ ~ E I ~ ( M ) G I ( M ) , which implies ~ E 79. Hence M is self-adjoint.

=

For any T E 1 G ( A J ) we define two linear functionals on Ad by

T f ( X ) = tr T X , f T ( X ) = tr Xt*T,

X E M.

L e m m a 1.9.7. (1) fT = T f for each T E ~ l ( d ~ ) . (2) fT is a strongly positive linear functional on AJ for each T c ~ 1 (.A/l)+. (3) Any fT, T E ~l(Ykd), is written as fT = fT1 -- fT2 + i(fTa -- fT4) whereby T j c G](A4)+,j = 1,... ,4. Proof.

(1) Let T = E

tn~n | ~

be a canonical representation of T.

n

Since T E G I ( A 4 ) , we have {G~}, {r/~} c 79, so t h a t X T = ~ _ t n X ~ n

and T X = E

t~

| Xtrl~ for each X E A4. Hence we have

n

T f ( X ) = tr T X =

~--~t~(~nlXtrl~) n

= ~

tn(X~,drln) ft

= tr X T

= fr(x)

N~-g~

1.9 Trace functionals on O*-algebras for each X E M . (2) Let T E ~ I ( . A / [ ) + and T =

Ethan|

35

~n a canonical representation

n

of T. Then it follows that tn >_ 0 and ~n E l) for each n E 1%1and fT(X) = t~(X~nl~n) for each X E .M, which implies that fT is strongly positive. n

(3) This follows from (2) and Lemma 1.9.5, (1). For any T E 1G(A4) the linear functionals T f and fT are well-defined, but the ~sertions of Lemma 1.9.7 are not true in general ~ seen next. L e m m a 1.9.8. The following statements are equivalent,: (i) fld is algebraically self-adjoint. (ii) T f ----fT for each T E I ~ ( M ) . P r o o f . (i) =~ (ii) Take an arbitrary T E 1~(2t4). Let T - - E t n f n

|163

n

be a canonical representation of T. Then {~,~}, {~?n} E :D*(Ad) and

fT(X)

=

tr Xt*T = E tn(xt*~nl~n), n

Tf(X)

:

tr T X

:

'}--~t.(~,~Ix*7/,d n

for each X E M , which implies that fT = Tf. (ii) =~ (i) Take arbitrary ~, ~ E :D*(~4). Then we have T = {| and

(x**r

: fT(X) : Tf(X)

=

E I~(M)

(r

for each X E M , which implies ( E :D**(M). Hence we have T)*(Ytd) -:D**(Ad). This completes the proof. R e m a r k 1.9.9. Suppose 34 is algebraically self-adjoint and T E 1~ (]~4) +. Then fT is a positive linear functional on 3,t. But, we don't know whether fT is strongly positive. We investigate the continuity of trace functionals fT, T E ~ I ( A d ) , with respect to some topologies: P r o p o s i t i o n 1.9.10. Every trace functional fT, T E ~ I ( M ) , continuous with respect to both topologies p and t~w.

on fld is

P r o o f . By Lemma 1.9.7 it is sufficient to show the continuity of fT, T G I ( M ) + . Let T E G I(A4)+ and T - - E t n ~ n | a canonical representation n

of T. Since

36

1. Fundamentals of |

f~(x) = X ~ ( x ( t . ~ . ) l ( . ) ,

x c M,

n

it, follows that. f r is continuous w.r.t, t ~ . Take an arbitrary A E 3,4+. VCe put. r]~ = v ~ , n E N. Then since E 7]~ | ~ converges absolutely w.r.t. .A4 and

Ifz(X)l

< E -

n

I(Xw~l,.)l = Y~ I(Xwnl~)t (At/nit/n) (A'q~lv.) l~t

_< (X~(A~j,~I~,J)~A(X) n

for all X E ~-[A = {X E 2~; ,OA(X ) < 0(3}, it follows that fT is continuous w.r.t, the topology p. This completes the proof. We next investigate the continuity of trace functionals with respect to the uniform topology T~. We define the locally convex topology Tc called precompact topology determined by the family of seminorms:

p~,~(x)=

sup

t(x~l~)l,

xcM,

where ~J~ and ~ range over the precompact subsets of T~[t~]. P r o p o s i t i o n 1.9.11. Suppose 73[t~] is a Fr6chet space. Then every trace functional fT, T E ~1(A4), is continuous with respect to the topology T~. Hence it is continuous with respect to T~. P r o o f . The projective tensor product topology on 79[t~] | :D[t~] is defined by the family of seminorms {]] ]Ix @~ ]] ]lY; X, Y E A/t}. Here the seminorm N Iix | II IIY is defined by n

II [Ix |

II Ily(T) = inf{ E IIX~kllllZ,kll}. k=l

where the infimum is taken over all representations of T E T)[t~] |

Z)[t~] =

n

.7-(/)) as a finite sum T = E

~k |

with {~1," "" , ~n} and {rh,- -" , ~,,} in D.

k=l

Hence every element T of G 1(A,I) belongs to the completion of the projective tensor product D[t~] | D[t~]. Therefore, by the Grothendieck result (K6the [1], w 4) T is represented as T = E t~G~| where E It. I < oe, and { ~ } n

n

and {Un} in D[t]~] converge to 0. Since E ( t n ~ n ) | ~ converges absolutely n

w.r.t. Ad, we have

1.9 Trace functionals on O -algebras *

37

cY

~t

_< (~-~ It~l)-P,~,~(x),

x E M,

n

where 93I = {~n} and r162= {r/~}. Clearly, 9Jr and r are pre-compact in ig[tM]. Therefore fT is continuous with respect to T~. This completes the proof. P r o p o s i t i o n 1.9.12. Suppose t h a t Ad contains the restriction N of the inverse of a trace-class positive operator on ~ . T h e n every trace functional fT, T E G I ( A d ) , is continuous with respect, to %. For the proof of Proposition 1.9.12 we prepare the following lemmas: L e m m a 1.9.13. Suppose T is a positive b o u n d e d operator on H with T ? - / C 2). T h e n T ~ T / c Z) for all i~ > 0. _ _

m

P r o o f , Let 0 < tz _< 1 . Since l) C 79(1 + X ' X ) C 79(X) for each X E AJ and 3,t is closed, we have 79 =

N

79(I + X ' X ) .

(1.9.3)

XEAd

Take an a r b i t r a r y X E A d . T h e n we have TT-/C 79 C 79((I + X ' X ) 5 )

= (I + X*X)-5?-/,

which implies t h a t (I + X ' X ) ~ T is bounded. Hence there exists a constant 3' > 0 such t h a t m

1

IITxll ~ ~ll(I+X*X) ~xll m

m

for all x E 7-/. Since 79(I+X*X) = 79(3'(I+X*X)), we can take ~ / = I without any loss of generality. T h e n it follows from the Kato-Heinz inequality (Kato [1]) t h a t

IIZ~xll ~ I1(1 + x * X ) - l x l l for all x E ?-f, which implies

~

c

(I + x * ~ ) - l ~

= 79(I + x ' X ) .

Hence it follows from (1.9.3) t h a t T ' ~ C 79. Let lJ > 1. Choose a positive n u m b e r n with n > i/. T h e n T n ~ c 79, and so T ~ = (Tr~)'/nTl C 79 by the above consequence. This completes the proof.

38

cy 1. Fundamentals of 0 * -al~,ebras

L e m m a 1.9.14. Suppose Ad contains the restriction N of the inverse of a positive trace-class operator on 7 / a n d T is a positive operator on ~ with T ~ C T). Then T" is a trace-class operator for each 0 < p _< 1. P r o o f . Let N = E

nl~l | ~ be a canonical representation of N. Since l

N -1 is a positive trace-class operator, it follows that {~k} is an orthonormal -1- <

basis in 7-I contained in :D and E I

oo.

Let T = E

nl

tn~|

be

a

n

canonical representation of T and U an isometry on 7-/ defined by U~n = 7In for each n E N. Take an arbitrary k E i%1. Since TT/ c :D, it follows that U*TUTl c l), which implies NkU*TU is bounded and we put ^/k = I]NkU*TU]]. Then we have

IIU*rVxil -- IIU*TUNkN-%]I

----- klIN-%ll for all x c ~ , which implies

Hence we have

for suitable k, and so 1

u

l

--

~

(X).

l

This means that T ~ is a trace-class operator on 7-/. This completes the proof. T h e p r o o f o f P r o p o s i t i o n 1.9.12: Take an arbitrary T E G I ( M ) + . Let T = E

t,~(n | ~ be a canonical representation of T. By Lemma 1.9.13

n

and 1.9.14 we have E

t,~ < oo for 0 < ,, _< 1

and

T~

c T~.

(1.9.4)

Hence XT" is bounded for each X c Ad and 0 < l/ <__ 1, and so T~/C is a bounded subset in :D[t~], where/E is the unit ball in ~ . We now have

[fz(X)l = E ( X T ( n ICdl n

I

<_ ft

n

1.9 Trace functionMs on O*-algebras

39

for all X E M , which implies that fT is continuous with respect to wu. By Lemma 1.9.7, (2), a general fT, T E G I ( M ) , is continuous with respect to w~. This completes the proof. Notes 1.1. A *-algebra of closable linear operators defined on a common dense domain in a Hilbert space was introduced by Lassner [1] and he called it an O~-algebra, and we here call it an O*-algebra according to the Schmiidgen book [21]. The notion of closed O*-algebras was introduced independently by Lassner [1] and Powers [1]. The notion of self-adjoint O*-algebras was introduced by Powers [1] in analogy with a self-adjoint operator. 1.2. The weak commutant first appeared in papers on quantum field theory: cf. Ruelle [1] and Streater-Wightman [1]. In the context of O*-algebras or *-representations weak commutants were first studied by Vasiliev [1], Powers [1] and Uhlmann [1]. 1.3. For O*-algebras 2r on 79 Powers [1] pointed out that there are some pathologies between A/I-invariant subspaces 9)~ of.~P and the projections P ~ , and he showed that if 2~4 is self-adjoint, then ~ is essentially self-adjoint if and only if P ~ c J~#w and P~:D = ~ t ~ . After that, Ikeda-Inoue [1] extended this result in case of non-selfadjoint O*-algebras. Powers [1] defined two different notions of cyclic vectors and strongly cyclic vectors for O*algebras. Schmiidgen [21] gave an example of a cyclic but not strongly cyclic vector, and in [21] he has called a cyclic (resp. strongly cyclic) vector a weakly cyclic (resp. cyclic) vector. Gudder-Scruggs [1] investigated the relations between them and the commutant. 1.4. Induced extensions of O*-algebras were studied by Borchers-Yngvason [1] in their approach to the decomposition theory, and introduced independently by Inoue-Ueda-Yamauchi [1] for the studies of the integrability of commutative O*-algebras and unbounded bicommutants of O*-algebras. After that, Schmiidgen [21] and Inoue-Kurose-Ota [1] studied induced extensions in detail. 1.5. It was first shown by Powers [1] that a self-adjoint commutative 0% algebra M is integrable if and only if the von Neumann algebra (,~/I~)' is commutative, and the further studies were done by Schmiidgen [21]. Let A and B be commuting symmetric operators in s The integrability of the polynomial algebra P(A, B) generated by A and B was first investigated by Inoue-Takesue [1], and Schmiidgen [21] studied the polynomial algebra P(A, B) in detail. 1.6. The uniform and quasi-uniform topology were introduced and studied by Lassner [1], and the p-topology and A-topology were defined and studied by Jurzak [1] and Arnal-Jurzak [1]. The other topologies Tw, 7-~, T*, r ~ and T*~ were introduced by Inoue [1] and Arnal-Jurzak [1]. More detailed studies about topologies of O*-algebras were done in Schmiidgen [21].

40

1. Fundamentals of O*-algebras

1.7. The notion of extended W*-algebras was first introduced by Dixon [1] and further studies was done by Inoue [1,2,3] for the study of unbounded Hilbert algebras. The notion of generalized von Neumann algebras was defined by Inoue [9]. 1.8. The studies of *-representations of *-algebras were first done by Vasiliv [1] and Powers [1]. The study of unitary equivalence of two *representations was done by Takesue [1], Inoue-Tkesue [1] and Ikeda-InoueTakakura [1]. 1.9. The Gelfand-Naimark-Segal construction (Theorem 1.9.1) for positive linear functionals on general *-algebras was first introduced by Powers[l]. Example 1.9.2 is due to Counter example I in W'oronowicz [i]. The spaces ~1(-/~), ~I(M)+, 1~(./~) and I ~ ( M ) + and trace functionals f T and T f , T E 1G(A/I)+ were introduced by L ~ s n e r and Timmermann [1]. Further studies were developed by Schmiidgen [2], [21]. Proposition 1.9.11 is due to Schmiidgen [2]. Proposition 1.9.12 is due to Lassner and Timmermann [1]. Let 7P be a dense subspace in a Hilbert space. We denote by s 7-{) the set of linear operators X such that ~D(X) = ~D, ~(X*) D Z). The set s (Z), ~ ) is a partial *-algebra with respect to the following operators: the usual sum X1 + X2, the scalar multiplication AX, the involution X --~ X t = X* [~D and the partial multiplication X l n X 2 : Xtl*X2, defined whenever X2~D C 59(X~*) and X~Z) C ~9(X~). A partial O*-algebra on Z) is a *-subalgebra M of s (7?, ~ ) , that is, 3// is a subspace of s (~9, ~ ) , containing the identity operator and such that X t c 34 whenever X E 34 and X l n X 2 E 3,4 for any X 1 , X 2 E Ad such that X1DX2 is well-defined. Partial O*-algebras have been studied in Antoine-Inoue-Trapani [1,2,3], Antoine-Karwowski [1] and Kiirsten [1].

2. Standard s y s t e m s and modular systems

In this chapter we introduce and study the notion of standard systems which makes it possible to develop the Tomita-Takesaki theory in O*-algebras. In Section 2.1 we define and study the notion of cyclic generalized vectors for an O*-algebra which is a generalization of that of cyclic vectors. Three commutants h c, M and h ~ of a cyclic generalized vector h are defined and investigated. In Section 2.2 we introduce the notions of three standard systems (Ad, h, hc), (2td, h, M) and (A/I, h, h ~) consisting of an O*-algebra M , a cyclic generalized vector h and the commutants h c, M and h ~ and in these cases the modular automorphism group {a~}t~ R of Ad is defined and the quasiweight ~x defined by h satisfies the KMS-condition with respect tp {O't~}tEN. The most important object in these is the standard system (A/l, h, hC), and then h is said to be a standard generalized vector for M . In Section 2.3 we define and study the notions of quasi-standard generalized vectors and modular generalized vectors which are able to apply the Tomita-Takesaki theory to more examples. Such generalized vectors are extended to standard generalized vectors. In Section 2.4 we investigate the standardness and the modularity of generalized vectors in special cases: A. Standard systems with vectors. B. Standard traeial generalized vectors. C. Standard systems for semifinite O*-algebras. D. Standard generalized vectors in the Hilbert space of Hilbert-Schmidt operators. E. Standard systems constructed by von Neumann algebras with standard generalized vectors. In Section 2.5 we introduce the notion of semifiniteness of generalized vectors and generalize the Connes cocycle theorem to standard, semifinite generalized vectors for generalized von Neumann algebras. In Section 2.6 we generalize the Pedersen-Takesaki Radon-Nikodym theorem to standard, semifinite generalized vectors for generalized von Neumann algebras. We construct the standard, semifinite generalized vector hA associated with a given standard, semifinite generalized vector h and a given positive self-adjoint operator A affiliated with the centralizer of h, and consider when a standard, semifinite generalized vector # is represented as such a hA. This is closely related to the invariance of # for the modular automorphism group {cr~ }teR and the properties of the Connes cocycle {IDa, : Dh]t}teR.

42

2. Standard systems and modular systems

2.1 Cyclic

generalized

vectors

In this section we define and study the notion of cyclic generalized vectors for an O*-algebra which is a generalization of that of cyclic vectors. Throughout this section let 3d be an O*-algebra on a dense subspace 7) in a Hilbert space

D e f i n i t i o n 2.1.1. A map A of A4 into 79 is said to be a generalized vector for A4 if the following conditions hold: (i) The domain 79(A) of A is a left. ideal of A4. (ii) A is a linear map of 79(A) into 79. (iii))~(AX) = AA(X) for all A E AA and X E 79(A). A generalized vector A for AA is said to be cyclic (resp. strongly cyclic ) if A(79(A)) is dense in T/ (resp. 79[tM]). Let A be a generalized vector for 3//. Then the closure of the O*-algebra A4 [~(z~(~)) on A(79(A)) in 7-/(A) - A(79(A)) is called the &-algebra defined by A and denoted by Ad(A). If A is strongly cyclic, then )A(A) = AA, but even if A is cyclic, then A4(A) r A4 in general. Let /~1 and A2 be generalized vectors for Ad. If 79(A1) C 79(A2) and AI(X) = A2(X) for each X E 79(/\1), then A2 is said to be an extension of A1 and denoted by A2 D A1. We give an important example of generalized vectors for O*-algebras. E x a m p l e 2.1.2. Let Ad be an O*-algebra on 79 in 7-/and ~ E 7-/. We put

{

73(A~) = {X E 2,4 ~ E 79(Xt*) and Xt*~ E 79}, A~(X) = Xt*~, X ~ 79(A~).

Then ,kr is a generalized vector for M . It is clear that Ar is cyclic (resp. strongly cyclic) if and only if {Xt*~ ; X E 79(Ar is dense in 7-/ (resp. D[tM]). We remark that, putting _ 79(~) = ( x c M ; ~ c 79(x) and X~ E 79},

[A(X)

Xr

Xe79(A),

79(),) is not necessarily a left ideal of ~4 because X + Y ~ X + Y in general, and so A is not a generalized vector for Azt. Suppose ~ E 79. Then it follows that D(Ar = Ad and Ar (X) = X~ for each X E A4. Hence Ar can be identieM with the vector ~; for example, A~ is cyclic (resp. strongly cyclic) if and only if ~ is cyclic (resp. strongly cyclic). Let us give an O*-Mgebra which has no strongly cyclic vector, but has a strongly cyclic generalized vector of this above type. Let

.M={Zfk k=l

~

C~(R) ; fkEC~

0
nEN}.

2.1 Cyclic generalized vectors

43

Then fl/l is a self-adjoint O*-algebra on C~(]I{) which has no strongly cyclic vector by the property of suport. Let ~0(t) = e - t ~ , t ~ I~. Since Af = - { f d ~ / d t ~ [ C ~ ( l ~ ) ; f ~ C~(I~), n ~ N} C :D(A~o) andAf~0 = C~r it follows that Ar is a strongly cyclic generalized vector for M . We next define commutants and bicommutants of cyclic generalized vectors. Let A be a generalized vector for A4. Suppose

(Ch M ' ~ c z~, (C)2 A(Z)(A)tTD(~)) is total in 7-/. We define three commutants of A as follows: z)0,') = {K c M'w ; 3~K

N

m

z~(x) s.t. K ~ ( X ) = X~K for all X E :D(A)}

M(K)=~K,

{

KED(M).

'z~(~ ~) = {K e M " ; 3~K

A~(K) = ~K,

{

['~ D ( X t*) s.t. K A ( X ) = X t * ( K Xe~(~) for all X E D ( I ) }

K e v(~).

v ( ~ c ) = {K e .Ad/w ; 3~K E ~) s.t. K/~(X) -~- X ~ K for all X e 7D(A)} AC(K) = ~K, K ~ ~(~c).

Then we have the following P r o p o s i t i o n 2.1.3. M,A ~ and Ac are generalized vectors for the yon Neumann algebra M~w and A c c M c A~. P r o o f . Since A(7?(A)tD(A)) is total in 7-/, it follows that ~K is uniquely determined for K E :D(M), D(M) is a subspace of AWw and A' is a linear map of D(M) into r] ~D(X). Take arbitrary C E Ad~ and K c D(M). xe~(~) Since X is affiliated with (Ad~w)' for each X c D(A) (by (C)1), it follows that CA'(K) e N 79(X) and C K A ( X ) = C X M ( K ) = X C A ' ( K ) for all xe~(~) X E :D(A), which implies C K E D(A') and A ' ( C K ) = C M ( K ) . Therefore, D(A') is a left ideal of M~w and A' is a generalized vector for M~w. We can similarly show that A~ and Ac are generalized vectors for 3d~. It is clear that Ac C M C A~. We define the notion of regular systems, and characterize it. D e f i n i t i o n 2.1.4. A system (M, A, A') (resp. ( M , A, A~), ( M , A, Ac)) is said to be regular if it satisfies the conditions (C)1, (C)2 and moreover

44

2. Standard systems and modular systems

(R) 79(M)* N I)(M) (rasp. 19(~k~ * n 7)(~~ 19(~c), n 7)(Ac)) is a nondegenerate *-subalgebra of M~w. A system (224, A, M) (rasp. (M, ~, ~k~), (A/f, A, Ac)) is said to be strongly regular if the conditions (C)~, (C)2 and the following condition (SR) hold: (SR) There exists a net {K~} in 19(A')* N D(A') (rasp. 19(A~)* N 19(~o), 19(~c), N 19(Xc)) for which Ks I I and K ~ K z = K~K~ for each O~ ]~. When (224, ~, ~c) is regular (resp. strongly regular), ~ is simply said to be regular (resp. strongly regular) . Since D(A c) C D(M) c 7)(~~ it, follows that if (224, ~,h e) is regular (resp. strongly regular) then (Ad, ~, M) and (3/l, A, )~r are regular (resp. strongly regular). P r o p o s i t i o n 2.1.5. Let A4 be a closed O*-algebra on 7) in H such that M'wD c 19 and A a strongly cyclic generalized vector for Ad such that ~(/)(A)tI)(~)) is total in 7-/. Then (M,~,AC)(resp. (M,A,M), (A4,)~,A ~) ) is regular if and only if there exists a net {[~} in i9 (resp. n D(X), xe~(;~) ~-~ 19(X t*) ) such that ]].k(X)]] = sup ]]xt*~]] for each X E 19()Q. P r o o f . Suppose (Ad,)~, )~c) is regular. Then there exists a net {K~} in 19()c), N 19(~c) such that 0 < K~ <_ I, v& and {K~} converges strongly to I. We put ~ = AC(K~), vc~. Then we have sup ]]X~]] = sup [[K~A(X)][ = HA(X)]],

Vx E 19(~).

Conversely suppose that there exists a net {{~} in 19 such that [[A(X)[] = sup []X{~[[, Vx E D(A). Then since ~ is strongly cyclic, there exists a net {Ca} in A/Vw such that C~.~(X) = X{~ for each X ~ 19(~) and a. We now put Ks = [C~],vc~. Then it follows that K~

C19(),C)*N19(Ac),

supllK~(X)ll

liK~ll <_ 1 and

= supllX(~ll

= ll~(X)ll,

VX

c

~(~),

which implies that {K~} converges strongly to I, and so :D()~c) * n 19(Ac) is nondegenerate. Similarly we obtain the similar results for (Ad, A, M) and (M,~,~). We next define the notion of cyclic and separating systems which is a generalization of that of cyclic and separating vectors.

D e f i n i t i o n 2.1.6. (Ad, A,A') (rasp. (M,A,A~

(M,A, Ac)) is said to be

a cyclic and separating system if it satisfies the conditions (C)1, (C)2 and

2.1 Cyclic generalized vectors

45

moreover (C)3 A'(lP(A')*lP(),')) (resp.)~~176 )is total in 7Y. A cyclic and separating system (34, A, A'), ((M, A, A~), (34,)~, Ac)) is said to be strongly cyclic if A is strongly cyclic. I n c a s e o f A 4 = 3 4 5 w e h a v e A ~= A~ = A c and so we simply call (M,A) a cyclic and separating system provided (A/l, A, A') is cyclic and separating. We remark that if (AJ, A, A'), (Ad, A, X~) and (Ad, A, Ac) are cyclic and separating systems then they are regular, respectively. In fact, it is easily shown that ICI 9 lP(A') whenever C 9 lP(A'), so that (/P(X')* A lP(A'))7/ D {ICI~ ; C 9 7?(A'),~ 9 7/} D A'(T)(A')*lP(A')). Hence, (M,A,A') is regular. Cases of (3d, A, X~) and (34, A, Ac) are similarly proved. Let (A/I, A, A') be a cyclic and separating system. Then A' is a cyclic generalized vector for the von Neumann algebra A4" such that A'(/P(A')*T~(A')) is total in 7/, and so by Proposition 2.1.3 three commutants (A')', (A') ~ and (A') C of A' are well-defined and they coincide. To emphasize the commutant of A' we use the notation (A')' (simply A") as the commutant of A', and then A" is defined as follows: lP(A") = {A E (A4~)' 3~A C Tl s.t. AA'(K) = K[A, VK C :D(A')}, A"(A) (A, d C lP(A"). When ( M , A, A~) (resp. (AJ, A, Ac)) is a cyclic and separating system, we can similarly define the commutant A~ (resp. Acc) of A~ (resp. Ac) as follows:

{

lP(A~ ) = {A e (A4~)' ; 3~A C 7/ s.t. AA~ A~176 = (A, m 9 lP(A~

= K~A, VK 9 lP(A~)},

lP(A cc) = {A 9 ( M ~ ) ' 3~A 9 7/ s.t. AAC(K) = K~A, VK 9 lp(Ac)}, ACC(A) ~A, A 9 lp(Acc). Then we have the following P r o p o s i t i o n 2.1.7. (1) Suppose (M,A,A') is a cyclic and separating system. Then ((M~)',),") is a cyclic and separating system satisfying =

A,.

(2) Suppose (Ad,)% A~) is a cyclic and separating system. Then ((M~)', A~ ) is a cyclic and separating system satisfying A~~176 = (A~~ ~ = A~. (3) Suppose (M,),, Ac) is a cyclic and separating system. Then ( ( M ~ ) ' , ) c c ) , ( ( M ~ ) ' , A") and ((M~w) ', A~ are cyclic and separating systems satisfying l~

CA"CA

ccandl cCA

c c c - (lcc) c C A ' C A

~.

P r o o f . (1) It is easily shown that A" is a generalized vector for (M~w)'.

46

2. Standard systems and modular systems

We show that ,V'(/)(A")*/9(A")) is dense in 7-/. Take an arbitrary X E /9(A). Let X = UIXI be the polar decomposition of X and IXI -spectral resolution of IXI. We put. En =

//

/0

tdE(t) the

dE(t) and X,~ = XF,,,~ for n E N.

Since X is affiliated with ( M ' ) ' , it follows that U, X~ E (Ad'w)' and

Xn,V(K)

=

U I X I E , A'(K) = U E n U * X A ' ( K ) = UGU*K;~(X) = KUEnU*~,(X)

for all K E Z)(A') and n E i%1.Hence we have

Xn ~ ~(~H) and A"(Xn) = UE,~U*A(X),

n E N.

(2.1.1)

Further, since (UU*A(X)I,V(K;K2)) = (UU*KIA(X)IA'(K2)) = (UU*-XM(K1)I.V(K2)) = (X.V(K~)IA'(K2)) = O~(X)]Y(K;K2)) for all K1, K2 E /:)(M) and A'(/:)(,V)*/:)(,V)) is total in 7-/, it follows that UU*A(X) = A(X), which implies by (2.1.1) that lim A"(X.) = A(X).

(2.1.2)

f t - - - ~ CX?

Further, by the definition of Xn we have lim X n ( = X~,

v~ E /9(X) and lim X,~r/= X*r],

?'t ---~ fiK~

Vr] E/:)(X*),

n ----~ O O

and so {A*A(Y) ; A E /:)(M'), Y E /~(A)} is dense for A(/P(),)t/:)(A)) and further, M'(/)(A")*/9(M')) is dense for {A*,~(Y) ; d E/:)(M'), Y E/9(,k)} by (2.1.1) and (2.1.2). Hence it follows that MI(~(M')*:D(A")) is total in/-/, so that M" =- (M')' is well-defined by

{

/~(A'") = {K E A4tw ; 3~g E 7-/ s.t. KA"(A) = A(K, Y"(K) ~K, K c z~(A'").

VA E/:)(A")},

It is clear that M c ,V'. Hence, ((Ad~w)', M') is a cyclic and separating system. We finally show M" = M. Take arbitrary K E/9(M') and X E /)(A). Then it follows from (2.1.1) and (2.1.2) that lim EnA'"(K) = Am(K) and lira XEnA"'(K) = lim K A " ( X n ) = KA(X). ~---+ ~0

n ----+ O(2

n---~ O0

2.1 Cyclic generalized vectors Therefore, M"(K) 9

47

N /?(X) and X,V"(K) = K A ( X ) for all x 9 ~(),), xez~(~)

and so K 9 :D(M). (2) This is proved similarly to the proof of (1) considering the polar decomposition of X t* and the spectral decomposition of Ix**l, x 9 z~(~). (3) It is clear that )~cc is a generalized vector for (A4")'. It follows from Proposition 2.1.3 that (]~d,A, A') and (3,t, A, ~r are cyclic and separating systems, which implies by (1) and (2) that ((3,t~)', A") and ((AJ~)', X~162 are cyclic and separating systems. Further, since ),r176C A" C Acc, it follows that ((AJ~)',)~cc) is a cyclic and separating system. By (1) and (2) we have )~c c ACcC c M c A~ This completes the proof. We remark that M" = M and ~r176 = A~ but Accc :/: )~c in general as will be seen in Proposition 2.1.11, (6) later. We difine and study the fullness of a cyclic and separating system. P r o p o s i t i o n 2.1.8. (1) Suppose (dr4,)% A c) is a cyclic and separating system and put

{

:D(I~) = {X E M ~ o ( x ) = ~x,

3~x E :D s.t. X l C ( K ) = K ~ x , V K C :D(Ac)},

x c z~(~c).

Then (Ad, he, Aec) is a cyclic and separating system such that/~ c ),e, ) c = Ac and

{

:D(A~) = {X E ~d ; 3{A~} c :D(t cc) and 3~x E ~ s.t. A ~ ~ x ~ , v ~ r :D and ACC(A~) ~ ~x}, ~(x)

= ~x,

x e z~(~).

(2) Suppose (rid, ~,)~) is a cyclic and separating system and put

/

c v(x**) and 3(x r 39 s.t. X**X~ x e z~(~).

v(x~) = {x c z4 ; ~(v(~))

[x~(x)

: &r

= K ~ x , VK 9 79(),~)},

Then (M, Ar A~) is a cyclic and separating system such that ,k C ,~r ,~r = ,~ and

/

/:)(,~r = {X r M ; 3{A~} C/9(A r176and 3~z 9 7) s.t. A ~ -* Xt*~, v~ 9 and Ar162 ~ ~x 9

I,~(x)

= ~x,

x 9 ~(~).

P r o o f . (2) Since Ar162162 is total in ~ , it follows that ,~ is well-defined. Further, since ( A + B) t* D A t * + B t* and (AB) t* D At*B t* for

48

2. Standard systems and modular systems

each A,/3 9 Ad, it follows that 7P(Ar is a subspace of 3.4 and )~ is a linear map of ~(A~) into/9. Take arbitrary A 9 3A and X E 7P(Ae). Then we have (AX)t*Ar

= At*xt*A~

= AKAr

= KAAe(X)

for each K 9 7P(A~), and so A X 9 ~D(A~) and Ae(AX) = AAe(X). Hence A~ is a generalized vector for M . It is clear that A C A~ and A~ = A~, so that. (Ad, Ae, A~) is a cyclic and separating system. We put. Z)(p) = {X 9 M ; 3{A~} c 7P(A~ ) and 3~x E 7? s.t. A ~ ~ Xt*~,v~ 9 T)(X t*) and .k~(A~) ~ ~x},

(,(x)

~x,

x 9 z~(,).

We show Xe = p. Take an arbitrary X 9 D(Ar Since (M,A~,A~) is a cyclic and separating system such that A~ = A~ we can show similarly to the proof of Proposition 2.1.5 that. X 9 /9(p) and p(X) = Ar The converse is easily shown. Similarly we can prove the statement (1). This completes the proof. R e m a r k 2.1.9. Let (Ad, A, A') be a cyclic and separating system. We put

{

~(Ae) = { x 9 M 3~x 9 • s t X ~ ' ( K ) = K ~ x , V K 9 ~(~')}, ~e(x) ~x, x 9 v ( ~ ) .

Then Ae is not necessarily a generalized vector for M . In fact, we don't know even which ~9(A~) is a subspace of M because X1 + X2 ;b X1 + X2 for X1, X2 E/9(Ar We can prove that if M is integrable, then A! = A~ and so (A/l, A~, A~) is a cyclic and separating system such that A c Ae and A' = A~. D e f i n i t i o n 2.1.10. A cyclic and separating system (dvl, A, Ac) ((A/l, A, A~ (Ad, A, A')) is said to be full if A = Ar In particular, if (Ad, A, Ac) is a full cyclic and separating system, then A is said to be full. We remark that if ( M , A, Ac) and (Ad, A, A~ are cyclic and separating systems, then they can be extended to the full cyclic and separating systems (M,)%, Ac) and ( M , ;%, A~), respectively, but this doesn't hold in case of (M,A,Y). For the commutants ,k~, A~ and A~ of the generalized vector Af for M associated with ~ E 7-/we have the following P r o p o s i t i o n 2.1.11. Let ( E ~ \ / 9 . Suppose (i) M ~ / 9 c / ~ , ! ! (ii) ~ is a cyclic and separating vector for (Adw) , (iii) { x t y t * ~ ; X, Y E/~(Af)} is total in ~ . Then the following statements hold:

2.1 Cyclic generalized vectors

49

(1) 19(I[) : M~w and A[(C) = C( for each C C 2t4~; I9 ( ,k(~ ) = (Adw) ' ' andA[~ (Ad~)'. Hence (3//, ~(, t [ ) is a full cyclic and separating system.

(2) 19(t~) = {K ~ •'w ; K~ c

[-]

19(X)},

V(X[) = {K ~ M " ; K ( ~ 19}.

(a)

in general.

(4) The following statements are equivalent: (a) (A,t, ,k~, l~) is a cyclic and separating system.

(b)~c

N

19(Y)

x~(~)

(c)19(~) = M'w. (d) ~ =

~.

If this is true, then (M, A{, X~) is full. (5) Suppose (Ad, A(, Ag) is a cyclic and separating system. Then A( is a full generalized vector satisfying A~c = A~' = l [ ~ and .,~),cC~'ccc~,,(= X~ = X~. P r o o f . The statements (1) and (2) are easily shown. (3) Suppose ~ c N I)(X). Then I E 19(),~), but I r I 9 ( ~ ) . Hence wehave ,,( xcC~,' ~-,4, S u p p o s e ~ r

~

- ~ , t C?,o Z~(X).ThenlCg)(.k~),andso.,~;~.,4

(4) (a) ~ (d) Since ~' ~ ~[~ by (3), it fonow~ from (1) that ~}' = ~[~, which implies by Proposition 2.i.7 that, I~ -- A[. (d) =~ (c) This follows from (1). (c) ~ (b) This is trivial. (b) ~ (a) Suppose ~ ~ N 19(X). Then I C 19(A~), and so X~ = A~. xE~)(),~) Hence it follows from (1) that (3-/, t~, A~) is a cyclic and separating system. (5) By (3) we have A ~ C A" C ) ~ c and so we show ~ c C A~. Take an arbitrary A ~ 19(~c). Since 19(),~)* (~ID(A~) is a nondegenerate .-subalgebra of Ad~, there exists a net {K~} in 19(A~)* ~l)(),~) such that 0 < K~ <_ I for each a and {K~} converges strongly to I. Then it follows that

{K~CK~} C

D(.k[)* M 19(A[), t [ ( K ~ C K ~ ) : lim K ~ C K ~ ( = C ( : A((C),

lira h[( (K~CK~)*) = lira KaC* K~( = C*( = A[ (C*) for each C ~ A/l~w, which implies AA~(C) = l i m A t ~ ( K ~ C K ~ ) = lira K~CK~XCC(A) = CACC(A) c,~ aft

50

2. Standard systems and modular systems

for each C E ~9(A~) = A4'w. Hence we have A E Z)(A~'~

and ~ ~ = A~C(A), and so A~c c A~. Thus we have A~c = A~ = . A ~ and further by Proposition 2.1.7 ,,~cC~ccc;~,,~= A~' = ~.~ We finally show (2t4, ~ , A~) is full. Take an arbitrary X E T)((A~)e) , that is, 3~z E Z) s . t , X K ~ = K ~ x , V K ff :D(/~). Then we have

for each ~/ E l:), and so ~ ~ :D(X t*) and x t * ~ = ~x E T~. Hence we have X E :D()~), which implies ,k~ is full. This completes the proof.

2.2 S t a n d a r d s y s t e m s a n d s t a n d a r d g e n e r a l i z e d v e c t o r s In this section we study standard systems which are able to develop the Tomita-Takesaki theory in O*-algebras. Throughout this section let (M, ~) be a pair of a closed 0*-algebra A4 on a dense subspace ~D in a Hilbert space and a generalized vector A for ~4 satisfying

(S)i MiD c 9, (S)2)~((:D(A)t M :D(A))2) is total in 7-{. Then it follows from (S)I, (S)2 and Proposition 2.1.3 that A' and )~c are generalized vectors for the yon Neumann algebra M'w and I 3 9 ( ~ ' ) : {K e Mw; ~

e

A

D(X) s.t.

K A ( X ) : X~K, V X E T~(A)t n ~9()~)}, (2.2.1)

79(/~c) = {K C 2t4~w;3~K E Z) s.t.

K:~(X) = x~K,Vx ~ ~ ( ~ ) t n z)(~)}. (2.2.2) In fact, the inclusion C is trivial. Conversely suppose K E A/l~w such that K , k ( X ) = X~K, VX E Z)(A)t n 59(A) for some ~K E N 79(X). Take an arbitrary X E :D(A). Since y t x it follows that

E ~9(A)t N Z)(A) for each Y E :D(A)t f'l :D(A),

(K.k(X)I~(YZ)) = (K),(Z*X)I~(Z)) = (ZtX~K I~(Z)) = (X~K I~(YZ)) for each Y, Z E :D(A)t ~ :D(A), which implies by (S)2 that K E :D(A') and A'(K) = ~g. Therefore the statement (2.2.1) holds. Similarly the statement (2.2.2) is shown. By Proposition 2.1.7 we have the following L e m m a 2.2.1. Suppose

2.2 Standard systems and standard generalized vectors

51

(S)~ ,V((2P(,V)* D :D(M)) 2) is total in 7-/. Then the following statements hold: (1) M(~P(,V)* fq T)(,V)) is an achieved right Hilbert algebra in 7Y equipped with the multiplication and the involution: ,V(K1)/V(K2) = A'(K2K1),

A'(K) ~ = ,~'(K*),

K, K,, t(2 ~ D(,V)* D ~P()~'),

and its right yon Neumann algebra equals Ad~w. (2) M'(TP(M')* D ~9(M')) is an achieved left Hilbert algebra in 7-/equipped with the multiplication and the involution:

~,"(A)~,"(B) =

~"(AB),

~"(A)# = ~"(A*),

A, B ~ ~(~")* n Z~(~"),

and it equals the commutant of the right Hilbert algebra M(Z~(sV)* n ~9(M)). ' '., 3~n E 7-/ s.t. A M ( K ) = K~A, VK E ~D(M)* n (3) :D(,V') = {A E (Adw) :D(,V)} and M'(A) = ~m for each A E T)(,V'). (4) ~P(,V) = :D(,V") = {K c 2L4'~;3~K E 7-I s.t. K,,V'(A) = A ( K , V A Z)(,V')* D T)(,V')} and ,V(K) = (K for each K E ~)(,V). Suppose (S)~ holds. By Lemma 2.2.1 the map M'(A) ~ M'(A*), A C ~D(),')* Q~D(M~) is a closable conjugate-linear operator in 7-/, and so its closure is denoted by S~,,. Since (A(X)IX'(K~K2)) : (;~'(K~K~)IA(xt)) for all X E ~)(~)t N~)(.~) and K1, K2 E D(,V)*~qD(M); it follows that the map )~(X) --~ X(XI), X E 7?(h) t N ~(,k) is a closable conjugate-linear operator in nl/2 T /I 1/2 T{, and so its closure is denoted by Sx. Let Sx,, = J~,,/___ix,, and Sx = jxz._~ be the polar decompositions of S),- and S~, respectively. L e m m a 2.2.2. Suppose M((Z)(M)* cl D(M)) 2) is total in 7-/. Then Sx c S,x,,. P r o o f . Take an arbitrary X E ~9(),) t • Z)(A). Let X = UIX [ be the polar decomposition of X, En =

/0

Ixl

--

fo

tdE(t) the spectral resolution of IXI and

-

d E ( t ) , n C 1~. We put Xn = -RE,~,n E 1%I.Then, by (2.1.1) and

(2.1.2) we have

x . e 9(~") and ~im ~"(X.) = ~ ( X ) Further, since

X ; X ' ( K ) = E~X--~a'(K) = K E ~ X ( X * ) for all K E ~D(M), it follows that

(2.2.3)

52

2. Standard systems and modular systems X,: E l)(.k") and /X"(X~) = En)~(xt),

n E •.

(2.2.4)

By (2.2.3) and (2.2.4) we have Xn E/9(~")* N l)(,k"), lilnoo)~"(X,~) = A(X) and li~no A"(X~) = A(xt). (2.2.5)

Therefore, ,~(X) E/?(Sa,,) and Sa,,~(X) = k(X t) = S'aA(X). This completes the proof. By the Tomita fundamental theorem (Takesaki [1], Tomita [1]) we have /

!

!

(2.2.6)

Ja,,(Mw) d~,, = A//w; -

c (M'w)'

(A

(M'w)'.t

]I{);

(2.2.7)

a~"(T)()~")* N T)(,k")) = T)()~")* n/~(~") and /!

I~"

A (at~ (B)) = A~t,,A"(B) "

A it

(B E lP(,V')* N T)(,V'), t E ]1{);

A-it

t

at~ (C) - ,-~x,,C,-~a,, E Mw "

at~'' (/9(~')* N/9(/V)) )d(at~"(K)) =

(C E r162

E 1~);

(2.2.8) (2.2.9)

/9(,V)* N/9(,V) and

Ai~t,,)d(K)

(K E :D()d)* NID()C),t E ~ ) .

(2.2.10)

Further, we have

at~"(Z)()~")) = T)(,V') and ,\"(atX"(B)) = Z~t,,,V'(B) (B E T~(/V'), t ff ]l{); (2.2.11) ~it

!

a t (T)(~)) = :D()~') and &'(at~ (K)) =

Ai~t,,~'(K) (K E :D(~'),t C ]~). (2.2.12)

In fact, the statement (2.2.11) follows from Lemma 2.2.1, (3) and by (2.2.10)

@" (B)~'(K) = z2~t,,BA~,~t~'(K) =

l!

(K))

: :

for all /3 E :D(A"), K E :D(A')* ~ 7P(A') and t E N. The statement (2.2.12) it follows from A" = A' and (2.2.11). By (2.2.7) the unitary group {z~,,}telt~ implements a one-parameter group of .-automorphisms of the von Neumann algebra (Adw) ' . But, we don't know how it acts on the O*-algebra A/l, and so we define a system which has the best condition: D e f i n i t i o n 2.2.3. A triple (3/l, A, X') is said to be a standard system if it satisfies the above conditions (S)I ~ (S)3 and the following conditions (S)~, (S)~ and (S);:

2.2 Standard systems and standard generalized vectors

(s)i A~',,Z~cV, t A it

(S)~

txA

-it

teR.

. . . . . . ~,, = M ,

(s); o~ (v(;~)*

n

53

tER.

z)(~)) : z)(;~)t

n

z~(~),

t~]~.

T h e o r e m 2.2.4. Suppose (54, A, t') is a standard system. Then the following statements hold: (1) & = &,,. it --it A~ (2) crX(x) ~_ Axxz21 x = a t (X) for each X E Ad and t E R and {atx}teR is a one-parameter group of ,-automorphisms of 54. (3) ~ satisfies the KMS-condition with respect to {@}rEX, that is, for each X, Y E/:)(A)t A I)(t) there exists an element fx,Y of A(0, 1) such that

fx,y(t) = (A(@(X))l,k(Y)) and fx,y(t + i) = (t(Yt)]A(c%~(xt))) for all t E ~ , where A(0, 1) is the set. of all complex-valued functions, bounded and continous on 0 _< Im z _< 1 and analytic in the interior. P r o o f . Take arbitrary X , Y E :D(A)* N I9(t). By (2.2.5) there exist sequences {Xn} and {Y,~} in D(A")* N D(,V') such that lim A"(X,0 = ,k(X), r e ----+ O4)

lira A"(X,~) = A(xt), ~%~oo

lira ,V'(Yn) = ,k(Y), n----~ o o

lim ) , , ( y 2 ) = ,~(yt).

(2.2.13)

n~oo

By Theorem 10.17 in Stratila-Zsido [1] and (2.2.8), for any n E l~l there exists an element fn of A(0,1) such that

A(t)

= (~"(~"(xn))t~"(y.)) =

(A~,,~,"(x,oI;v'(Y,d),

f~(t + i) = (A"(Y*)IA"(g~"(X*))) = (,k"(Y*)IA~,,,k"(X*)) (2.2.14) for all t E ]~. Since A ( ~ " ( X ) ) is well-defined by (S)~ and (S)~ and by (2.2.10)

(~(~" (X))I~'(K~K2)) = (o~" (X):,'(KI)I~'(K2)) = (A~,,XA'(~_'; (K1))IA'(K2)) = (KIA~t,,A(X)IA'(K2)) =

(A~,,:~(X)I:,'(K'fK2))

for all K1,K2 E D(A')* A:D(A'), it follows that A(o~"(X)) = all t E ]~, which implies by (2.2.13) and (2.2.14) that

A~t,,A(X) for

54

2. Standard systems and modular systems sup Ifn(t) - (A(a{"(X))]A(Y))I tE~

_< l l ~ " ( x n ) - ~(X)Wlll~"(Yn)ll + II~(X)llll~"(Y~) - ~(Y)I] -~ 0(n ~

~),

sup If~(t + i) - (A(Y t) ~k(a{ ' ( X t ) ) )

tE~ -< II~,"(Y,,*) - ~,(Y*)III[~,"(x,~)II + II~,(Y*)I[II~,"(x,~) - ~,(x*)tl 0(n -~ o0). Hence there exists an element f x , y of A(0, 1) such t h a t

f x , y ( t ) = (A(a~t"(X))I)~(Y)) and f x , y ( t + i) = ()~(Yt)l)~(a~t"(Xt))),

t 9 R. (2.2.15)

We next show S~ = S~,,. L e t / C be the closure of {/~(X); X ? = X e D(A) t N :D(A)} in 7-/. T h e n / C is a closed real subspace of 7-/. Since A(7)(A) t n 7:)(A)) c K: + iK: and it is dense in 7-/, we have (K: + iK~)• = {0}. Further, we have /CNi~ = {0}. In fact, take an a r b i t r a r y ~ E/CAi/C. T h e n there exist sequences {An} and {Bn} in 7P(A)tAD(A) such t h a t gtn : An, B t n : Bn, l i r a A(An) : and lim A(Bn) : - i ~ , and then we have n - - ~ o<)

(~]A'(K~K2)) = lim ()~(A,~)]A'(K~K2)) = lim (AnA'(K1)IA'(K2)) n--+

oo

n---~ OG

=

lim ( ~ ' ( K ~ ) I K 2 ~ ( A n ) )

= (~'(K;K1)]~)

for all K ] , I42 9 D(),~) * N D(A'), which implies

= ( ~ ' ( K ~ K 1 ) ] - i~)

= i()C(K~K1)I~) = i(5]A'(I4;K2)), so t h a t by the totality of/V((73(~')* N T)(A')) 2) we have ~ = 0. Therefore it follows t h a t S~ equals the closed o p e r a t o r S defined by S(~ + irl) = ~ - iv,

~, 7 / 9

(2.2.16)

Further, it follows from (2.2.15) and (S)~ t h a t the o n e - p a r a m e t e r group it {Z~x,,}teR of u n i t a r y operators satisfies the KMS-condition with respect to /C in the sence of Definition 3.4 in Rieffel-Van Daele [1] and z~kt,,3C c / C for all t E ~ , so t h a t by T h e o r e m 3.8 in Rieffel-Van Daele [1] and (2.2.16) A~t,, = Z~kt for all t 9 ~ . Therefore it follows t h a t S~,, = S~, which implies

2.2 Standard systems and standard generalized vectors

55

by (S)~ and (2.2.15) that A satisfies the KMS-condition with respect to the one-parameter group {crt~}teR of *-automorphisms of Ad. This completes the proof. We next proceed to the standardness of the system (A/I, A, Ac). We almost have the same results as Lemma 2.2.1 and (2.2.6) ~ (2.2.12) for Acc. L e m m a 2.2.5. Suppose (S)~ ,kC((D(~C) * n Z)(Ac)) 2) is total in /-/. Then the following statements hold: (1) Ac(l)(Ac) * N 1)(),c)) is a right Hilbert subalgebra of the right Hitbert algebra A'(:D(A')* n :D(A')). (2) Acc(I)(Xcc) * N D(),cc)) is an achived left Hilbert algebra in T / c o n taining A"(D(,V')* n D(A")). (3) Let S~cc be the closure of the involution ;~CC(A) --+ )~CC(A*) (A 9 A 1/2 D(,kcc)* N D(Acc)) and let S a c c = laxccz-axcc be the polar decomposition of Sxcc. Then Sx c S~,, c Sacc.

!

!

!

(4) &cC(Mw) Jx c = Mw. Ait aA-it (5) at~CC(A) = ,--~xcc~,--~cc E (Ad'w)' for each A 9 (A4;)' and t 9 R ,

~t~cc(7)(Acc)) = 1)(Acc)) and ;~cc(atxCC(B)) = A~ccAU CC(B) for each B 9 D(A cc) and t 9 N, e it

A~ccA

cc

= v(a

c)* e

and

=

(B) for each B 9 l)(;~cc) * N 1)(A CC) and t 9 N.

(6) crt~CC(C) - A, - ~i tc c ~~-' ~~Ax c- ict

! 9 AJ w for each C 9 3/i'w and t 9 R ,

at~cc(z)()~ccc)) = D(A ccc) and )~ccc(at~cCC(K)) --- z ~ c c c A c c c ( K ) for each K ~ D(A ccc) and t 9 1~, a~ccc (D(Accc)* N D()~ccc)) = D(Accc) * n Z)(A ccc) and =

for each K 9 D(Accc) * n D(A cCc) and t 9 It~. R e m a r k 2.2.6. Let ~ E 7-/. Suppose the conditions (S)I and (S)2 for the generalized vector A~ and the condition (S)~ for ,k~ hold. Then (A/l, A~, A~) is a cyclic and separating system, and so by Proposition 2.1.11 A~' = A~c. Hence the right Hilbert algebras Ac(7)(Ac) * N :D(Ac)) and )~'(D(A')* n 19(A')) are equivalent. But, for general generalized vector A we don't know whether their right Hilbert algebras are equivalent, or not. it

By Lemma 2.2.5, (5) the unitary group {Z~acc}tei ~ implements a oneparameter group {at~cc} of *-automorphisms of the von Neumann algebra / ! (2t4w) , but we don't know how it acts on the O*-algebra 3d, and so we define a system which has the best condition:

56

2. Standard systems and modular systems

D e f i n i t i o n 2.2.7. A triple (34, A,A c) is said to b e a n essentially standard system if it satisfies the conditions (S)~, (S)2, (S)~ and the following conditions (S)~ and (S)~: (S)~ AxcoT? " c i9, vt E IR.

(S)~

Air

t ,cA -it ~..~)~ccJv~z-a;~cc = 34, V t C ~ . Further, if (S)g ~)cc(z~(,X)~ n ~(.X)) = Z)(.X)t n Z~(,X), vt ~ R , then ( M , A, Ac) is said to be a standard system.

T h e o r e m 2.2.8. (1) Suppose (M, A,A c) is a standard system. Then ( M , A, A') is a standard system and S~ = S'~,, = S~cc. (2) Suppose (34, A, Xc) is an essentially standard system. Then (34, A~, Ac) is a standard system, where Ae is the extension of A in Proposition 2.1.8. P r o o f . (1) We can prove S~ = S~cc in the same way as in Theorem 2.2.4, and so by Lemma 2.2.5, (3) we have S~ = S~,, = Sxcc. Therefore (34, A, A') is a standard system. (2) Since A c Ar and h C = Ac by Proposition 2.1.8, it follows that Acc (34, Ae, Ac) is an essentially standard system. We show a t (~D(Ae)t N ~D(A~)) C ~D(Ae)t A ~D(Ae) for each t c JR. Take an arbitrary X E ~D(A~)t D ~D(Ar Since X~ c ),~c = A c c it follows from (2.2.5) that there exists a sequence {X~} in ~D(Acc)* A 7?(Acc) such that X~ _~. X, lim X c c ( x n ) = A~(X) and limooAcc(x:) = Ae(xt). By Lemma 2.2.5, (5) we have that for each t E l~ Acc

{m

(x~)} c z~(~cc) * n z~(ac~),

~cc

~t

" A~ccZ) c I9,

Acc

(Xn) ~

~t

(X),

lim Acc(@ ~ ( x n ) ) = lira A a~t. c c ~ ~ (X~) = A x~c c A ~ ( X ) , 7%---*oo

n --* oo

limooAcc(atxCC (X~).) = A ~ito , ~ e ( x t ), )~cc

which implies by Proposition 2.1.8 that a t (X) E ~(.Xe)t N ~(Ae) and ACC it ACC A~(a t (X)) = ZS~ccAe (X). Therefore it follows that a t (59(A~)tDz)(Ar C :D(Ae) t N/?(A~) for each t E ]~, which implies (34, Ae, Ac) is a standard system. This completes the proof. R e m a r k 2.2.9. (1) It seems meaningless to define the notion of essentially standardness of systems (A4, A, A') Off it satisfies the conditions (S)1, (S)2 and (S)~ ,-~ (S)~ without the condition (S)~). Because we can't construct a standard extension of (3,/, A, A') as the standard extension (34, he, Ac) in general (Remark 2.1.9).

2.3 Modular generalized vectors

57

(2) It. seems also meaningless to consider the standardness of (Ad, A, A~) by the following reason: Suppose ~~ N79(~r 2) is total in 7-t. As seen in Proposition 2.1.8 the extension ~r for (A/l, ~, ~ ) is possible, but the closed operator S x ~ defined as the closure of the involution A~176 ~ &r and Sa don't have any relations in general. As seen in Remark 2.2.9 it is more natural and useful to treat with the standard system ( M , A, Ac) than to do the systems (3,t, ~, A') and (M, A, A~), and so we mainly study the standard system (A/I, ~, Ac), and introduce the following notions: D e f i n i t i o n 2.2.10. When (Ad, A, Ac) is a standard (resp. essentially standard) system, we call simply A the standard (resp. essentially standard) generalized vector.

2.3 M o d u l a r g e n e r a l i z e d v e c t o r s Weakening the conditions (S)~ and (S)~ in Definition 2.2.7, we define and study the notions of quasi-standard generalized vectors and modular generalized vectors which are able to apply the Tomita-Takesaki theory to more examples. Throughout this section let (Ad, A) be a pair of an O*-algebra A~ on 79 in 7-I and a generalized vector for A4. D e f i n i t i o n 2.3.1. A system ( M , ~, Ac) is said to be quasi-standard if

(S)~ M ~ v c 79, (S)2 A((79(,k) t n 79(h))2) is total in 7-/, (S)~ ,kc((79(,kc) * N 79(,kc)) z) is total in 7-f, c 79 for each t E Ii~. (S)1 A~,cc79 ~ And then )~ is said to be a quasi-standard generalized vector for ~4. T h e o r e m 2.3.2. Suppose (d~4, ~, ,W) is a quasi-standard system and then put

{

79(X) = {X E Add,c; 3~x E 79 s.t. XAC(K) = K ( x , V K E 79(Ac)}

X(x)

~x,

XEV(X).

Then X is a standard generalized vector for the generalized von Neumann algebra Z4~c such that ~ c X, ~c = Xc and 79(N) = {X E Adwc, 3{A~} c 79(~cc) and 3~x E 79 s.t. A ~ ~ x ~ , v ~ E 79 and )~CC(A~) -~ ~x} H

L~(x) = ~x,

~

x ~ 79(X).

58

2. Standard systems and modular systems

P r o o f . It is shown similarly to the proof of Proposition 2.1.8 that A is a generahzed vector for M w c such that A C A, A = A and o

H

"

=

I C

C

" 9~ { A ~ } c i P ( A Mwc,

~ c ) a n d 3 ~x 9 s.t. A ~ -+ x~,V~ 9 19 and ACC(A~) ~ ~x}

19(X)={x 9

IX(x)

I

~x,

x

9

19(~).

Hence (M,A,-~ c) satisfies the condition (S)1, (S)2 and (S)~. Further, since it Acc . t " Axcc19 C 19 and ~r~ (Mw) C M'w for each t 9 ]~, it follows that. it

-it

A

it

v

ACclr€

C

it

--it

A ) , c c X A ) , c c C ~ = ~),cc~O-_ t t,.,}~),cc~ = c A ; , c ~ x A ) , c c ~ tl

A it

I

vA

-it

for each X 9 Mwc , C 9 A/lw, ~ 9 T) and t 9 ]~, which implies ~-~xcc~,-~xcc G ~CC [ ~ll X II ~4~c for each X 9 M~c and t 9 JR. Hence we have ~t UVtwc) = 2M~,c for each t 9 1~. It follows from the definition of A that A is full, and hence at~cc(19(-X) t n 19(X)) = 19(~)t A 19(~) for all t 9 l~. Thus ~ is a standard generalized vector for M ~ . This completes the proof. A

We next define another generalization of standard generalized vectors as follows: D e f i n i t i o n 2.3.3. A system (f14, A, Ac) is said to be modularif the conditions (S)1, (S)2 and (S)~ in Definition 2.3.1 and the following condition (M) hold: (M) There exists a dense subspace s of 19[tM] such that

(M)I ~(D(A) f n 19(A)) c 8, (M)2 {AC(KIK2); Ki 9 19(Ac) * n19(A c) s.t. AC(Ki),AC(K *) 9 C, = 1, 2} is total in the Hilbert space 19(S~c), (M)3 M s C s it

(M)4 z~)~ccs C s for all t 9 ]~.

And then A is said to be a modular generalized vector for M . L e m m a 2.3.4. Suppose (Ad, A, Ac) is a modular system and then denote by i/:)M the subspace of 19 generated by U Y, where 5~ is the s e t of all s

subspaces of 19 satisfying t h e conditions (M)I "-~ (M)4 of Definition 2.3.3. Then the following statements hold: (1) M [ ~ M is an O*-algebra on 19M such that (.M[z)M)~ = J~4~ and M t w'-q-~M %X C 19M (2) (A4 [ v M ) ~ is a generalized von Neumann algebra on 19M over (A4~)'. ~CC.

.

It

(3) {fit }telRIS a one-parameter group of *-automorphisms of ( M FIDM)wcP r o o f . (1) It is clear that 19M is a subspace of 1l? which is the largest element of ~-, so that M [~M is an O*-algebra on 19M. Since :DM is dense in

2.3 Modular generalized vectors

59

T)it~4] , we have (A/l[DZ) ~ = .MI~. Since dVlw/) ' C D and cr,; ~ c c . . (d~/Iw) , . = AA,W AA~ "r~ for each t ~ ~, it followsthat the subspace of/9 generated by, ~,w~;14 sat.isties the conditions (M)I --~ (M)4: which implies by the maximum of T)M that (2) It followsfrom (I) and Lemma 2.4 in Inoue [9] that (L4[~,)~c is an O*-algebra on ~M such that

(J~[vM)'~ = {X E s

M)

X is affiliated with ( ~ ) ' ' }

and A-it A i~ccWV~/vM),~ca.a~cc r ," , ,~ ~-

,,~

H = (AJ[V~)~r for all t E ]l~.

(2.3.1)

We show that the O*-algebra (JUI[~M)'~ on /)M is closed. In fact, it. is easily shown that the completion :D~ of the locally convex space :D[tl~4fvM)~r satisfies the conditions (M)I ~ (M)4, so that we have /:)~ = ID~ by the maximum of i/9M. Thus (,4,1 [ v ~ ) ' r is a generalized von Neumann algebra on V M over (J~d~)'. (3) This follows from (2.3.1). L e m m a 2.3.5. Suppose (A4, A, Ac) is a modular system and then put.

{

19()~79~) = { X [ ~ y ; X E/)(A) s.t. )fiX) C D ~ }

Then )~v~ is a quasi-standard generalized vector for the O*-algebra J~d [z>M satisfying (i) ~ M ((19(),v~)t (~ D(~,lV~)) 2) is total in 7-/; (ii) v ( ~ ) : {K e ~(~c);~c(K) e 9~'} and ~ ( K ) : ~c(K) for each K E :D()~M); ~c * N v ( ~ . ) = v ( ~ o c ) * n v ( ~ ~ (iii)),~c is well-defined and/:)(Av~) P r o o f . It is clear that Av~ is a generalized vector for ~I[vM. By (M)I we have

{ x [ 9 ~ ; x ~ v(~,)t n ~(A)} c v ( A v ~ ) t n z~(A~,~),

(2.3.2)

and so , ~ ((:D(,~z~~ )t r~/9(XI>M ))2) is total in ?t. The statement (ii) follows from (2.2.4) and (2.3.2), and the statement (iii) follows from (ii) and (M)2. Further, it follows from the above (i) ~ (iii) that )~v~ is quasi-standard. This completes the proof. By Theorem 2.3.2 and Lemma 2.3.4, 2.3.5 we have the following

60

2. Standard systems and modular systems

T h e o r e m 2.3.6. Suppose (34, A, Ac) is a modular system and put

79(As) - 79(Av~)

L A s ( x ) _-- ~ ( x )

,, 3

= ~x,

x 9 z)(As).

Then As is a standard generalized vector for the generalized yon Neumann M It C algebra (34 [z>x )wc on 79M over (34'w)' such that A~ff C As, Av~ = A~ and 79r~.,.,s J n 79(As~) = ~ ( A ~ ) * n I)(Ac~:)

We remark that it is meaningless to consider the notion of modularity of systems (3d, A, A') because the extension theory for (34, A, A') does not succeed as seen in Remark 2.1.9.

2.4 Special

cases

A. S t a n d a r d s y s t e m s associated w i t h vectors Let A4 be a closed O*-algebra on 79 in ~ and ~ C 7-t. We consider when (A/l, A~, A~) is a standard (or modular) system. P r o p o s i t i o n 2.4.1. Suppose ( S ) l M/w ~) C ~:),

(S)2 tlxtxt*~'l 2 ~,~ E 79(Xf*) N79(X*) and xt*~,X*~ E 79, i = 1,2} is total in ~ , (S)~ {K1K2~; Ki C JVl~ s.t. Ki~, K*( c 7:), i = 1, 2} is total in ~ , tti~ It (S)~ z21~ 79 C 79, vt E 1~, which A~ is the modular operator of the achieved left Hilbert algebra (A4~)'~. Then A~ is a quasi-standard generalized vector for M . Further, suppose 1lit

r,

it

(s)gA~ MA~= M , vt e R. Then A~ is a standard generalized vector for Ad. P r o o f . It follows from (S)l, (8)2, (S)~ and Proposition 2.1,11 that (A/I,A~,A~) is a cyclic and separating system such that 79(A~c) = (Jtd~)' / ! and A~C(A) = A~ for each A E (Adw) , and so (M'w)'~ is an achieved left II Hilbert algebra in 7-/ and A~ = Z~x~c. Hence the condition (S)~ implies that A~ is a quasi-standard generalized vector for AJ. Further, suppose the condition (S)~ holds. Since A~ is full, it follows that A~ is standard. P r o p o s i t i o n 2.4.2. Suppose the conditions (S)1, (S)2 and the following condition (M) hold, then A~ is a modular generalized vector for 3.4:

2.4 Special cases

61

(M) There exists a dense subspace 8" of ~9[tz4] such that (M)I {Xt*~;~ E 2)(21.) N 2)(X*) and X**~,X*~ r 2)} c 8,; (M)2 {K~K2{; K~ c 34'w s.t. K ~ , K~*{ ~ 8,, i = 1, 2} is total in ~ ; "it

(M)3 z ~ 8 , c 8 , , (M)4 MS, C 8,.

vtcR;

P r o o f . We put

9.1 = {K{; K E A4" s.t. K{, K*~ C ~gv~ }. Then it follows from (M)2 that 91 is a right Hilbert algebra in 7-/ whose commutant 91' equals the achieved left Hilbert algebra (M~w)'{, which implies 912 is total in the Hilbert space 2)(5"~c)(= Z)(S~,),~)). Therefore ~ is a modular generalized vector for Ad. D e f i n i t i o n 2.4.3. Let { E D. If ~ is standard (resp. quasi-standard, modular), then { is said to be a standard (resp. quasi-standard, modular) vector for A4. For the standard (quasi-standard, modular) vectors we have the following C o r o l l a r y 2.4.4. Let ~ E 2). If the below conditions (S)1, (S)2, (S)3 and (S)4 hold, then ~ is a quasi-standard vector for A4; and if the further condition (S)5 holds, then ~ is a standard vector. If the below conditions (S)1, (S)2, (S)3 and (M) hold, then ~ is a modular vector for A4: (8)1 J~tw~) C ~[~. (S)2 Ad~ is dense in 7-/. ( s ) 3 Adw~ ' is dense in 7-/. "it "it

"

it

( s ) s A ~ M A c- = M, vt c N. (M) There exists a dense subspace 8, of 2)[tz4] such that "it

(M)3 a ~ 8, C 8, vt c R; (M)4 .Ads C 8,. P r o o f . Suppose the conditions (S)I , (8)2 , (8)3 and (M) hold. Then it is easily shown that the linear span of Ad'~s satisfies all of the conditions (M)I --~ (M)4 in Proposition 2.4.2, so that A~ is modular. Hence ~ is a modular vector for M . The other assertions follow from Proposition 2.4.1. B. S t a n d a r d t r a c i a l g e n e r a l i z e d v e c t o r s Let 2r be a closed O*-algebra on 19 in 7-/such that 3.Iw2) , c 2). A generalized vector p for Ad is said to be tracial if (p(X)]p(Y)) = (p(Yt)lp(Xt)) for each X, Y E 2)(p)t A D(p). Here we consider when a tracial generalized

62

2. Standard systems and modular systems

vector p for AJ is standard. We first introduce standard tracial generalized vectors constructed by the Segal LP-spaces. E x a m p l e 2.4.5. Let Ado be a v o n Neumann algebra on a Hilbert space 7-/ and #o a s t a n d a r d tracial generalized vector for Ado. T h e n P0 (:D(p0)* NZ)(po)) is an achieved Hilbert algebra in 7-/, and so the natural trace %v~,o on (3/10)+ can be defined by %V~o(A) =

Sil#o(B)ll 2 if A = B*B for some B c D(po)* N D ( p o ) , if otherwise.

We denote by LP(%v,o ) (1 < p < oo) the Segal LP-space with respect to %V,o (Segal [1]). For each ~ ~ 7-/we put r

-: r~(po(A))~ = J , oA*J,o~,

A ~ :D(#o)* N ~ ( # o ) ,

where ~r~ is the right regular representation of the Hilbert algebra po(~9(po)*N ~ ( # o ) ) and J~o is the unitary involution on 7-/ defined by J,o#o(A) = #o(A*), A ~ Z)(po)* n Z)(po). It is well known in Inoue [2] and Segal [1] that 7ro(~)* = =o(J,o~),

r ~ 7-/,

(2.4.1)

which implies

=o(~) +=o(v) - =o(4) + =o(v) = =o(~ + v), 9 =o(~) = / o ,

),

=

o

for each 4, r/C 7-/and A c C. We now put 7fP o = {~ e ~ ; 7ro(~) e LP(%vuo)}, 1 _< p < oc ;

n

2
Take arbitrary ~,~ E ~ ( -

7ro(r 7ro(~) e L'~(~uo) -=

7-/~o). By (2.4.1) we have r~o(~)- 7to(T/) = n

LP(~uo)' and so 7ro(~) 9 7to(T/) = ~ro(r

for

2
some ~ ~ ~ , which implies r D(~r0(~)) and ~ = 7ro(~)7/. Hence it follows from (2.4.2) t h a t ~ is a subspace of 7-/and it is a *-algebra equipped with the multiplication and the involution : ~r] -- rro(~)~? for ~, ~ C 7-/~, so t h a t we put

and

~* = J~o~

2.4 Special cases

63

=

By (2.4.1) the closure 7r~ of Ir~ is an integrable representation of 7-/~ and AJ ~ - 7r~(?-/~) is an integrable O*-algebra on 79(7r~), and further the O*algebra 2bl~ generated by Ado [z~(~-~) and Ad"~ is an EW*-algebra on 79(~r~) co by over Ado. We define a generalized vector p ~ for Ado~

=

4

and define a generalized vector #~ for M ~ by the restriction p~ - p ~ [z4~ of p ~ to AJ ~. Then it follows that p~ and p ~ are standard tracial generalized vectors for Ad ~ and Ad~o , respectively, such that (#.,)cc = ~o cc = # o . P r o p o s i t i o n 2.4.6. Let AJ be a closed O*-algebra on 79 in 7-/ such that Adw79 c 79. Suppose # is a tracial generalized vector for Ad such that p((79(p)t N 79(#))2) is total in ~ . Then the following statements are equivalent: (i) # is standard, that is, (3d, g, pc) is a standard system. (ii) #c((79(#c), M 79(#o))2) is total in ~ . (iii) (Ad,#, p;) is a standard system. (iv) #'((79(#')* A 79(p,))2) is total in ~ . I IJ , = 3,tw, I (v) J ,(Adw) where J , is the unitary involution on ~ defined by J~p(Y) = # ( y t ) for each Y E 79(#)t n If this is true, then J , = J , cc = J,,, and A], = A , c c = z ~ , , = I, and further y t * = ~ for each Y C 79(p). P r o o f . (ii) ~ (i) Since #cc(79(pcc). N 79(#cc)) is an achieved left Hilbert

1

algebra in 7-/and # is tracial, it follows that S~cc = J ~ c c A ~ c c D S~ = J~, which implies that A ~ c c = I and J~,cc = J~. Hence p is standard. (i) ~ (iii) This follows from Theorem 2.2.8. (iii) ~ (iv) This is trivial. (iv) ~ (v) We can prove in the same way as in the proof of (ii) ~ (i) that , ,j A , , , = I and J / , = J~, which implies that J,(Adw) , = 2%4,w. (v) ~ (ii) Let X c A/l, and let X = UxIX] the polar decomposition of and IXI =

//

t d E x ( t ) the spectral resolution of IXI. We put

E x ( n ) = fO n d E x ( t ) and X n = X E x ( n ) ,

n c l~.

Then we have U x , E x ( n ) , X n C (AJ~w)' for each n E N. Take arbitrary Y E 79(#) and n E N. Then, by the assumption (iv) we have J~Y~ J~p(Z) = J ~ E y ( n ) # ( Y t Z t) = Z J ~ E y (n)J~#(Y)

64

2. Standard systems and modular systems

for each Z 9 79(p)t C~79(p), which implies

juy~ ju 9 79(pc) and pc(JuY~ J,) = J~,Ey(n)J~t~(Y),

(2.4.3)

and hence lira p c ( j u y , j , ) = p ( y ) . n--+

(2.4.4)

o~

For any Y 9 79(p)t n 79(p) we have

JuY,~Jup( Z) = JuVy E y ( n ) V { Y Jup( Z) = Z J,, Uy Ev ( ~ ) v r*~ J . u ( Y ) t for each Z 9 79(p)t n D(p), and hence by (2.4.3)

j~,y,~j, 9 79(pC), r? 79(pC), Uc(JuYnJ,) = JuUyEy ( n ) V *; J u u ( Y )t, pC(JuY*J,~ ) = J~,Ey(n)Jup(Y ).

(2.4.5)

Further, by (2.4.5) we have

(JuUyU{#(Y)Ip(Z~Z2)) = l i m (uC(JuY,~Ju)lp(Z{Zz)) = lira (JuYnJup(Z1)IP(Z2)) n~O0

= (Jup(YZ~)tp(Z2)) = (.(Y*)I#(ZIZ2)) for each Zl, Z2 9 79(p)t f) D(#). Since # ( ( D ( p ) t n 79(p))2) is total in ~ , it follows t h a t U~.U{#(Y) = #(Y), which implies by (2.4.5) t h a t

linapC(juy,~ju) = p ( y t ) . Since p((79(p)t n 79(p))2) is total # c ( ( D ( # c ) * n 79(/1c)) 2) is total equivalent and z:~,, = z~ucc = I We finally show t h a t y t * = y

(2.4.6)

in 7-/, it follows from (2.4.3) ~ (2.4.6) t h a t in 7-/. Thus the statements (i) ~ (v) are and J~,, = J~cc = J~. for each Y 9 79(#). We first show

79(pc), n 79(pc) __ { j u A , ju ; A 9 91}, #c(juA*J,) = poe(A), # c ( J , A J , ) = #CC(A*), A 9 9.1, (2.4.7) where 9.1 ~ {A e D(pCC) * n 79(pcc) ; #CC(A), pCC(A, ) 9 79}.

(2.4.8)

In fact, take an arbitrary A 9 9..[. T h e n we have J~,AJ~, J~,A*Ju 9 3.4~ and by (2.4.5) and (2.4.6)

2.4 Special cases

65

(J, A J , ) p ( Y ) = lira J, ApC(J, YnJ,) n ----~ o r

=

lim J, JuYnJM, CC(d) n~oo

=

lim YnpCC(A *) n---* oo

= ypCC(A*), ( J , A * J , ) p ( Y ) = lira JuA* pC(j~YnJ,) ~ ---* o o

= ypcc(A) for each Y c 1)(p)t n 1)(p), which implies t h a t J~AJ~ E D ( p c ) * N 1)(pc), pC(j~mJ~,) = pCC(d*) and pc(j~,A*J~) = pCC(A). Conversely, take an a r b i t r a r y K C 1)(pC)* N 1)(pc) and put. A = J~K*J,. Then we have / ! A,A* 9 ( A / w ) ,

Ape(K1) = JuK*JupC(K1) = KlpC(K) and A*#C(K1) = KlpC(K *) for each K1 9 D ( p c ) * N / ) ( p c ) , which implies t h a t A 9 9~, p e t ( A ) = p C ( K ) and pCC(A*) = pC(K*). T h u s the s t a t e m e n t (2.4.7) holds. Take an a r b i t r a r y Y 9 1)(p). By (2.4.7) we have

7ro(p(Y))pCC(d) = 7r'o(pCC(d))p(Y) = JuA* g,p(Y) = ypcc(m), and further by (2.4.4)

7ro(J,p(Y))pCC(d) = J , d * p ( Y ) = lim J , A 9p c (J~,Y~9 g,) n .-..~ o o

= lim Y ;9p cc (A) n--.-.~ o o =

ytpcC(A)

for each A 9 9,/, and further since 9,/is a Hilbert algebra in 7-/by (2.4.7), it follows t h a t 7ro(p(Y)) C Y and 7Co(J~,p(Y)) C y t , which implies by (2.4.1) that ~ 0 ( p ( r ) ) C r C r t * C ~ 0 ( J , p ( r ) ) * = ~0(p(Y)). Hence we have y t * = y = 7ro(p(Y)) for each Y 9 1)(p). This completes the proof. B y P r o p o s i t i o n 2.4.6 we have the following C o r o l l a r y 2 . 4 . 7 . Let A4 be a closed O*-algebra on 7) in 7-/ such t h a t A 4 ~ D C 7) and (o c 1). Suppose ~0 is a cyclic tracial vector for AA. T h e n the following s t a t e m e n t s are equivalent: (i) (0 is standard. (ii) A,ffw~0 is dense in 7-/.

66

2. Standard systems and modular systems

(iii) J~o(Mw) ' ' J~o = Mw. ' If this is true, then J~0 J~'0' A~o = A "~o I and M is an integrable O*-algebra on /9. Further, M is a *-subalgebra of the *-algebra L'~(aJ~o) = N LP(wfo) equipped with the strong sum, strong scalar multiplication, :

l
strong product, and adjoint, where LP(co(o) is the Segal LP-space with respect ! ! to the vector trace ~ o on (Adw) . C. S t a n d a r d s y s t e m s for s e m i f i n i t e O * - a l g e b r a s We treat with a standard system ( M , K'p, (K'p)') constracted by a standard tracial generalized vector p and a non-singular positive self-adjoint operator K ~, and consider when a standard system (M, t , A') is unitarily equivalent to such a standard system (Af, K'/I, (K'p) 0. Let M be a closed O*-algebra o n / 9 in 7-/such that M'w/9 C/9, tt a standard tracial generalized vector for AA and K ' a non-singular positive self-adjoint operator in 7-/affiliated with M'w whose d o m a i n / 9 ( K ' ) contains p(/9(p)). Let K' = and K - J ~ K ' J , =

/5

tdE'(t)

tdE(t) be the spectral resolutions of K' and K, re-

speetively and let E'(n) = here put

/5

/0

dE'(t) and E(n) =

/0

dE(t) for n E N. We

/9(K'.) =/9(#!, ( K 'v) ( X ) = K , ( X ) ,

X c/9(.).

Then K ' # is a generalized vector for M . In fact, let K " = K ' E ' ( n ) and Kn = KE(n), n E N. Then, for each A E 3,/ and X c / 9 ( p ) we have lim K ' p ( X ) = K ' p ( X ) and lira AK'~p(X) = K ' p ( A X ) . rt--~ oo

n -~ oo

Hence we have

K' # ( X ) e / 9 and A K ' p ( X ) = K ' A p ( X ) , which implies that Ktp is a generalized vector for A/[. For the standardness of the generalized vector K ' p we have the following P r o p o s i t i o n 2.4.8. Let M be a closed O*-algebra o n / 9 in ~ such that I Mw/9 C / 9 and # a standard tracial generalized vector for Ad. Suppose K ' is a non-singular positive self-adjoint operator in 7-/affiliated with Ad~w such that #(/9(#)) C / 9 ( K ' ) , K ' p ( ( / 9 ( p ) t n/9(p)) 2) is total in 7-/and K ' p ( / 9 ( p ) t n/9(p)) is dense in the Hilbert s p a c e / 9 ( K 9 K ' - I ) . Then K ' # is a generalized vector for A4 satisfying the following conditions: (i) ( K ' # ) ' ( ( / 9 ( ( K ' p ) ' ) * n / 9 ( ( K ' p ) ' ) ) 2) is total in 7-(.

2.4 Special cases

(ii)

SK,~, = S ( K , , ) , , = J ~ K

67

9 K '-1.

K'p, ( K ' p ) ' ) is a s t a n d a r d system if and only if Kit:D C i/:) and K i t Y K -it [:D 9 :D(p) for all Y e T)(p) and t 9 ~ .

Further, (Ad,

P r o o f . Since # is standard, there exists a net { A s } in :D(#CC)* A :D(#cc) which strongly* converges to I. Let. Y 9 :D(p)t A :D(p) and let Yn = YEy(n),n 9 N. T h e n it follows from Proposition 2.1.7 t h a t {Yn} C T)(pcc) * NT)(Pcc), Yn --~ Y strongly*, nlim pCC(yn) : #(Y) and l i m p cc (Yn) = p ( y t ) . Take a r b i t r a r y C 9 :D(#c), Y 9 :D(#) t A :D(p) and m, n E I~. T h e n we

have

E'(n)CE'(m)(K' p)(Y)

=

lim

E'(n)CE'(m)K' pCC(Yk)

k ~o~

=

lira

E'(n)CJ, KE(m)pCC(Y~)

k---* o c

=

lim k ----~o o

=

c~

lira lira k ----*o o

=

limE'(n)CJ, A,~KE(m)pCC(ys*)

lim

E'(n)Cj~#CC(d,~KE(m)Y~*)

c~

limE'(n)C#CC(ykKE(m)A*)

k----* o c

=

lim k ---* or

= lim

limYkE'(n)KE(m)d*#C(C) c~

YkE'(n)KE(m)#C(C),

k----* c ~

and so

(Yt~IE'(n)KE(m)#C(C))

=

:

lim (Y~TllE'(n)KE(m)#C(C)) lira

(~]YkE'(n)KE(m)#C(C))

k---~ o o =

which implies

(~l[E'(n)CE'(m)(K'p)(Y)),

E'(n)KE(m)#C(C) C :D(yt*) = :D(Y) and Z E ' ( n ) K E ( . ~ b c ( C ) = E'(n)CE'(.~)(K'u)(Y).

Hence we have

F,'(n)CF_,'(m) 9 ~((K'~)')* n ~((K'~)'), (K'H)'(E'(n)CE'(m)) : E'(n)KE(m)HC(C), VC 9 ~ ( ~ ) * n ~ ( ~ ) , vm, n 9 N.

(2.4.9)

Similarly, we have

E'(n)CE'(m)K '-1 9 T)((K' N)')* r-] T ) ( ( K ' # ) ' ) , (K'p)'(E'(n)CE'(m)K '-1) = E'(n)E(m)#C(C), (K%)'(E'(.~)K'-'C*Z'(~)) : E'(~)K'-'E(~)K~(C) v C 9 ~(~o). n ~(~o), m, n 9 N.

(2.4.10)

68

2. Standard systems and modular systems

By (2.4.9) and (2.4.10) we have

E ' ( n ) q E ' ( 0 C ~ E ' ( - ~ ) ~ ; '-1 9 ( ~ ( ( K ) 0 ' ) * n ~((K'~,)')) 2 and

lira m

(K'p)'(E'(n)CIE'(1)C2E'(m)K'-I)

,72--~o c

:

lim S ' ( n ) q S ' ( O ( K % O ' ( S ' ( O C y ( ~ ) K m,n~

:

'-~)

c~

lim E ' ( n ) C 1 E ' ( l ) E ( m ) p C ( C 2 )

m~n--~o~

= pc(c1c2) for each C1, C2 9 Z)(#c) * NZ)(#c), which implies since pc((~9(pc)* NT)(pC)) 2) is total in ~/ that ( K ' p ) ' ( T ) ( ( K ' p ) ' ) * N D ( ( K ' p ) ' ) ) 2) is total in 7-(. We show J,K K '-1 = S K , , = S(K,,),,. Since 9

K ' p ( l ) ( p ) * N :P(t,)) is densely contained in the Hilbert space T)(K 9 K '-1) and further

JuK . K'-IK'p(Y)

= J u K p ( Y ) = K ' p ( Y t) = S K , u ( K ' p ) ( Y )

for each Y 9 :D(p)t NZ)(p), we have J u K 9 t ( '-1 C SK,tt. By Lemma 2.2.2 we generally have SK,u C S(K,u),,, and hence Jt, K 9 K ' - 1 C S K ' t L C S(K,t~),,. Conversely we show S(K,u),, C J , K 9 K t-1. It is shown similarly to (2.4.9) that E ' ( n ) C E ' ( m ) 9 T)((K'p)')* n T)((K'p)'), ( K ' p ) ' ( E ' ( n ) C E ' ( m ) ) = E'(n)KE'(m)pCCC(C), VC 9 ~ ( ~ ) *

n :D(~'~'~), Vm, n 9 N

and clearly J~,A*J~, 9 T)(p ccc) and pCCC(j,A*J~) = p e t ( A ) ,

VA 9 I ) ( p ~ ) * n ~(p~c), so that E ' ( n ) J ~ A * J v E ' ( r n ) e 7P((K'p)')* N :D((K'p)'), (K'p)'(E'(n)J~,A*J~,E'(m)) = K E ( m ) E ' ( n ) p C C ( A ) , VA c Z~(~cc) * n Z)(~cc) Hence

we have

2.4 Special cases

69

(KK'-I K'Et(rz)E(m)#CC(A)I(IC~O"(B)) : ((K'~O'(E'(rz)J.A*J.E'(m))I(K'#)"(B))

= (S(K,.),,(K'Iz)"(B)IS~K,u),,(K',O'(E'(n)JuA*J.E'(m))) = (S(K,u),.(K'#)"(B)I(K'#)'(E'(rn)J~AJ~E'(n))) = (K'#)"(B) IKS(n) = (S(K,.),, ( K ' # ) " ( B ) I J . K ' E ' ( n ) E ( m ) # C C ( A ) ) =

(K'#)"(B))

for each A E 2)(#cc). N 2)(#cc) and B E 2)((K'11)")* A 2)((K'p)"), and further it follows from (2.4.11) that {K'E'(n)E(m)gCC(A) ; A E 2)(#cc). N 2)(#cc) and m , n E N} is total in the Hilbert space 2 ) ( K . K ' - ~ ) , which implies S(K,.),, C J . K 9 K '-1. Thus we have SK,~ = S(K,.),, = J . K 9 K '-1, and hence

Jg'. = J(g'.)" = 4 and A K , ~ = A(K,~),, = K

9 K '-I.

(2.4.11)

It follows from (2.4.12) that (Ad, K ' # , (K'#)') is a standard system if and only if Kit2) c 2) and K i t Y K -it [2) E 2)(#) for all Y E 2)(#) and t E IR. This completes the proof. We consider when the condition in Proposition 2.4.8 "Kit2) c 2) and

K i t Y K - i t [2) E 2)(p) for all Y < 2)(#) and t E ~ " holds. C o r o l l a r y 2.4.9. Let (A4, #, K') be given in Proposition 2.4.8. Suppose # is full and K/~ [D E A4 for all t E ]~. Then (A/l, K'#, (K'#)') is a standard system. P r o o f . Take arbitrary Y E 2)(#) and t E N. Then we have

( K i t y K - i t # c ( c ) 14) = ( J~,K'-it#c(C*)lYt K-it~) = l i m ( J . C ~ K ' - i t # c ( c *) I Y i K - i t ( )

:

= (cK

14)

for all C E V(#C)*n2)(# c) and { c D, where {Ca} is a net in V(#c)*n2)(# c) which converges strongly* to I. Hence it follows form the fullness of # that K i t y K - i t E 2)(#~) = 2)(#). By Proposition 2.4.8 ( M , K ' m ( K ' # ) ' ) is a standard system. We next consider the converse of Proposition 2.4.8:

When is a standard system ( Ad, A, t') unitarily equivalent to such a standard system (N', K' #, ( K' II )') in Proposition 2.4.8?

70

2. Standard systems and modular systems

An O*-algebra 34 is said to be semifinite if (A4'w)' is a semifinite yon Neumann algebra. P r o p o s i t i o n 2.4.10. Let M be a closed semifinite O*-algebra on D in 7-/such that ~#wT) C T). Suppose A is a generalized vector for 7%4 such that (i) A((D(,~) t n D(A)) 2) is total in 7-/; (ii) A'((ID(A')* Q :D(/V)) 2) is total in 7-{; (iii) S~ = S~,,; (iv) Y c L2(7 ") for each Y C T~(A), where r " is a faithful normal semifi! ! nite trace on (Mw) . Then there exist a standard tracial generalized vector p for a closed O*-algebra N" in L2(r '') and a non-singular positive self-adjoint operator K ' in L2(r '') affiliated with JV~wsuch that (p, K') satisfies all of the conditions in Proposition 2.4.8 and ), is unitarity equivalent to the generalized vector K'p, that is, there exists a unitary operator U of L2(r ") onto 7-{ such that U * M U = A[, U*:D(A)U = D(p) and X(Y) = U(K'p)(U*YU) for each Y E ~(A). P r o o f . By the assumption for A, A"(D(A")* n I)(A")) is an achieved left Hilbert algebra in 7-/ whose left yon Neumann algebra equals the semifinite t t yon Neumann algebra (A4w) , so that the following results have been shown by Takesaki [1]: (2.4.13) We put HoA"(B) = B, B E ~D(A") N cJ~,,. Then H0 is a closabte operator of the dense subspace A"(Z)(A") N r onto the dense subspace I)(A") nr in L2(r '') whose closure/7 is non-singular. (2.4.14) Let /7 = VT' be the polar decomposition o f / 7 . Then V is a unitary operator of 7-/ onto L2(r '') and T ' is a non-singular positive selfadjoint operator in 7-{ affiliated with A4'w such that z ~ , , = T -1 9 T', where

T = J~,,T'J~,,. (2.4.15) Let P0 be the left regular representation of ( M ~ ) ' on L2(T '') defined by po(A)B = AB, A E ( M ~ ) ' , B ff r Then the unitary operator V implements a spatial isomorphism between (A/I'w)' and p0((M~w) ') such that V B V * : po(B) for each B E Z)(A") N r (2.4.16) A't(I)(A")* N T)(An) Q r is dense in the Hilbert space D(T'). Let A be the inverse o f / 7 and A = U K ~ be the polar decomposition of A. Then we have U = V* and K ~ = U * T ' - I u . It follows from (2.4.14) that U is a unitary operator of L2('r ") onto 7-/and K ~ is a non-singular positive self-adjoint operator in L2(r ") affiliated with the von Neumann algebra po((M~)')'. We put

A; = U*MU, :D(#) = U*D(A)U and p(U*YU) = Y c L2(T"), Y E D(A). Then Af is a closed O*-algebra on U'T) in L2(T ") such that APw = U*MwU' and (Aft')' = U*(M~)'U = po((AJ'w)'), and by (2.4.15) p is a tracial gener-

2.4 Special cases

71

alized vector for N'. We show A(Y) E 79(H) and H A ( Y ) = #(U*YU), Y E 79(A). In fact, let X E 3,4 and let. X = Ux [XI be the polar decomposition of and IX[ =

/0

tdEx(t) the spectral resolution of IXI. Take an arbitrary

Y E D(A). Then it. is shown that Y E v ( n ) E 79(A") Neff~,, and A " ( Y E y ( n ) ) = E y , (n)A(Y), n E N. Hence we have

l i m I l E y t ( n ) A ( Y ) - A(Y)II = 0, liln IIHEyt(n)A(Y) -- P(U*YU)II = lira r " ( ( I - Eg(n))l~:l 2) = O, n~*oo

m---~ o o

which implies A(Y) E 79(H) and HA(Y) = p(U*YU). Hence we have p(79(p)) c 79(K'). Since K'~L(U*YU) = U*A(Y) for each Y E 79(A) and A((79(A) t D79(A)) 2) is total in ~ , it follows that (K'p)((79(K'p) t N 79(K'p)) 2) is total in L 2 ( r ' ) . Further, since we have (K'~,)(u*vu)

= u*A(Y),

( K 9 K ' - I ) ( K ' p ) ( U * Y U ) -- U * T -1 9 T ' A ( Y ) : U * J x A ( Y t)

for each Y E 79(A) t N 79(A), and further Sx -- S~,, = J~T -1 9T' by (2.4.14) and A(:D(A) t C~79(A)) is dense in the Hilbert space 79(Sx,,) = 79(T -1 9 T'), it follows that K'p(79(p) t (1 79(p)) is dense in the Hilbert space 79(K 9 K ' - I ) . Thus the pair (p, K ~) satisfies all of the conditions in Proposition 2.4.8. It is clear that 79(p) = U*79(A)U and A(Y) = U(K'p)(UYU*) for each Y E 79(A). This copmletes the proof. By Propositions 2.4.8, 2.4.10 we have the following T h e o r e m 2.4.11. Let M be a closed semifinite O*-algebra on 79 in 7-/ such that 2tdwD ~ C l), and let A be a generalized vector for 2M. The following statements are equivalent: (i) ( M , A, A') is a standard system such that Y E L2(r ") for each Y E 79(A), where r " is a faithful normal semifinite trace on (3,1~)~. (ii) There exists a closed O*-algebra A/" on 8 in h5, a standard tracial generalized vector p for J~ and a non-singular positive self-adjoint operator K ' in 1C affiliated with JV~ such that (ii)l p(79(p)) c 79(K') and K'p((79(p)* N79(p)) 2) is total in ]C; (ii)2 K'p(79(p) t N 79(p)) is dense in the Hilbert space 79(K 9 K ' - I ) , where K =- JuK'Jv; (ii)3 K i t e c $ and K i t Y K - i t [$ E 79(p) for all Y E 79(p) and t E ]~; (ii)4 A is unitarily equivalent to the generalized vector K~p, that is, there exists a unitary operator U of JC onto 7-/ such that U*]MU = N', U*79(A)U = 79(p) and A(Y) = U(K'p)(U*YU) for each Y E 79(A).

72

2. Standard systems and modular systems

C o r o l l a r y 2.4.12. Let 2t4 be an integrable O*-algebra on 79 in 7-/with a standard tracial vector 40, and let ~ E 79. Then ~ is a standard vector for Ad if and only if there exists a non-singular positive self-adjoint operator K I in 7-/ affiliated with A//'~ such that (a) ~0 E 79(K I) and Kl~o E 79; (b) K1~4~o is dense in the Hilbert space I ) ( K 9 KI-~), where K =- J4oKIJ4o; (c) K it IT) E A,{ for each t E 1~; (d) 4 is unitarily equivalent to Kt~o . D. S t a n d a r d g e n e r a l i z e d v e c t o r s in t h e H i l b e r t s p a c e o f H i l b e r t Schmidt operators For physical applications, it is customary to study the Hilbert space ~| of all Hilbert-Schmidt operators on 7-/, together with the natural representation 7c of an O*-algebra A/I on D in "H on ~ | 7-/. In this section we show how every positive Hilbert-Schmidt operator s determines a generalized vector An for lr(A4), and we investigate under which conditions An is standard or modular. Further, we prove that a positive self-adjoint unbounded operator /2 also defines a generalized vector A~, and we ask the above questions as before. We denote by H @ 7-/the Hilbert space of all Hilbert-Schmidt operators on a separable Hilbert space 7-/ with the inner product < SIT >--- t r T * S , S , T E ~ @ ~ , and the norm IITIt2 - - < TIT > 1 / 2 T E ~ | We define some operators on ~ | ~ : Let H and K be closed operators in 7-/ and put

79(Trl'(H))

= {T E 7-/@ ~ ; T ~ c 79(H) and H T E ~ | ~ }

[Tr//(H)T = HT,

T C 79(~/'(H)),

(7c'(K)) ~ {T E "]-I| 7-l; T K is closable and T K E 7-l | TI} [7~ ( K ) T = T K , T ~ 79(Td(H)). Then we have the following L e m m a 2.4.13. (1) 7r'I is a *-homomorphism of/3(T/) onto a von Neum a n n algebra on 7-/| 7-/ and 7r' is an anti *-homomorphism of/3(7i) onto the c o m m u t a n t of the yon Neumann algebra 7r'(B(Tt)), and they have the relation: 7/1(B(7-/)) = Jzcf(13(7-l))J, where J denotes the isometry on 7-/@ defined by J T = T*, T E 7-t @ 7-/. (2) 7r'(H) and TJ(H) are closed operators in 7-/ @ 7i affiliated with 7r1'(~(7-/)) and 7d(B(7-/)), respectively, and J79(z~"(H)) = 79(Tr'(H)) and

7r"(H) = J ~ ' ( H ) J . (3) Suppose H and K are (positive) self-adjoint operators in 7-/. Then, 7r'(H) and l d ( K ) are (positive) self-adjoint operators in T/| 7-l, and further 7J'(H)zr'(K) and 7r'(K)Tr"(H) are (positive) essentially self-adjiont operators in 7-/@ 7-/and 7r"(g)zr'(K) = 7r'(K)rc"(H). Let 34 be a closed O*-algebra on 7) in 7-/such that 2r

c 79. We put

2.4 Special cases

' z)(~) =

n

73

z)(~"(~)),

X E.A4

T h e n we have the following L e m m a 2 . 4 , 1 4 . Let M be a closed (resp. self-adjoint) O*-algebrk on ID in H such that, MwID' c i/). T h e n 7r is a closed (resp. self-adjoint) *representation of Ad on 7 / | ~ with the domain :D(rc) = ~ 2 ( M )

C D and X T ff ~H|

~ {T G 7-g | ~ ; T ~

for all X G AA}.

Suppose 3,4" = C I . Then, rr(M)'w = rc'(B(~)) and (rc(M)'w)' = r/'(B(~)). P r o o f . It is not difficult to show the first part. We show the l ~ t part. Suppose 3A'w = C I . It is clear t h a t r/(13(7-t)) c rc(AA)~w. Conversely take an a r b i t r a r y 6 6 rr(A4)'w. Let ~ and r? be any elements of 9 Since the continous sesquilinear form on 7-/x ~ is defined by

( z , y ) ~ H x ~ - - ~ < ~5(z | rl)ly | r >, it follows from the Pdesz theorem t h a t there exists a b o u n d e d linear o p e r a t o r F(~, r/) on H such t h a t

< ,~(x e w)ly | ~ > = (c(r .)xly) for each x, y E ~ . Further, we have

(c(~,~)x411(2) = < ~ ( x ) ( C l | ~)1(2 | 1 6 2> = < 6(r | ~ ) l x t ( 2 | >

= (F(~, 71)r 1X~r for each X E 3A and (1,(2 6 ~ . Hence it follows from 2M~ = C I t h a t /'(~, r~) = c~(~, 77)I for some c~(~, r/) E C. Further, since ~ can be extended to a eontinous sesquilinear form on ~ x ~ , there exists an element A of B ( ~ ) such t h a t ct(~, r/) = (Ar for each ~, 7? 6 :D, which implies

< ~(~ |

|162> =

(Ar

= < ~-'(X)(x | ~)ly | ~ > for each x, y E 7-/ and (, r/ E 2). Hence we have 5 = rc'(A) E rr'(/3(~)). T h u s we have rr'(/3(~)) = rr(M)'w. This completes the proof. T h o u g h o u t the rest of this section let M be a closed O*-algebra on 2) in such t h a t 3 , t " = C I . We now show t h a t every positive Hilbert-Schmidt o p e r a t o r ~2 on ~ determines a generalized vector An for rr(Ad). Indeed, for s _> 0 E 7-/| 7-/let us put

74

2. Standard systems and modular systems

{

79(~9) = {7r(X); X r Jk4, J? ~ 79(7c(Xt) *) and ~r(Xt)*f2 ~ ~ ( 2 t 4 ) } , ~(~(x)) = ~(xt)*~, ~ ( x ) r 79(~).

Then At? is a generalized vector for 7r(M) and

{

79()~) = {~r(X);X ~ M , XT'H C 79(X t*) and X t * n 9 ~2(.M)}, ~ ( ~ ( X ) ) = x t * ~ , ~(x) ~ 79(~).

(2.4.17) We search sufficient conditions for A~ to be a standard or a modular generalized vector for 7c(M). For the condition (S)~ in Definition 2.2.7 we have the following L e m m a 2.4.15. Let f2 > 0 E ~ | such that f2?t is dense in ?t. Suppose there exists an orthonormal b ~ i s {~n} in ~ such that { ~ } C 79 and ~, | ~ r M for n, rn 9 N. Then ~x? is a cyclic generalized vector for re(M) such that Xx~((79(~) t n Z ) ( ~ ) ) ~) is total in ?-t | ~ . Furthermore, if {~,} is total in 7)[t~], then ~ is strongly cyclic. P r o o f . We put g

{}--~ ~k~k |

~k,/31 9 C}.

k,l

Then it is clear that $ C (79()~)t N 79(A~))2 and

k,l

k,l

and since {~n} is an ONB in ~ and s is dense in "H, it follows that /kx?(s is dense in {x | ~; x, y 9 "H}, which implies that A~((~D(,k~)* N ~ ( ; ~ ) ) 2 ) is total in ?t | 7-/. When { ~ } is total in 79[t~], it is similarly shown that . ~ is strongly cyclic. For the regularity of A~ we have the following L e m m a 2.4.16. Suppose S? _> 0 E 7Y | ~ such that A~((79(Ag)t N ID(Aj?)) 2) is total in IX | ~ . Then the following statements hold: (1) A9 is regular if and only if there exists a net {Ks} in B(TY) such that 0 _< K s _< I, Ko --~ I strongly and Y2K~ E ~2(Jt4) for every ~. (2) /k~ is strongly regular if and only if there exists a net {Ks} in B ( ~ ) such that. 0 _< K~ _< I, K s I I strongly, $2K~ G ~2(JM) for every c~ and K ~ K z = K z K ~ for every c~,/3. (3) Suppose that F2g c /9 for some dense subspace g of 7-/. Then A}~ is strongly regular.

2.4 S p e c i a l c a s e s

75

P r o o f . (1) Suppose ),~ is regular. Then there exists a net {Tr'(K~)} in 7r'(B(7"t)) such that O < ~'(K~) _< I, ~'(K~) --* I strongly and ~'(K~)A~(Tr(X)) -- ~(Xf)*A~(~'(K~)) for all X E D(Ag). It is clear that O < K~ _< I and K~ -* I strongly. Since

= < 7r'(K~)Tr(Xi)*/2l),s~(w(Y))

>

= < ~r(X*)*/21~:(Y)A~,(~'(K,~))

>

= < /21~r(XfY)A~(Tr'(K~)) > =
>

= < zr'(K~,)F2lAa(zr(X~Y)) >

for all X , Y ~ ~(An)* O~(An) and ~,~((~(An)~ O~(A9))2) is total in ~| it follows that /2K~ = ~r'(K~)/2 = A~(:r'(K~)) ~ ~2(A4). The converse is trivial. (2) This is shown in the same way as (1). (3) Since 7-/is a separable Hilbert space and $ is dense in 7-/, there exists an n

ONB {~n} in 7-/contained in ~. Since/25 c D, the sequence { Z ~ |

n 9

k:l

N} satisfies the conditions in (2). Hence A9 is strongly regular. L e m m a 2.4.17. Let /2 > 0 c ?-I | ~ such that /2-1 is densely defined. Suppose A~((D(A~)f A D(),n)) 2) is total in ~ | 7-/, and An is regular. Then A~((Z)(A~)* A D(A~)) 2) is total in 7-/| ~ and A~c(T)(A~c) * 7/D(A~c)) is an achieved left Hilbert algebra in 7-I | which equals ~" (B(T/)) f2. Its modular conjugation operator J;,cc coincides with the anti-isometry J : T -~ T*, T C 7-/| ~ and its modular operator z ~ c c coincides with the positive self-

adjoint operator

7rt(/2-2)Trtt(/22).

P r o o f . By Lemma 2.4.16 there exists a net {K~} in B(7-/) such that O < K~ _< I, K~ --* I strongly and /2K~ C G2(A4) for each c~. Then we have (2.4.18)

7 / ( K ~ A K ~ ) E T)(A~)* N T)(A~). Since II~'(K~AK~)/2

- ~'(A)/2112

II~'(K~)Tr'(A)Tr'(K~)~

- ~r'(K,~)Tr'(A)/2tl2

+ll~'(K~)~'(A)/2

- ~'(A)/2112

<_ I t ~ ' ( A ) ( ~ ' ( K g ) - Z)/2il2 + I I ( ~ ' ( K ~ ) - Z)~'(d)/2112

for all ~,/3 and A r B ( ~ ) , we have

76

2. Standard systems and modular systems

lim II~'(K~AKz)~2 - r r ' ( A ) n h = 0, c~,~

lim Ilrc'(KzA*K~)f2 c~,~

-

~r'(A*)OIl~ =

0

(2.4.19)

for all A 9 B ( ~ ) . Since rc'(B(7-l))f2 is dense in ~ < 9 ~ ; it follows from (2.4.18) and (2.4.19) that A~(2)(A~)* N2)(A~)) is dense in ?t<97-L Furthermore, since rr'(K.r)rr'(K~AKz) 9 (Z)(A~2)* n ~9(A~)) 2 for every a, ;3, "7 and A 9 g ( ~ ) , it follows that A~((2)(A~)* N D(Ao) c ) 2 ) is total in 7-/<9 ~ . By (2.4.18) we have

{~'(K~AKe)O; A 9 t~(~), ~, 9} c A~,(>(A~,)* n >(AS)) c ~ ' ( S ( ~ ) ) O , and so it. follows from (2.4.19) that A~c(D(A~C) * N D(A CnC )) is an achieved left Hilbert algebra in ~ <9 T{ and it equals the achieved left Hilbert algebra rd'(B(~))f2. For any T E 7-{ <97-t we have

(2.4.20) Let { ~ } be the ONB in T/consisting of eigenvectors of non-zero eigenvalues {wn} of ~. We now put P~ = s

~k <9 ~k, n E N. Since

k=l

Ilrc"(P,~APn)n- rr"(A)nlle <_Ilrc"(P,~AP,dn - rr"(P~A)nll2

+ [Irr"(P~A)n - rr"(A)~ll2 -< IIAIIIIP,~O- nil2 + IInA*Pn - nA*ll~

--tIAII (k

n< l,Ve~kllOl/2+ (k:n< ll~?A*~kllO1/2

for each A E B(TI), we have lira r~" ( P n A P n ) n = r r " ( A ) n

n---+oo

(2.4.21)

for each A E B(T/). Since P ~ A P n E 7-/(9 7-/ for each A E B(~/), it follows that 7d'(~ <9 7-{).(2 is dense in the Hilbert space :D(S~,~c). By (2.4.20) we have

&~o c J~'(n-~)~"(n). Since T ' " ( ~ ' ~ - - l ) f f t ' ( ~ ) 2.4.13, we have

is a

(2.4.22)

positive self-adjoint operator in 7-/<9 ~ by L e m m a

< f 2 l r r " ( A ) J r d ' ( A ) J ~ > = < S:,ccrr"(A)f2lJrr"(A)~ > = < Jrr'(D-1)rr"(J'2)rd'(A)~iJrr"(A)Q > = < rr"(A)D[rr'(~-l)rr"(f2)rr"(d)~ >

>_0

2.4 Special cases

77

for each A E B(Tt). It hence follows from Theorem 1 in Araki [3] that J =

J~g~.

By (2.4.22) we have ~ c c

c 7c'(g?-~)~r"(s

By the maximality of

1

self-adjoint operators, we have ~ c c

= 7c'(f2-1)rc'(g?)- This completes the

proof. T h e o r e m 2.4.18. Let 31t be a closed O*-algebra on D in 7-t such that Ad~w = C I and let ~ ~ 0 E 7~ | ~ . Suppose (i) F2-1 is densely defined; ( i i ) ) , 9 ( ( D ( ~ 9 ) t n ~D(~)) 2) is total in 7-t | ~ ; (iii) ~9 is regular; (iv) f2i~D C D and ~ i t M ~ - i t = J~ vt E JR. Then ~ is a standard generalized vector for 7r(~4) such that Ja~ = J and P r o o f . By (iv) we have X ~ 2 i t T = Jr~it(~'2-itx~'~it)T E ~-~ |

for all X E 3/l, T E G2(Ad) and t E N. Hence it follows that 7r"(~2u)G2(Ad) c G2(2t4) for all t E N, which implies by Lemma 2.4.17 that

A ~ G 2 ( M ) = ~'(x?-2i~)~r"(r174 it

--it = 7r( ~ 2 i t M ~ _

c ~2(M),

2it) = 7r( ~ )

for all t E ~ . Further, for any 7~(X) E D ( $ 9 ) t N D(),9) and t E ~ we have

F27-~C Z)( ~22itx t * ~2-2it) and ~2it x~* J~- 2it ~ ~ it

~2(.A/[). --it

Hence it follows from (2.4.17) that z ~ c ( D ( ~ 9 ) t N D ( ~ 9 ) ) / l ~ c = ~9(~9) t N D ( ~ ) for all t ~ ]~. Therefore ~9 is standard, and by Lemma 2.4.17 J~z = J and Z ~ z = 7/(~2-2)~'(t92). This completes the proof. C o r o l l a r y 2.4.19. Let H be a positive self-adjoint operator in 7-t, D =

( - ~ D ( H n) and F2 ~_ 0 c 7 ~ |

Suppose Y2-1 is densely defined

n=l

and ~2H C HF2. Then An is a standard generalized vector for 7r(Et(D)). P r o o f . Let us take an ONB { ~ } in ~ contained in D. Since ~ | ~,~ E s for n , m E N, it follows from Lemma 2.4.15 that A~((D(),~) t n D()~9)) 2) is total in ~ | ~ . Since F2H c H ~ , it follows that J2[z), f2u~z~E s for all t C JR, so that ~ is standard by Lemma 2.4.17 and Theorem 2.4.18.

78

2. Standard systems and modular systems

We now look for sufficient conditions for An to be a modular generalized vector. T h e o r e m 2.4.20. Let Aft be a closed O*-algebra on :D in 7-/ such that MIw = C I and let f2 >_ 0 E 7-[ | ~ . Suppose (i) f2 -1 is densely defined; (ii) An((T)(A~) t ;-1T)(A~)) 2) is total in 7-/| ~ ; (iii) there exists a dense subspace g of ~D[t:~] such that (iii)~ AdS c 8. (iii)2 ~28 c S. (iii)3 ~ u g c 8 for each t E ~ . Then An is a modular generalized vector for ~r(A4). P r o o f . It follows from (ii), (iii)2 and Lemma 2.4.16 that An is regular, which implies by Lemma 2.4.17 that A CnC (Z)(ACnC ) * V/ Z)(A~c)) is an achieved left Hilbert algebra in T{| and it equals ~r"(B(7-/))f2 and z~xcc = 7r'(~-2)lr"(f22). We denote by /C the linear span of {( | ~; ( E E, y E 7-/}. Since g is dense in :D[tAa], it follows that /(2 is dense in G2(A4)[t~(A4)]. We next show that { A ~ ( K ) ; K E :D(X~)* n :D(A~) s.t. A~,(K),A~(K*) E/C} 2 is total in the Hilbert space "D(S*),cc). (2.4.23) In fact, let us take an ONB {(n} in 7-/contained in S and put Pn = ~

~k|

k=l

n E N. Then we have ~'(PndPn)

E (l)(A7~)* n

Ag(Tr'(P,~APn)) =

D{Ac~'~2 , .. ,

~~,(AC~jl~k)f2~k | ~ E IC

( by (iii)2)

k=l j=l

for all A E B(7-/). Furthermore, we have lira

II~'(PnAP~)~Q

- ~r'(A)~2[]2 = 0,

n~m----+~

lira n~m-----~o0

II~'(PnAPm)*~Q - zr'(A)*~ll2

= 0.

Thus the statement (2.4.23) holds. By (iii)2 we have 7r(J~4)K: c K, and by (iii)3, A~,t~clC c K: for all t E ]~. Thus, An is modular. This completes the proof. For the standardness and the modularity of a vector s E G2(A~) we have the following

2.4 Special cases

79

C o r o l l a r y 2.4.21. Let Ad be a closed O*-algebra on /9 in 7-{ such thatA4~ = C I and tet Y2 >_ 0 9 G 2 ( M ) . Suppose s -1 is densely defined and 7r(A4){2 is dense in 7~ | ~ . Then the following statements hold: (1) Suppose f2 9 G2(s and Y2itZ) C /9 for all t 9 ]R. Then X2 is a modular vector for re(A4). (2) Suppose that there exists an element N of s such that N -~ 9 7-/| ~ , and [2itTP C / 9 for all t ~ 1~. Then f2 is a quasi-standard vector for ~r(M). Further, if J~2itM~-d-it .A~ for all t 9 ~ , then f2 is a standard vector for ~r(A4). =

P r o o f . (1) Since 7c(Ad)S2 is dense in ~ | and g?TP c :D, it. follows from Lemma 2.4.16 that An is regular, so that by Lemma 2.4.17 ACnC (:D(ACnC ) * n :D(A~c)) = 7r"(B(TY))g2 and it is an achieved left Hilbert algebra in 7-I | such that A a c c = 7r'(y2-2)Tr"(Y22). Let {G~} be an ONB in 7-/consisting of eigenvectors of non-zero eigenvalues {w~} of X2. Then {G~} c D and s = w~{n | {~. We denote by g the linear span of {{n | ~ ; n, m 9 1%1}.Then n=l

since s C ~2(s follows that G 2 ( s

C G2(Ad) c Z ) | and s is dense in G2(Ad)[G(~)], it is dense in ~ 2 ( M ) [ G ( M ) ] . Since f22it F~ 9 s for

all t 9 R, it follows that z 3 ~ t c c 6 2 ( s clear that 7r(Ad)G2(s c G2(s vector for 7r(A4). (2) Since XT = N-I(NXT)

c G2(s for all t 9 R . It is Thus Y2 is a modular generalized

9 (7-{ | ~)B(?-t) = Tl |

for all X 9 s ) and T 9 @ ~ , it follows that. G 2 ( s = ~2(Ad) = :D | 7-{. Hence statement (2) follows from (1). This completes the proof.

We next investigate the standardness and the modularity of a generalized vectors An defined by a positive self-adjoint unbounded operator ~2. We put

{

:D(An) = { : r ( X ) ; X E M and Xt*,.Q E G2(AJ)},

M(~(x)) = ~ ,

~(x) c v(A.).

Then An is a generalized vector for 7r(A4). L e m m a 2.4.22. Let J~I be a closed O*-algebra on /9 in 7-/ and K2 a positive self-adjoint operator in 7/such that Y2-1 is densely defined. Suppose there exists a subspace $ of I) N 9 such that

(i) {~ | 7; ~, 7] C g} c A/I, (ii) $ is a core for J2. Then An((:D(Ag) t n :D(An)) 2) is total in ~ | ~ . Furthermore, if g is dense in T)[t~], then An is a strongly cyclic generalized vector for 7r(~'I).

80

2. Standard systems and modular systems

P r o o f . Since {~ | ~; ~, r] r g} C A d and g" is dense in T/, it follows that 2t4~w = C I . It is easily shown that {~r(~ | ~); ~, r/E s C (79(A~?)* ~ 79(/~a)) 2 and A~(~r(~| = ( | Y2r/for each ~, 7/E g. Since Y2g is total in 7-/, it follows that {(~ | r/)f2;~,r/ E s is dense in {4 | f ; 4 , r/ ~ 79}, and further, since {~ | f; ~, r / r 79} is total in ~2(fl4), it follows that A~((79(Aa)* C~79(Aa)) 2) is total in 7-/| 7-{. When g is dense in 79[t2a], we can similarly show that Ax? is strongly cyclic. This completes the proof. T h e o r e m 2.4.23. Let Ad be a closed O*-algebra on 79 in 7-I such that Ad~w = C I and X2 a positive self-adjoint operator in 7-(. Suppose (i) t ? - t is densely defined and 79 N 79(g?-1) is a core for Y2-1; (ii) there exists a subspace Af of 2t4 such that 7r(AY) c 79(~x?), A/'t79 c lP(X?) and the linear span of N ' t D is a core for .O; (iii) ~ x ? ( ( 9 t (-/7P(~)) 2) is total in 7-/| ~ . Then the following statements hold: J'79(flS) = {Tr'(A); A E/3(7-{) s.t. A T / C 79(Y2) and f2A E O2(M)}, (1) [. AS(rc'(A)) = f2A, 7r'(A) E Z)(A~) and A~((:D(A~)* N :D(A~)) 2) is total in T/| 7-{. ~79(A~c) = {~r"(A); d E/~(T/) and AY2 r 7-{ | ~ } , (2) [ x~c(rr''(d)) = ~ , rc"(d) E 79(.X5c) CC CC * and A~ (79(A~) A 79(A~c)) is an achieved left. Hilbert algebra in 7-( | 7-(. (3) S:,cc = JTr"(Y2)rc'(f2 -1) = JTr'(Y2-1)Tr"(f2), and so J;,cc = J and

A ~

= ~,,(~)~,(~?-1) = ~'(~-~)~"(t?).

(4) Suppose Y2it79 C Z) for all t E ~ . Then A~ is a quasi-standard generalized vector for zr(Ad). (5) Suppose ~it79 c 79 and X?itYtdY2-it = A/I for all t E]l{. Then )~x? is a standard generalized vector for rr(2t4). P r o o f . (1) Take an arbitrary ~r'(A) E 9 T of G2(Jt4) such that

~(X)T

= ~'(A)a.(~(X))

there exists an element

=

X**S?A

for all 7r(X) E Z?(Xs~). For each X E JV, x E 7-/and sc E 79 we have

(AxI~X*~) = (xtA*~?X*~)

= (Xi*~Ax[~)

=

(XTxlg),

and since the linear span of A/t79 is a core for .(2, it follows that A x E 79(Y2) and s = T E G2(Ad). Conversely, suppose that A E B(T/), AT/ c 79(Y2) and Y2A E ~2(A4). Then we have rr(X)Y2A = X U ? A ) = X t * S ? A = zc'(A)),~(~r(X)) for all ~r(X) E 79(A~), and so zr'(A) E 79(A~) and A~(~r'(A)) = Y2A. Thus we have

2.4 Special cases

{

79(A~) = {Tr'(A); A 9 B ( ~ ) s.t. A ~ C 79(s A~(Tr'(d)) = s

81

and t2A 9 O2(Ad)},

7r'(d) 9 79(A~).

(2.4.24) C C We next show that A~((79(A~) follows from (2.4.24) that

?l.-'(,f~--l~

|

,.Q--I/,]) 9

(~D(.~5)*

n

*

C 2 ) is total in Tt | ~. In fact, it N 79(A~))

/~2(,Kt(,(2--1{

~)()~))2,

|

,t2-1~)) = ~ | ~(~--1~ (2.4.25)

for all 4, r] 9 79 N 79(s Since s N 79(s is dense in 7-{, it. follows C C * from (2.4.24) and (2.4.25) that. A~((79(A~) R 79()~))2) is total in 7-/| 7-{. (2) Take an arbitrary zr1~(A) 9 79(A~CC ). Then there exists an element T of 7-/| 7-/ such that 7ff~(A)A~(Tr'(B)) = 7r~(B)T for all 7r'(B) 9 79(A~?). By (2.4.24) we have A(s

Vzr'(B) 9 79(A,~).

= TB,

(2.4.26)

By (2.4.25) and (2.4.26) we have At2(s

| s

= T(s

| s

for all ~, 7] 9 79 n 79(t2-1), and so A~ = Tt2-1~ for all ~ 9 79 N 79(ff2-1). Since 79 n 79(s is a core for Y2-1, it follows that A4 = Ts for all 9 79(t2-1), and so As = T~ for all ~ 9 79(t2). Hence, As is closable and At2 = T 9 7-I | ~ . Conversely, take an arbitrary A 9 /3(7-/) such that At2 9 7 / | 7-/. Then it follows from (2.4.24) that 7r"(A)A~(zr'(B)) = A(s

=

At2B = 7r'(B)As

for all 7r'(B) 9 79(A~), so that ~r"(d) 9 79(A~c) and ,~c~C(~r"(A)) = As (3) By Lemma 2.4.13 7r"(t2) and ~r'(s -1) are positive self-adjoint operators in 7-/| 7-/ affiliated with the von Neumann algebras ~r'(B(~)) and ~r'(B(7-/)), respectively, and ~r"(s -1) and ~r'(t2-1)~r"(t2) are positive, essentially self-adjoint operators in T / | 7-{ and ~r"(A) 9 79(A~c) * n 79(A~c). Let s

=

fo ~ dE(t),

/j n

tdE(t) be the spectral decomposition of s and put En =

9 N. Then

we

have

lim ~r'(En)~r"(Em)As

= As

lim zr"(s

n~m---* oo

=

lim rr"(Em)~r'(E,~)s

n,m---+ oo

Hence, it follows that As 9 79(Tr"(s163 s which implies by (2) that

and 7r"(s

= t2A. -1) AY2 --

82

2. Standard systems and modular systems Jzr"(f2)Jrt(~Q-1)Jrt'(A)f2

=

JTrtt( f~))7rt(J~-1) A n

= (f2A)* = A ' f 2 = Sx'cc~r'(A)f2. Hence we have Sagc C JTr"(~?)Tr'(f2-1). Conversely, take an arbitrary T E /~(Tr"(f2):r'(f2-1)). Then, T = A n

for some A E D(Tr"(~)). Hence we have

7r"(~2)Tr'(f2-1)T = 7r"(f2)A : f2A c 7-/O ~ ,

and so (f2A)* : A ' J ? r 7-{ | ~ . Hence, rr"(A) E / ) ( A nC C ) * ~ 7)(A~c). Thus we have T = 7r"(A)g? E D(Sxcc) and S x c c T = JTr"(g?)rc'(f2-1)T, and hence Jrc"(f2)Tr'(S? -1) C Sxgc. Thus we have S~cc = J~r"(~)Tr'(S?-l). The statements (4) and (5) follow from (3). This completes the proof. We give examples of standard generalized vectors for O*-algebras on the Schwartz space S(I~). E x a m p l e 2.4.24. Let 8(]~) be the Schwartz space of infinitely differentiable rapidly decreasing functions and {f,~}n=0,1,.-. C 8 ( ~ ) the othonormal basis in the Hilbert space L2(]~) of normalized Hermite functions. Let .4 be the unbounded CCR-algebra, :r0 the Schrhdinger representation of .4 and A/[ an O*-algebra on 8 ( R ) containing lr0(A). Then ~r(M) is a self-adjoint

O*-algebra on 3 | L~( - 8(]1{) | L~(]l{)) in L 2 | U ( -

L2(]~) | L2(]I{]) such

that 7r(fl4)'w = :r'(B(L2(~))) and (Tr(A4)'w)' = :r"(B(L2(~))). In fact, as OG

oo

wen-known, S ( ~ ) = [-] D(Nk), where N = Z ( ~ k=l

+ 1)fn | •

in M, and

n=0

hence it follows that

XT = N-I(NXT)

C (L2| ~ ) B ( L 2 ( ] ~ ) ) = n 2 |

for each X C M and T C S | L 2, which implies G2(AJ) = 8 | L 2. Since 7r0(A) is a self-adjoint O*-algebra on 8(]i{) such that 7r0(A)~w = C I , it follows by L e m m a 2.4.14 that 7r(Ad) is a self-adjoint O*-algebra on 8 | L 2 such that 7r(Ad)'w = 7F(B(L2(~))) and (zc(Ad)~)' = # ' ( B ( L 2 ( ~ ) ) ) . We put /2 = { { ~ } E/2;a,~ > 0, n = 0 , 1 , 2 , - . . } , s+ = {{a~} C 1 2 ; s u p n k a ~ < oo for each k c N}, n oo

r~=0

Then Y2{~} E S | L 2 for each {c~n} E s+. But, for a general {a,~} E 12 S2{a~} is not a vector for ~r(Ad), and so we need to consider the generalized

2.4 Special cases

83

vector ) ~ ( ~ t for 7r(A4). Let {an} C I~_. We have the following results for the standardness of A{~,} : (1) Suppose { y x t * y 2 { ~ . } ; X , Y c : D ( A ~ ( ~ ) t Cl : D ( A ~ , ) ) } is total in L 2 | L 2. Then A~(o,~ is a quasi-standard generalized vector for 7r(A4). (2) Suppose .M D {f,~ | ~ ; n, m C 1%10 {0}}. Then Ag(o,~ is a quasistandard generalized vector for ~(A4). (3) A ~ ( ~ is a standard generalized vector for 7r(s We first show the statement (1). Since t2{~}S c 8(I~), where s is the dense subspace of L2(~) generated by {f,~}n=0,1,..., it follows from Lemma 2.4.16 that s is strongly regular., so that by Lemma 2.4.17 A cc~(o~) is a standard generalized vector for the von Neumann algebra 7r"(B(L2(~))) such that the modular conjugation operator J;~cc equals the involution T ~ L 2 | ---~ L --~ T* a L 2 | ~L and the modular operator A ~ c c

t-2{ a n }

T(t (f2(~})zr --2 tt (f2{~}). 2

equals

Further, since n it(~.) N C Nf2 ~{~}, vt ~ ]~ and S ( ~ )

0<3

N 79(N~)' it follows that f2}t }8(]~) C 8(]~) for vt 9 ]~, which implies k=l

z~c

S | ---~ L C S | ~L for vt 9 ~ . By Theorem 2.4.18 A~to~ is quasi-

~{c,n}

standard. We show the statement (2). Since {~r(fn | C T)(A~(~,.~) t Iq " D ( A ~ } ) and {Tr(Y)~r(Xt*)Y2{~}; :D(),~(,,~)} is total in L 2 | L 2, it follows from (1) standard. The statement (3) follows from (2) and the

~ ) ; n , m ~ i~U {0}} X,Y 9 ~)(.~.~)t ffl that Ag(~,~ is quasiequality:

Let {c~n} c s+. Then the following statements hold: (4) Suppose T~(A4)Y2{~} is dense in L 2 | L 2. Then f2{~} is a quasistandard vector for 7r(A4). (5) Suppose cn~ < ? e - ' ~ , V n E N for some 3' > 0 and/~ > 0. Then f2{~} is a quasi-standard vector for 7r(h4). In particular, for any/3 > 0 f2{~-~} is a standard vector for 7r(Tro(A,)). This proof and more results will be given in Chapter IV. We shall apply the results of this section to physical examples (namely the BCS-Bogolubov model of super conductively, a class of interacting boson models in Fock space and the quantum field theory) in Chapter IV. E. S t a n d a r d s y s t e m s c o n s t r u c t e d b y v o n N e u m a n n a l g e b r a s w i t h standard generalized vector Here we consider the following question: Let Jvlo be a yon Neumann algebra on a Hilbert space 7-{ with a full standard generalized vector Ao. Do there exist a generalized yon Neumann algebra

84

2. Standard systems and modular systems

Ad on a dense subspace 79 of ~ over Ado and a standard generalized vector for Ad such that Ace = ~o 7

Let A x o and Jxo be the modular operator and the modular conjugation operator for the achieved left Hilbert algebra Ao (79(Ao)* Ch79(~o)), respectively, and let T(Ao) be the *-subalgebra of Ado such that Ao(T(Ao)) is the maximal Tomita algebra equivalent to the achieved left Hilbert algebra Ao(79(Ao)* N

T h e o r e m 2.4.25. Let Ado be a yon Neumann algebra on 7t with a full generalized vector ~o. Suppose there exists a dense subspace g of ~ such that (i) M'og c E; it (ii) A~,og C E, Vt C ~ ; s T2 (iii) Ado Ns 7s o ~ { A ~ g ; A c . A , los.t. A g c g a n d A'8C8}; (iv) &o((79(a~)*C379(AoE))2) is total in ~ , where 79(Ag) = {A E 79(&o); As C g and ,ko(A) E 8}; (v) &0(79(Ag)* N 79(Ag)) is dense in the Hilbert space 79(S~o); (vi) {K E 79(&;)* N 79(~;); A;(K), ~;(K*) E 8} is a nondegenerate *subalgebra of Ad~; (vii) {)~;(K); K C 79(X~)* N79(A;) s.t. A;(K), X~(K*) E g} is dense in the Hilbert space 79(S~o). Then, for any {@o }tcR_invarian t subset T of (Adg" ~s (79))\Adg there exist a generalized von Neumann algebra Ad(EJ) in ~ and a standard generalized vector a(e,~) for Ad(e,:r) such that ( A d ( e J ) ) ~ = M'o and at2 ) = a o P r o o f . By (i) [JMoE]s" = M oE,~ N s is an O*-algebra on g and since 9 N 79(Ao~) C AdoE, it follows from (iv) that (2t4oE)'' = Ado. Hence we have by (iii) that g s* = , -AAg~rAABlS* [Mo]b ,o~t,-,oJ c {X ~ s

is affiliated with Mo}.

(2.4.27)

Let 2r be any {ot~~ subset of [Mo~]s" \ AdoE. We denote by Ad(e,~-) the closere of the O*-algebra [3,loe, T] on 8 generated by 3,loe and T, that is, 79(E,~-) =

N

79(X) and AdlE,~-l = {XF~(e.r);X c [3,toE, T]}.

We simply write 79(e,~-) and Ad(E,T) by 79 and 3,4, respectively. By (2.4.27) we have [AdoE, T]~ = 3A~, and by the {at~~}-invariance of T and (ii) we have

so that

2.4 Special cases A/I c {X E s

z3 itXo79 C 79, it

is affiliated with Ado};

v t 9 R;

--it

(2.4.28) (2.4.29)

Vt

AXoAdA~ o = Ad,

85

E •.

(2.4.30)

We next define a generalized vector A(e,~-) for the O*-algebra Ad ~ follows: '79(A(E,=r)) = {X 9 Ad; 3~x 9 79 s.t.

A(e,~)(x) = ~x,

XAC(K) = K~x for VK 9 79(A~) s.t. A'o(K ) 9 g}, x 9 79(~(e,~-)).

We simply write A(E,7 ) by s By (iv) and (vi) A is well-defined and by (2.4.27) and (iv) A is a generalized vector for Ad such that s163 A79(A))2) is total in 7-/. We next. show that {K 9 79(A;)* N 79(X;); s A~(K) = he(K),

9 g} C 79(Ac) * N 79(AC) C 79(s

VK 9 79(~;)* A 79(A~) s.t. A~(K) 9 g.

A 79(A~), (2.4.31)

In fact, it is clear by the dcfinition of A that for any K 9 T)(A~)* N 7?(A~) s.t. A~(K) 9 $, K 9 79(Ac) * N 79(Ac) and AC(K) = A~(K). Take arbitrary K 9 79(Ac) * A79(Ac) and A 9 79(Ao)* A79(Ao). By (v) there exists a sequence {An} in 79(Ao~)*A79(AoE) such that lim ~o(An) Ao(A) and lim Ao(A~) = Ao(A*), ----

n~c~

n~-~oz

and since (79(AoE)*N79(Aoe))[79 C 79(A)t N79(A), we have KAo(A~) = dnXC(K) for Vn 9 N, so that for each K1 9 79(A~)* A 79(A~) we have (KAo(A) IA;(K1)) = limo (KA0 (An)IX~ (K1)) lira (AnAC(K)IA'o(K1)) n---~ oo

lira ( ~ C ( K )

n---~ oo

* ' IAnAo(K1))

~im (XC(K)IKlXo(A:)) (AC(K) IKI.Xo(A*)) (AC(K) IA*A~(K1))

(AAC(K) IA'o(K1)) and similarly

(K*Ao(A)IA'o(K1)) = (AAC(K*)tA~(K1)). Hence we have K E 79(A~)* C379(A~). Thus the statement (2.4.31) holds. By (vi), (vii) and (2.4.31) we have Ac((79(Ac) * C3 79(Ac)) 2) is total in 7-/ and Accc = A~. Therefore we have Acc = Ao, which implies by (2.4.29) and (2.4.30) that A is a standard generalized vector for Ad. This completes the proof.

86

2. Standard systems and modular systems

C o r o l l a r y 2.4.26. Let (Ado, Ao) be in Theorem 2.4.24. For any subspace g of 7-/ satisfying the conditions (i) ~ (vii) in Theorem 2.4.24 there exist a generalized yon Neumann algebra AdE on /gE over Ado and a standard generalized vector he for Ads such that A~c = AoP r o o f . This follows from Theorem 2.4.24 by taking (AdoC~ Ms (/:)))\AdoE as T in Theorem 2.4.24. We denote by S(Ado, Ao) the set of all subspaces 8 of 7-/ satisfying (i) (vii) in Theorem 2.4.24. C o r o l l a r y 2.4.27. Let Ado be a yon Neumann algebra on 7-/with a cyclic and separating vector 40. Suppose there exists a dense subspace s for 7-I such that (i) Ad;g C g; it c g, vt c ~ ; (ii) A~og (iii) 3,4oE~* M s

# Ado~;

(iv) (AdoC) '' = Ado;

(v) 4o c 8. Then 8 E 8(Ado, A~o), and so [o is a standard vector for the closed O*-algebra Ad(a,7-) for any {ct~~

subset T of (~4o~ " ;3 s

\ AdoE.

P r o o f . By (v) we have/:)(A[o)* N/9(,~[o ) = AdoE, which implies by (iv) and (v) that the conditons (iv) and (v) in Theorem 2.4.24 are satisfied. Further, since/7(~o ) = A/I; and A~o(K) = K[o for each K C Ad~, it follows from (i) that the conditions (vi) and (vii) in Theorem 2.4.24 are satisfied. Thus we have 8 c 8(Ado, A~o). This completes the proof. E x a m p l e 2.4.28. Let Ado be a yon N e u m a n n algebra on t-I and Ao a tracial standard generalized vector for Ado. Then/-/~o in Example 2.4.5 belongs to 8(Ado, Ao). In fact, this follows since A~ o = I and/9(Ao~ ~

= ~

,k 0 "

E x a m p l e 2.4.29. Let .Ado be a v o n N e u m a n n algebra on 7-I with a cyclic and separating vector ~o and H a positive self-adjoint operator in 7-/ affiliated with the fixed point algebra ""0AA~~ -- {A E M0; at~~ (A) = A, Vt E ~ } of {e~o} such that 4o E 19~176 Then I:)~(H) E 8 ( M o , A~o). In fact, it is clear that the conditions (i), (ii) and (v) in Corollary 2.4.26 are satisfied. Let H =

/5

tdEH (t) be the spectral resolution of H. Take an arbitrary A c 3/I0.

Since E H ( n ) A E H ( n )

E "h "A '0D ~ 1 7 6

,

Vn

C 1~ and lim E H ( n ) A E H ( n ) = A 72 -'-~ O0

strongly, it follows that the condition (iv) in Corollary 2.4.26 holds. Further, we have H n E g ( m ) E "AAT~(H) "o for Vn, m E 1~ and H n E H ( m ) [ , D ~ ( H ) > m~oo

2.5 Generalized Connes cocycle theorem

87

Hn[~O~(H), strongly*, and hence H ~ [ ~ ( H ) C (Ado ~ ( H ) ~ : - ~ N s176176 \ Ad~~176 0 , and so the condition (iii) in Corollary 2.4.26 holds. Thus we have

~~

c S(Ad0,~o).

E x a m p l e 2.4.30. Let 3//0 be a yon Neumann algebra on 7-/with a cyclic and separating vector ~0.

When can we construct such a positive self-adjoint operator H in 7-[ as in Example 2.4.28? The above question is affiwmative in cases of the following (i) and 5@ ^ A ~ ~ is infinitely dimensional. (i),v,0 (ii) Ado is semifinite and the spectrum Sp(A~o ) of A~o is an infinite set. ^A~176 is infinitely dimensional. Then there is a sequence {E~} Suppose ~v'0 of mutually orthogonal projections in ~.,0 AA~176 such that ]tE~01] < 1 and log ] [ E ~ 0 ] ] - log I]E,~+l~0]] > 1 for n E N. Then it is easily shown that oo H -- ] ~ ( - l o g I]En~olI)E~ is a positive self-adjoint operator in 7-/ affiliated n=l with r^A~176 such t h a t ~0 c Z)~176 Suppose the statement (ii) holds. By 0 Theorem 14.2 in Takesaki [1] there exists a positive self-adjoint operator K in affiliated with Ado such that z~o = K -2 9 K '2 where K ' = J~oKJ~o. Then K is affiliated with ,"0^~~176 . Since Sp(A~o ) is an infinite set, it follows that

Sp(K) is an infinite set, which implies that A d ~ o is infinitely dimensional. Therefore, the question is affirmative by the above argument.

2.5 Generalized Connes cocycle t h e o r e m In this section we generalize the Connes cocycle theorem for von Neumann algebras to generalized yon Neumann algebras. Let A4 be a closed O*-algebra on 79 in 7-/. Let ~ 4 be a four-dimensional Hilbert space with an orthonormal basis {?/~j}~,j= 1,2 and M2 (C) a 2 • 2-matrix algebra generated by the matrices Eij which are defined by E~j?/kz = 8ik?/kl. Identifying 4 = 41 <9?/11 J-~2 <9 ?/21 -}~3 <9 ?/12 -[- ~4 <9 7/22 ~ ~'{ <9 }(~4 with 4 = (41,42, ~3, 44) C ~.[4 ___ ~ (~ ~.~ (~ ~.~ (~ 7-/, Ad | M2(C) is regarded as the matrix algebra on 7:)4 =
Xl l X 1200) / / X21 X22 0 0 0 0 X l l X12 0 0 "~-21 X22

"Xij

C J~

'

Suppose >, and p are generalized vectors for M . We put

.

88

2. Standard systems and modular systems

~)(OA,p) =

oX11x12~ } ]

X = [ L 21 X22

0

. Xll~X21 E ~)(A)

0 X u X 1 2 'X~2, X 2 2 c g ( # ) 0 X21 X22 J

'

/~( X l l ) A(X21)

Then 0~,, is a generalized vector for A4 | M2(C). \Ve consider when 0~,, is a standard generalized vector for A/I | M2(C). W'e here need the notion of semifiniteness of generalized vectors: D e f i n i t i o n 2.5.1. A generalized vetor A for A/~ is said to be semifinite if there exists a net {Us} in D(A) t N :D(A) such that I1~11 -< 1 for each a and {Us} converges strongly to I. L e m m a 2.5.2. Suppose A is a semifinite generalized vector for dVl. Then the following statements are equivalent: (i) A is cyclic. (ii) A((/9(A) t A/9(A)) 2) is total in 1-{. P r o o f . (i) ~ (ii) Take an arbitrary X E/P(A). Since A is semifinite, there exists a net {U~} in/P(A)tA:D(A) such that II~ll -< 1, Vc~ and {Us} converges strongly to I. Then we have v~x ~

~(A)t n ~(~), v~,

l i ~ II:
=

o,

and hence it follows since A is cyclic that A(D(A) t n/9(A)) is dense in ~ . Repeating the same argument, we can show that A((/P(A)* N :D(A)) 2) is total in 7-/. (ii) ~ (i) This is trivial. L e m m a 2.5.3. Let AJ be a closed O*-algebra on /P in 1-/ such that ! A/lw/9 C :D and let A and # are semifinite, cyclic generalized vectors for A/I. Then the following statements hold: (1) A(/9(A) D Z)(#) t) and #(T)(#) ffl T)(A) t) are dense in 7-/. (2) 0~, u is semifinite and cyclic, and so 0~,v((/p(~,,)t A:D(0~,u))2) is total in 1-I4.

2.5 Generalized Connes cocycle theorem

(3) ~:)(0~,~)

:

{ I 1~176 K : //~2 K011i(022 K12 /(21

//~C(Kll) [~(K,2)

/AC(K21)

O~,~,(K) =

,

K E

} " /I~11'K21 C ~)('~c) ' K12, t(22 E "D(p c)

1

89

'

/(22 ]/

~(0~,,~).

(2.5.1)

\ ~(K=) in

(4) Suppose Ac((l)()~c) * n:D(AC)) 2) and #c((39(pc)* n ~(pC)) 2) are total Then 0~,~,((:D(0~#,) t A T)(0x,,) c ) 2 ) is total in 7_[4 and

?-/.

/ A n A12 0 A = [A21 A22 0 All 0 A21

CC

:V(O~,,~,) --

/Oo o

/

0 0 A12 A22

9 A u , A21 E ~()~cc) ( ' A12, A22 E "D(#cc) / '

~CC(Au) CC

Ox,~,(A) =

pC~(A12)

, A E "D(0~,~,,).

(2.5.2)

S~(A22) P r o o f . Since A and p are semifinite, there exist nets {U~} and {Vz} in :D(A)t M :D(A) and ~D(#)t A :D(#) such that [[ff~[[ < 1, va and H~[I -< 1, v/3 and (U~} and (Vz} converge strongly to I, respectively. (1) Take an arbitrary X E :D(A). Then we have V~X E 39(A) N ~9(#)t for each /3 and limzA(VzX ) : lim VzA(X ) : A(X), which implies that A(:D(A) A :D(p)t) is dense in 7-/. Similarly, p(T~(#) A :D(A)t) is dense in 7-l. (2) It is easily shown that

z~(0~,,)t n ~(0~,,) =

and

{

X = / x2~ l 0

2 0

/

O0 .x2~Ez~(X) nz~(.)t,

X l l X12

o x21 x22]

' X12 E ~)(]A)Cl"~(,~)t,

'

x22 E D(,)* n ~(#)

(2.5.3)

is a net in :D(0~,,) t n :D(0~,~) which converges strongly to. I, which implies that 0~,, is semifinite. By (2.5.3) and (1), 0~,, is cyclic. Hence it follows from Lemma 2.5.2 that 0~,,((:D(0~,,) t A :D(0~,,)) 2) is total in 7-I4.

90

2. Standard systems and ,nodular systems

(3) This is easily shown. (4) By the assumption Xc and #c are semifinite and cyclic, and by (2.5.2) it is proved similarly to the proof of (2) that 0cx,€ is semifinite and cyclic. Hence it. follows from Lemma 2.5.2 that. 0~,,((~9(0~,~)* n 79(0~,,)) c 2 ) is total in 7-t4. It is not difficult to show the statement (2.5.3). This completes the proof. P r o p o s i t i o n 2.5.4. Let Ad be a generalized yon Neumann algebra on 79 in 7-{ and let A and p be semifinite generalized vectors for M . The following statements are equivalent. (i) A and p are (full) standard generalized vectors for JM. (ii) Ox,, is a (full) standard generalized vector for 3,l | M2(C). P r o o f . (i) ~ (ii) By (2.5.2) and (2.5.3) we have

/Sxcc 0 0 i ) Socc= [ i 0 S~cc~cc , S, cc~cc 0 ' 0 0 St,cc and so

(

zS~cc

Ao c =

0

A.oC CC

0

o

A

0 I F

0

~o

,

(2.5.4)

zS,cc

Sxcc,cc is the closure of the conjugate linear operator pCC(A) -~ ACC(A*), A 9 D(# cc) A D(ACC)* and S~cc,cc = Jacc~ccZ~cc,cc is the polar decomposition of S~cc,cc, and S,cc~cc, J, ccxcc and Z~,cc~cc are where

operators defined similarly. Since it

t

I

--it

t

p

zS0fc((Adw) | M2(C)) zS0fc = (Adw) N M2(C),

te]~,

it follows from (2.5.4) that Acc

at fit

(All) =

it

--it

Z~cct, ccA11z~cct~cc,

A . A -it ACCZ~12z-attCCACC

=

A it ~ A -it ~-a)~CCttCCZal2X..al2CC ~

z~itt t c c l c c A 2 1 ~A--ai At c c

=

n it A n -it z_attCCZ~21~_aAccttcc

t~c c

cr t

(A22) :

Air ~ A-it. ~..a~ccAcc~t22~_tttccAcc

~

(2.5.5) (2.5.6) (2.s.7) (2.5.8)

for all All, A12, A21, A22 E (AJ~,)' and t E ~ . We now denote by [D#co : DAc~ the Connes cocycle associated with the weight ~ , c c with respect to the weight {~cc (Stratila [1]), that is,

[D#cc : DlCC]t = ~-~cc,-~cc~cc, A it A-it

t

6

~.

(2.5.9)

2.5 Generalized Connes cocycle theorem

91

By (2.5.7) we have [Dp cC : DACC]t =

it A-it ~-Jucc~cc~-~cc E A

, , (flAw) , t E 1~

(2.5.10)

and by (2.5.5) and (2.5.8) Air

A-it

~cc~-~xccucc,

it

it

--it

A , c c A u c c x c c E M~w,

t E R.

(2.5.11)

it

Since A ~ c c D c 79, Aucc79 c 7:) (Vt E ]~) and ,~4~wD c 79, it follows from (2.5.11) that it Air A-it Air q-~ it t Aucc~cc79 : ~-~uccL-~ucc~-~.cc~cc~ C AuccA,lw79C

79,

(2.5.12)

and similarly it

A~,cct, ccD c 79,

(2.5.13)

which implies by (2.5.9), (2.5.10) and (2.5.4) that A it q-~4 [Dp cc : DACClt[79 E M and ~-aoccw C 794 t E ~ . A,p

(2.5.14)

Furthermore, it follows from (2.5.5) ~-- (2.5.8), (2.5.12) and (2.5.13) that Acc

O"t

9

(Xll)

.

= /~iAtccpccXll/~;ctcpcc

Air ~ A-it ACCA12Z-~IzCCACC :

,

A it v A -it z-aACCpCCA12Z..alzCC

Air v A -it A it v A -it IzCCACCA21Z.-IACC ~- Z-.aI~CCA21Z._IACCI~CC tz CC

O'~

/~it

V

A-it

(X22) = z_alzccAccA22g_alzccAc C

for each X l l , X 1 2 , X21 , )(22 E .A/[ and t E ]~, and they belong to A4 since M is a generalized von Neumann algebra. Hence we have arIzCC ( X u )" = [DpCC : DACC]r arA CC"(X-11) [DpCC : DA CC ]~* , Xl 1 E ./~,

and so

t E l~,

(2.5.15)

92

2. Standard systems and modular systems

atxCC(xii) D.),at (Xl/) ~CC(X22) D.XcrxtCC(x21) i . pCC O 0

0 0

~,

0

0

I

. atxCC(xII) atACC(Xi2)D.~

CC ~,CC(X22) crt~ (X21)D~..x a t

D~.Xo"t (Xi2) ACC ~CC(X22) i Dz.xat O (X2i) at ~CC 0 0 0 crtxCC(xii)

,

0 0

O

0

0

O

I

,~cc(Xi2) atxCC(xii) D*~>,~ CC D.xo'{CC(x21) a~ (X22) 9 M

|

M2(C)

2.5.16)

for each X 9 M Q M 2 ( C ) and t 9 ~ , where Dpl =-- [D# cc : DACC]t. Therefore 0A,p is a standard generalized vector for Ad | M2(C). (ii) ~ (i) This is trivial. It is easily shown by (2.5.2) that 01,p is full if and only if A and p are full. This completes the proof. R e m a r k 2.5.5. Suppose A and p are standard, semifinite generalized vectors for AJ. We put

S p i A ( X ) = p(X*), X 9 V(;~) n V(p)*. &p,(x)

= ~ ( x * ) , x e z)(~) n D(,)*.

Then Spl and Sip are closable operators in 7-{ whose closures are denoted by 1 the same SpA and SAp ,respectively. Let Spi -- Jpl A 89 pl and Sip -- JApA~,p be polar decompositions of SpA and S~p, respectively. By Proposition 2.5.4 0A,p is a standard generalized vector for A4 | M2(C), and so by Theorem 2.2.8 S e c c = S0x,,. Therefore, we have

SAcc = Sl,

Spcc = Sp,

SpccAcc = S,A,

Siccpcc = SAp

and so

Hence we have

[D# Cc : DACC]t = A"--% itA -it i t A - i t , t 9 I~. '--~Ip = A "--%i'-'A

(2.5.17)

T h e o r e m 2.5.6. ( G e n e r a l i z e d C o n n e s c o c y c l e t h e o r e m ) Let Ad be a generalized yon Neumann algebra on /:) in /-/. Suppose A and # are full

2.5 Generalized Connes cocycle theorem

93

standard, semifinite generalized vectors for M . Then there uniquely exists a strongly continous map t E R --~ Ut E A4 such that (i) Utt is unitary, t E ]~ ; (ii) Ut+s = Ut~yt~(gs), s, t E ~ ; (iii) crg(X) = Utcrt~(x)utt, X E Ad, t ~ R ; (iv) for each X E 7)(t,) n 7)(A) t and Y E D(A) A 7?(t~) t there exists an element F x , y E A(0, 1) such that F x , y ( t ) = ( t ( U t ~ ( Y ) ) t "k(xt)), F x y ( t + i) = (l~(X) [ t,(U~crg(Yt)) for all t E ~ . P r o o f . We put

u~ = [D~,~ : D ~ %

[~), t E ]t{

Then it follows from Proposition 2.5.4, (2.5.15), (2.5.16) and Theorem 3.1 in Stratila [1] that t E ][{ -~ Ut E M is a strongly continous map satisfying (i) ~ (iv). We show the uniqueness of {Ut}tE~{. Let t E I~ --~ Vt E Ad be a strongly continous map satisfying (i) ~ (iv). We put 6t

(

/x,~ x,~ o o~ [ X2] X22 0 0 0 X I , X12

v,ot x~)

0

~(x~) 0

=

0 x21 X22/

00 /

o Ort~(Xl 1) V; G~ (X12)

0

Vt@(X~l)

c~(X22) /

, X E M | M2(C), t E E . Then {6t } is a strongly continous one-parameter group of .-automorphisms of M| such that 6t(D(0)t AT?(0)) c D(0)t AT?(0) for each t E It~, where 0 -- 0A,., and 0 satisfies the KMS-condition with respect to {6t}. Further, we have dr(X) = W t X W ~ for each X EAd | M2(C) and t E I~, where

A~t

Wt=

( \

89

0/ 9

V~ A

it

. Zi.

,

tEN.

Hence it follows that the one-parameter group {Wt}te~{ of unitary operators satisfies the KMS-condition with respect to ;C ( - the closure of {0(X); X t = X E Z)(0) t AZ)(0)}) in the sense of Definition 3.4 in Rieffel-Van Daele [t] and WtlC C K~ for each t E ]~, so that by Theorem 3.8 in Rieffel-Van Daele [1] Wt = A~ t for each t E ][{. Hence it follows from (2.5.16) that. Vt = [Dp cc : DlCC]~ [D = lit for all t E ]i{. This completes the proof. Let ,k and p be full standard, semifinite generalized vectors for J.4. The map t E ]~ --* lit E ]M, uniquely determined by the above theorem, is called the cocycle associated with # with respect to I, and is denoted by [Dp : DA]. For every standard vector ~o for Ad A~o is full and semifinite, and so the Connes cocycle [D,k,o : Dido ] associated with A~o with respect to t~o is

94

2. Standard systems and modular systems

defined for every standard vectors ~0 and 770. We simply denote it by [Dr/o : D~0], and also denote it by [DWvo : Dw~o] according to the usual Connes cocycle associated with the state coy0 with respect to the state co~o. Suppose standard, semifinite generalized vectors )` and/~ are not necessarily full, then we put [D# : DA]t = [D#~ : D)`~]t, t 9 1~. Then t --+ [ D # : DA]t is a strongly continuous map satisfying the conditions (i) ~-- (iv) in Theorem 2.5.6 and it is called the cocycle associated with p with respect to )`. This equals the Connes cocycle [Dp cc : DACc] associated with #cc with respect to )`cc. L e m m a 2.5.7. Let )`, Pl and #2 be full standard, semifinite generalized vectors for Ad. Suppose [D#I : D)`]t = [D#2 : D)`]t for all t 9 ]~. Then Pl = # 2 P r o o f . By Corollary 3.6 in Stratila [1] we have p~c = p~, and so p~ = #~. Take an arbitrary X 9 D(pl). By there exists a sequence { X , } in :D(#~~) -- :D(#~~) such that n -l- -i+mOOX n ~ = X~ for each ~ 9 2) and lim #~(X,~) = lim # ~ ( X n ) = ttl(X). Hence we have ~'t - - + OO

~1,--'+ OO

K p l ( X ) = lim K#2C C (X,~) = lim Xn#~(K) n - - - + OO

?'~--4 ~

= X~(K)

for all K 9 :D(#~)* • 19(#~), which implies by the fullness of #2 that Pl C #2. Similarly we can show #2 c Pl. We define the notion of relative modular pair (~1, "~2) of semifinite, modular generalized vectors A1 and ),2 to apply the generalized Connes cocycle theorem to more examples. D e f i n i t i o n 2.5.8. Let Ad be a closed O*-algebra on :D in 7-/ such that M'w/) C :D. A pair ()~1, A2) of semifinite generalized vectors for M is said to be relative modularif the conditions (S)2 and (S)~ in Definition 2.3.1 and the following condition (RM) hold: (RM) There exists a dense subspace $ of :D[t~] such that (RM)I)`i(~)(~i) t D T)()`i) ) c S, i = 1, 2; (RM)2 {),C(KIK2); K1, K2 e l)(Ac) * N/:)()`c) s.t. )`C(K1), )`C(K~'), )`C(K2), AC(K~) C S} is total in the Hilbert space :D(S~cc), * i -- 1, 2; (RM)3 AdS c s (RM)4 c s and

c s, vt 9

Similarly to Section 2.3 we can show the following

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

95

L e m m a 2.5.9. Let A4 be a closed O*-algebra on 2) in ~ such that ~4'w29 C 29, and A1 and A2 semifinite generalized vectors for M . Suppose (A1, A2) is relative modular. We denote by 29M the subspace of 29 generated by [ J c, where ~- is the set of all subspaces ~ of 29 satisfying the $6J:

conditions (RM)I ~ (RM)4 in Definition 2.5.8. The following statements hold: M Hc is a generalized von (1) A4w29(;~l,x2)/ M C 29M(.h,A=), and so (A4 [29(.h,;~2))w i !. Neumann algebra on 29~,,x2) in H over (AJw) (2) kl and A2 are modular generalized vectors for ~.4 such that 29~,;~2) C D AMt N 29M As " (3) (z~I)RS ~ (/~l)'Dgl,;~2) and (A2)Rs - (A2)v~,~2, are full standard, semifinite generalized vectors for the generalized yon Neumann algebra ( d ~ I"/'~M

/~(,M,,M) ~" Jwc"

By Lemma 2.5.9 and Theorem 2.5.6 we have the following C o r o l l a r y 2.5.10. Suppose a pair (s tors is relative modular. Then

A2) of semifinite generalized vec-

[D(A2)Rs : D(A1)Rs]t29~l,x2) C D (M,A2), M ACC

at

ACC

,

(X)~ = [D(A2)Rs : D(A1)Rs]tat 1 (X)[D(A2)Rs : D(A1)Rs]t~

for each X E A/t, ( c 29~1,~) and t E l~. As seen above, for any relative modular pair (A1, A2) of semifinite generalized vectors we can apply all results obtained for the full standard, semifinite generalized vectors (A1)Rs and (A2)Rs for the generalized von Neumann algebra (A/l[29(~,a=))wc M ,, to it.

2.6 Generalized theorem

Pedersen

and

Takesaki

Radon-Nikodym

In this section we construct the standard, semifinite generalized vector ~A associated with a given full standard, semifinite generalized vector A and a given positive self-adjoint operator A affiliated with the centralizer of A, and consider when a full standard, semifinite generalized vector # is represented as the full extension of such a AA. Let Ad be a generalized yon Neumann algebra on T) in a Hilbert space and A a standard generalized vector for A/I. We put

M ~ ={AcM;AA~

it t Dz~A ,

Vtc]I{},

M crb ~ = A d b n A & "~.

96

2. Standard systems and modular systems

Then .Ad~ and 3/1~ ~ are O*-subalgebras of .A4. L e m m a 2.6.1. Let A be a full standard generalized vector for Ad. Then the following statements hold. (1) Suppose A r Adb such that z 3 ~ A t B ~ 89 is bounded. Then X A E 1

1

:D(A)tDT~(A) and )~(XA) = J~zJ~Atz~2~Jx~(X) for each X r 7)(~)tA~(~). (2) Suppose X E 7?()0tNTP(A) and A E ~4 such that X A E Z)(A)tAz)(~). 1

1

Then A(XA) = JxzJ~ At zfi2~ J~A(X). (3) Suppose A E Ad~ ~. Then X A r 7P(A) and ),(XA) = J:,A*J~A(X) for each X r :D(A). P r o o f . (1) Since

A)AtA~ 89is bounded,

it follows that

A t A ( x t) e T)(S~) and S:~AtX(x t) = J x A ~ A t A ~ 8 9 which implies

(XAAC(K) I AC(K1)) -_ (Ac(K) I AtxtAc(tQ))

(),C(K~K) I A(AtX*)) (SI~C(K*K1) I :qmtxt))

= (KJ~A~AtA; 89

Is

for each K, K1 E f)(Ac) * N T)(Ac). Hence we have

XAAC(K) = KJ~z:]~, dtA~; 89J~A(X) 1

for each K E Z)(Ac) * D T)(Ac). Since J~z~,A* 1

1

and A is full, it 1

follows that X A E "D(),) and A(XA) = Jx A~, At A , ~ J~A(X). (2) This follows from

(A(XA) J AC(K*K~)) = (XAAC(K) I AC(K~)) = (AC(K;K) JAtA(Xt)) = (S~,AC(K*tfl) IAtA(xt)) = (AkAtA(xt) IAC(K*K1)) = (J~A~AtA; 89

IAC(K*KI))

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

97

for each X E Z)(/~)t N/}(A) and K, K1 E Z)(Ac) * Q %)(tc). (3) We first show

AAC(K) E 79(S~) and S ] A ~ C ( K ) = J x A ~ x ~ c ( K *)

(2.6.1)

for each K E Z)(Ac) * n/:)(Ac). This follows from

(s

a(y) i A C(K)) =

I

= (A~ 89 = (s163 = (J:,AJ),AC(K*)IA(Y)) for each Y E T)(A)t N Z)(A). By (2.6.1) we have

( X A A C ( K ) I A(Y)) = (AAC(K) I A ( X * Y ) ) = (~(YtX) IS~AX~(K)) = ()~(ytx)lJa-~J~&C(K*)) = (A(X) I J : , A J x K * A ( Y ) ) = (KJ),A*J),A(X)IA(Y)) for each K E :D(AC)* n Z)(Ac) and Y E 79(A)t N 79(~), which implies by the fullness of ~ that X A E T)(A) and A ( X A ) = J),A*J),)~(X). T h e o r e m 2.6.2. Let Ad be a generalized von Neumann algebra on T) in 7-/and )~ and p full standard, semifinite generalized vectors for Ad. Then the following statements are equivalent. (i) /9(#) is {atx}-invariant and Ilp(at~(X))][ = []p(X)[[ for all X E D(p). (i)' Z)(A) is {a~}-invariant and II~(a~(x))[I = II~(X)ll for all X E /?(A). (ii) [Dp ' DA]t E Ad ~", Vt E ]~.

(ii)'

[Dp : D/~]t

E

M ~)', Vt

E

]I~.

(iii) {[Dp : D~],}tEII~ is a strongly continous one-parameter group of unitary elements of M . P r o o f . The equivalence of (ii), (ii)' and (iii) follows from Theorem 2.5.6. (i) =~ (ii) We now put Ut = [Dp : DA]t, t E 1~. Take an arbitrary t E and put A = a"_,(Ut). By Theorem 2.5.6 we have

X A = X~r"_t(Ut ) = a"_~.(ag(X)Ut) =

for all X E %)(p)t N :D(p), so that by the assumption (i) that

98

2. Standard systems and modular systems

XA, Y A r ~P(p)f O Z)(#) and (#(X) I ~(z)) = (.(XA) I p(YA)) for all X, Y E /9(.)t n/9(p), and further by Lemma

(2.6.2)

2.6.2, (2)

Ib(X)ll--II.(~(Z))ll--b(u;~(z)u,)ll = Ib(XA)II = IIJ.A 3 A , A -3

G.(X)II

for all X r Z)(.) t N/9(.). Hence, Y~Z~3 A t A - 3 J , is bounded. Furthermore, sinee/9(,)* r-I/9(.) is {~}-invariant and {r }-invariant, it follows from Theorem 2.5.6 that

XU 2

=

U;*~"~( ~ ( X ) ) ~ /9(p) t n/9(p)

for all X C/9(.)* n/9(p) and s E ~ , which implies

x ~ /~ (G)* c/9(.)* n D(.) for all X r

n/9(p) and s E JR. Hence, by (2.6.2) we have

X A t, y A t r

n :D(p) and (#(X) [ p(YA)) = ( p ( X A t) I P(Y))

for all X, Y E /9(#)t N :D(#), which implies by Lemma 2.6.1, (2) that 89 t A -89d~,p(Y)) = (p(X) Ip(YA)) (#(x) I J~,A~,A =

(#(XAt) Ip(Y))

= (J,A~AA-~3J~,a(X) I#(Y)) = (,(X) I ( J , A ~ A A ; 3 J , ) * p ( Y ) ) for each X, Y r

J,A~AtA; which implies A A ,

n/9(#). Hence we have

89J~, = (J~,A~AA; 3 J,)*, c A , A . Therefore it follows that U~ E A/W" for all

tE~. (ii) ~ (i) It follows from Theorem 2.5.6 and Lemma 2.6.1, (3) that

~(x)

:

v ; o f ( x ) v , e z)(,)

and II~(G~(x))I[ = Ib(U;~f(x)u,)l[

= IIY.U;J.~(~f(X))ll = Ib(X)ll

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

99

for each X 9 D(p) and t E ]~. (i)' ~=~ (ii) This is proved similarly to the proof of the equivalence of (i) and (ii). This completes the proof. If the equivalent conditions in Theorem

2.6.2 are satisfied, we say that p

commutes with A. If # commutes with A, then

But, the converse is not necessarily true even in the bounded case (4.15 in Stratila [1]). We next present the canonical construction and the properties of the generalized vector AA associated with a given full standard, semifinite generalized vector A and a given positive self-adjoint operator A affiliated with the centralizer of A. We investigate when a full standard, semifinite generalized vector # for A4 which commutes with ), is represented as (AA)e. Let )~ be a full standard, semifinite generalized vector for A/I and ~4~ ~ the set of all non-singular positive self-adjoint operators A in 7-I satisfying {EA(t);--oo < t < oo}" FT~ c .h//~ ~, where {EA(t)} is the spectral resolution of A. Let A 9 J ~ x and put D(AA) = {X E D(A); A(YX) 9 D(JxAJ~) for all Y E M } , AA(X) = J ~ A J x A ( X ) ,

X 9 D(AA).

Then we have the following Lemma satisfying

2.6.3. AA is a standard, semifinite generalized vector for A/I

a~ A (X) = A2ita;~fX~A -2it [D)~A : D;~]t - [D(,kA)e : DA], --- A2UFD, X E A4, t E ]~. P r o o f . It is clear that AA is a generalized vector for A4. By Lemma 2.6.1, (3) we have

E A ( n ) X E A ( m ) 9 D(A) t A D()~), )~(EA(n)XEA(m)) = J~EA(m)J~EA(n))~(X) for each n , m 9 I~ and X 9 :D()~)t A :D(A), which implies E A ( n ) X E A ( m ) E :D(AA) t N:D()~A). Further, since A is semifinite, it follows that )kA is semifinite. Since

{ E A ( n ) X A - 1 E A ( m ) ; X 9 :D(A)t A D(A), m, n, 9 N} C ~)(AA) t A ~D(AA)

100

2. Standard systems and modular systems

and

..kA(EA(n)XA-1EA(m)) = EA(n)J),EA(m)Jx/k(X) ~

/k(X)

(/T/,~ n

---e 0 0 ) ~

it follows t h a t /~A is cyclic, SO t h a t by L e m m a 2.5.2

AA((79(..kA) t A 79()~A)) 2) is dense in H.

(2.6.3)

We put

h2 = { K 9 79(Ac); AC(K) 9 79(A) N 79(JxA-1J),) and AAC(K) 9 l)}. T h e n we have h2 C 79(A~) and A ~ ( K ) = AAc(K),

V K 9 K.

(2.6.4)

In fact, we have by L e m m a 2.6.1, (3) lim AEA(n),kC(K) = A,kC(K), n ---* o o

lim XAEA(n),,kC(K) =

n---~ oo

=

lim K,,k(XAEA(n))

n~oo

lira K J ~ A E A ( n ) J ~ A ( X )

n---~ oo

= KJ;~AJ;~,~(X) = K),A(X)

for each X E 79(~kA) and K E/C, which implies the s t a t e m e n t (2.6.4) is true. We p u t

Km,~ = J),EA(m)J),KJ),EA(n)J;~ for K 9 79(Ac) * N 79(,k c) and m, n 9 1%I. T h e n we have

KronA(X) = (J),EA(m)J),)K(J),EA(n)J;~))~(X) = (J~E,A(m)J~)K/~(XEA(n)) = (JxEA(m)J;~)XEA(n))~C(K) = X(J~EA(m)J;~)EA(n)AC(K) and

K~n,~,,k(X) = X(J;~EA(n)J;~)EA(m),,kC(K *) for each X C 79(~), so t h a t Kmn C/C A/C*,

),C(Kmn) = (J~F~A(m)J;,)Ea(n);~c(K), ),c(K~n) = ( J ~ E . 4 ( n ) & ) E A ( m ) ; ~ ( K * ) .

(2.6.5)

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

101

Hence we have

C~nKm~ 9 (t,2 N K*) 2, tim )~c(CmnKm~)= m,n--~oc

Cm~IC(Km~)

lira rn,n--*

=

oa

lira m,n--~oo

C~,(J~EA(m)J~,)EA(n)~C(K)

CXC(K)

=

:

:~C(CK),

t c ( ( C ~ n K ~ ) * ) = lc((CK)*)

lim m,m---*oo

for each C, K E :D(tC) * N T)(94

which implies t h a t

i c ( ( ~ N ]C*) 2) is total in the Hilbert space 79(S~,). For each K r

(2.6.6)

A ]C* and n r N we have

K~ : KJ),A-1EA(n)J), E K~ NK:*, Ac(Kn) A-1EA(n)IO(K), :

:,C(Kr : &A-IEA(~)&~,C(K *) and so by (2.6.4) lim

:d(CK,d

:

n---~ (~)

lira C ~ ( K . )

:

.---~ o~

lira C E A ( ~ ) ~ C ( K ) n---~ oo

=

CAC(K)

for each C 9 ~NK:*. Hence it follows from (2.6.6) t h a t i~(()~AK:*) z) is total in 7-/ , which implies by (2.6.4) t h a t A~((7:)(A~)* N :D(A~)) 2) is total in 7-/. For each K 9 1 6 2n )U* we have by (2.6.4) and (2.6.5)

lira ~ ( K m n ) =

m,n---,oo

Hi

m,n~cyo

AF~A(n)&EA(,n)&;,~(K)

= d l C ( K ) = A~(K), lim m,n.--~oa

c 9 )= tA(Kmn

lim

m,n--.-~o~

AEA(m)J~EA(n)J;,IC(K *)

= :d(K*). F u r t h e r m o r e , for each C 9

5 7 ) ( I ~ ) and m, n 9 1%1we put

Cmn = (J~,EA(m)J~,)C(J~,EA(n)J),). T h e n we have

CmnA(X) = : = C*nt(X) =

(J:,EA(m)J;OCtA(XA-IEA(n)) (J~EA(m)J;OXA-1EA(n)ICA(C) X(J),EA(m)J)OA-1EA(n)tcA(C), X(J~EA(n)J~)A-1EA(m)tcA(C*),

(2.6.7)

102

2. Standard systems and modular systems

and so by (2.6.4) and (2.6.5) Cr~n 9 K n K * , lira A ~ ( C m n ) =

lira

(&Ea (m)&)EA(n)iX(C)

=

lim ;~A(Cm~ c 9 ) = ~(C*).

m,n---*oo

Therefore it follows that {A~(Kmn) ; K e l C n ~ * ,

rn, n e N }

is total in the Hilbert space :D(S*),cc). (2.6.8)

For each K 9 K A K:* and m, n 9 l~l we have by (2.6.4) and (2.6.5)

S*),ccA~(Kmn) = AEA(n)J)~EA(m)J)~.XC(K *) = AEA (n) J),EA (m) J),S~,)~C(K) -_ S*),J)~AEA(n)J)~EA(m)AC(K) = S~,J),AEA(n)J),A-1EA(m).~CA(K), and so

,kCA(K) 9 7)(S~,J),AJ)~A -1) and S~,J:,AJ)~A-1)~(K) = ~,),ca c o * ACIK~~A J for each K 9 K M K*. By (2.6.8) we have

S~cc c S~,J~AJ),A -1.

(2.6.9)

S~ C S*~ccJ~,A-1J~,A.

(2.6.10)

Similarly we have

By (2.6.9) and (2.6.10) we have *

1

S)~cac = S~ J)~AJ;~A -1 = J)~ A ~ J),AJ)~A-1.

(2.6.11)

II 0"~ Since A is affiliated with (Adw) , it follows that the two self-adjoint op1

erators z~l ~ and J;~AJ:,A -1 are strongly commuting, that is, the spectral projections of the two self-adjoint operators are mutually commuting, and so

z~ 89 -1 is self-adjoint and it equals J),AJ~A l z ~ 8 9 Hence, it follows from (2.6.11) and the uniqueness of the polar decomposition of S~cc, it follows that

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

J~c = J~and A~c =

103

/ l ~-1-~ J ~ A J ~ A - ~ = J : ~ A J ~ A - ~ / I ~ 8 9

which implies

Z]~cc = J ~ A - 2 i t j ~ A 2 ~ t Z ] ~

and cr#Cff ( X ) = A 2 u a ~ ( X ) A -2u

for X C M and t E ~ . Hence it. follows from L e m m a 2.6.1,(3) that ~cc (7t A (~)()~A)t n ~:)(/~A)) C ~ ) ( ~ A ) ~ n ~)(-~A),

t ff R .

Therefore hA is a s t a n d a r d generalized vector for A// . Further, it follows from T h e o r e m 2.5.6 that [D)~A : D.k]t ~ [D()~A)e : D)qt ---- A 2it [i/:) for t E ~ . This completes the proof. Let A and p be full standard, semifinite generalized vectors for M . Suppose # commutes with A. Then it follows from Theorem 2.6.2 that {[Dp : D)k]t}tER is a strongly continuous one-parameter group of unitary ! tO-'k operators in (Mw) , and so by the Stone theorem there exists a unique non-singular positive self-adjoint operator A~,~ affiliated with (M~w) ' ~ such t h a t [Dp; DA], = ~.~,~,a2uFT) for all t E R. By L e m m a 2.5.7, 2.6.3 we have the following generalized Pedersen-Takesaki Radon Nikodym theorem : T h e o r e m 2.6.4. Let M be a generalized yon Neumann algebra on ~D in and A and p full standard, semifinite generalized vectors for A//. Suppose A~,u E M g ~. Then p = (AA~,,)e. C o r o l l a r y 2.6.5. Let M be an E W * - a l g e b r a on T) in 7-/and A and p full standard generalized vectors for A//. Then p commutes with A if and only if # = (AA)e for some non-singular positive self-adjoint operator A affiliated I

with (Mw)

Ja"A

9

P r o o f . Since A// is an EW*-algebra o n / 9 in 7-/, it follows that for every full standard generalized vector A for M , {A[Z); A c ~D(Acc)} c T)(A), which implies t h a t A is semifinite. Suppose p commutes with A. Since M is an E W * algebra on :D in 7-/, we have A~,t, c ~4 n , and so p = (AA~,,)e by Theorem 2.6.4. The converse follows from L e m m a 2.6.3. T h e o r e m 2.6.6. Let ~ / be a generalized von Neumann algebra o n / ) in T / a n d A and p full standard, semifinite generalized vectors for f14. Then the following statements are equivalent. (i) # satisfies the KMS-condition with respect to {ate}. (ii) a~ = at~ for each t E ]~. (iii) There exists a non-singular positive self-adjoint operator A affiliated with the center of ( M ~ ) ~ such that p =- (,kA)e.

104

2. Standard systems and modular systems P r o o f . (i) ~ (ii) This follows from Corollary 4.11 in Stratila [1]. (ii) => (i) This is trivial. (ii) ~ (iii) By T h e o r e m 2.5.6 we have

atX(X) = a f ( X ) = A 2it a)'rX~A -2i~ which implies t h a t A - Aa,, is affiliated with the center of ( M ~~) ~, s o t h a t we can show similarly to the proof of L e m m a 2.6.3 t h a t An is a s t a n d a r d , semifinite generalized vector for 2t4 such t h a t [D)~A : DP~]t = A 2it [79 for all t E ]~. Hence it follows from L e m m a 2.5.7 t h a t p = (AA)~. (iii) ~ (ii) Since A is affiliated with the center of (Ad'w)', it follows from o-X L e m m a 2.6.3 t h a t A E 2t4,7 , AA is a standard, semifinite generalized vector for 2M and a ~ ( X ) = ~t _(an)~,~ vJ , = A 2 u @ ( X ) A -2~ = ~ ) ( x ) for all X r 3,l and t E It{. This completes the proof. A generalized von N e u m a n n algebra Ad is said to be spatially semifinite if there exists a standard, semifinite, tracial generalized vector for A/l. P r o p o s i t i o n 2.6.7. Let Ad be a generalized von N e u m a n n algebra on 79 in ~ . T h e following s t a t e m e n t s hold. (1) Suppose Ad is spatially semifinite. Then, for each full s t a n d a r d , semifinite generalized vector A for 3,t there exists a non-singular positive selfadjoint o p e r a t o r A affiliated with (2bfw)'~ such t h a t crt~(X) = A 2 i t X A - 2 { t for all X e 3/I and t E ~ . (2) Conversely suppose there exist a full standard, semifinite generalized vector ,k for M and a non-singular positive self-adjoint o p e r a t o r A E Ad~ ~ such t h a t ~rtX(X) = A 2 i t X A -2it for all X r M and t E R . T h e n M is spatially semifinite. P r o o f . (1) Since 3/I is spatially semifinite, there exists a full s t a n d a r d , semifinite generalized vector # for M such t h a t A u = 1. Hence it follows from L e m m a 2.6.3 and T h e o r e m 2.6.4 t h a t ate(X) = ~*t,,~'t~2it ,,u(y~n-2it~j~u,x = A 2it X A -2it for all X E A d and t E N[. tt,.k

tt,.k

(2) Since A E M no. -k , it follows from L e m m a 2.6.3 t h a t # - ~A is welldefined and a ~ ( X ) = A 2 u u_ t~A~,~ ~lv~-2it = X for all X E M and t E R . Therefore, # is tracial, and so M is spatially semifinite.

Corollary 2.6.8. An E W * - a l g e b r a A4 is spatially semifinite if and only if there exist a full s t a n d a r d generalized vector A for M and a non-singular positive self-adjoint o p e r a t o r A affiliated with (A/ffw)' such t h a t a t ( X ) --A 2 i t X A - 2 i t for all X E M and t E ~ . If this is true, then for any full

2.6 Generalized Pedersen and Takesaki Radon-Nikodym theorem

105

s t a n d a r d generalized vector t/for 3,t there exists a non-singular positive selfadjoint o p e r a t o r A , affiliated with (M'w)' such t h a t ~ ' ( X ) = A2)tXA22it for allXEM andrE]I{. P r o o f . This follows from Corollary 2.6.5 and Proposition 2.6.7. We finally note some results for the case of s t a n d a r d vectors: L e m m a 2.6.9. Let M be a generalized yon N e u m a n n algebra on Z? in and ~0 and r/o s t a n d a r d vectors for A4. Suppose A - Aa~o,x,o E Ad~or(0 . T h e n ~0 ~ ~ ( A ) and 7/o = A~0. P r o o f . For each X E . M and n E l%t we have

Jr AJ4o X E A (n) ~o = X AE, A (n) ~o, and so

EA(n) C :D((Ar Since

Ano =

(()~{o)A)e,

and (A~o)A(EA(n)) = AEA(n)~o.

it follows t h a t

lim EA(n)~ 0 = G0 and lim AEA(n)~o = lim EA(n)rlo = n--+

oG

n~(~3

770 ,

n ~ o o

which implies ~0 E ~ ( A ) and 7/o = A~0. C o r o l l a r y 2 . 6 . 1 0 . Let Ad be a generalized yon N e u m a n n algebra on ~3 in ~ and ~0 and ~/0 s t a n d a r d vectors for M . (1) T h e following s t a t e m e n t s are equivalent: (i) Wvo is {crf~ (i)' COCois {at~~ (ii) [Dwno : DW~o]t E M ~ ~ vt c 1[{. (ii)' [Dw~o : Dwno]t E M ~176 v t r N. (iii) {[Dcovo : Dw~o]t}tE1E is a strongly continous o n e - p a r a m e t e r group of u n i t a r y elements of M . (2) T h e following s t a t e m e n t s are equivalent: (i) cO,jo satisfies the KMS-condition with respect to {~fo }. (ii) a~ ~ = ~ t, o v t EN. (iii) There exists a non-singular positive self-adjoint operator A affiliated with the center of (3/I~) ~ such that q0 = A~0. Proof. (I) This follows from Theorem 2.6.2. (2) This follows from Theorem 2.6.6 and Lemma 2.6.9. For the case of EW*-algebras 2.6.5 and Lemma 2.6.9:

we have the following result by Corollary

106

2. Standard systems and modular systems

C o r o l l a r y 2.6.11. Let M be an EW*-algebra on T) in 7-{ and G0 and r]0 standard vectors for 31t. Then A,o commutes with A~o if and only if r]0 = AGo for some non-singular positive self-adjoint operator A affiliated with (Ad'w) '~176.

2.7 Generalized

standard

systems

In Section 2.1 ~ 2.6 we have treated with the standard systems and modular systems under the assumption (S)1 : The weak c o m m u t a n t JM~w of an O*-algebra Ad on i9 satisfies always the condition ]VVw:D c ID. In the Wightman quantum field theory in Chapter IV O*-algebras jr4 whose c o m m u t a n t s A/I~ are not even von Neumann algebras have appeared. So, we consider to generalize the notion of standard systems. Let (A4, A,.4) be a triple of a closed O*-algebra ~4 on a dense subspace 7:) in a Hilbert space ~ , a generalized vector A for M and a yon Neumann algebra .4 on 7-/. Suppose (GS) 1 A((T)(A) t D T)(/~)) 2) is total in 7-/;

(GS) Then

..4' c

we put

~P(AA') = {K E A'; 3GK E D(eA,(~4)) s.t.

KA(X) = e.a,(X)GK for all X E T)(A)}, = GK,

K c

where eA,(Ad) is the induced extension of Ad defined in Section 1.4. Then AA' is a generalized vector for the von Neumann algebra .4'. Further, suppose (GS)3 AA,((T)(AA,)* A I)(AA,)) 2) is total in T/. Here we put :D(AA) = {A E A;3GA E 7-{ s.t. AAA,(K) = KGA for all K E :D(AA,)} AA(A) = GA,

A C lP(AA).

Then we have the following L e m m a 2.7.1. AA is a generalized vector for the von Neumann algebra ,4 and AA(lP(AA)* N~P(AA)) is a full left Hilbert algebra in ~ equipped with the multiplication A4 (A)Aa (B) = A.4 (AB) and the involution AA (A) --* AA (A*). P r o o f . It is easily shown that AA is a generalized vector for the von

2.7 Generalized standard systems

107

Neumann algebra A. Take any element, X of 79(,\)t n 79(,x). Let eA,(X) =

UleA,(X)l be the polar decomposition of eA,(X), leA,(x)l =

/7

spectral resolution and E~ =

/7

tdE(t) the

dE(t), n 9 N. Since CA, (X) is affiliated

with A, it is shown similarly to the proof of Lemma 2.2.2 that x.

- eA,(X)E~

9 79(~A)* n 79(~A),

~A (X~) -- U E ~ U * : ~ ( X ) ,

~A(x~)

=

E~(xt),

lira s

= ~(X),

lim s

n----* oo

n--*

= s

(2.7.1) (2.7.2) (z7.3) (2.7.4)

o~

Take arbitrary X, Y 9 79(A)t n 79(~). By (2.7.1) ~ (2.7.4) we have

{YmXm}

C

(79()~A)* n 79(/\A)) 2, m, n 9 l~,

lim lim I A ( Y m X . ) = m

-.-~ ~

n ---* o e

lim lim Ym)~A(X,~) m

- - * o o n ----* o o

lim YmA(X)

=

m---* ~

= z:~(x)

= )~(YX). Hence it follows from (GS)I that ~A((79(AA)* A79(AA)) 2) is total in T/, which implies that )~A(79(AA)* n 79(AA)) is a full left Hilbert algebra in 7-/. Let S~ A = J~.~z ~ be the polar decomposition of the involution h A (A) --* AA(A*). By the Tomita fundamental theorem we have

J : ~ A J x 4 = A'

(2.7.5)

A"xA"-" AA-" xA = A, t 9 1~.

(2.7.6)

Further, it follows from (GS)3 that the involution A(X) --* A(Xt), X 9 1

79()~)t n 79(A), is closable and its closure is denoted by Sx. Let Sx = J ~ z ~ be the polar decomposition of S~. By (2.7.1) ,-~ (2.7.4) we have the following L e m m a 2 . 7 . 2 . S~ C S~ A.

D e f i n i t i o n 2.7.3. A triple ( M , A , A ) is said to be a standard system if it satisfies the above conditions (GS)I, (GS)2 and (GS)3 and the following conditions (GS)4, (GS)5 and (GS)6: (GS)4 z~ it ~A7) = 7 9 f o r a l l t E N .

(GS)5

A ~x•3/l" A - ~"

= 3 , t for a l l t E ~ . (GS)6 Z:]~tA(79(A)t O 79()~))A~-: t = 79(A)t n 79(A) for all t E R.

108

2. Standard systems and modular systems

T h e o r e m 2.7.4. Suppose ( M , A,.4) is a standard system. Then the following statements hold: (1) Sx = S,~A. (2) {cr{}tE R is a one-parameter group of *-automorphisms of M , where it

-it

c~tX(X) -- Z]~ XZ~x for X E A4 and t E JR. (3) ,\ satisfies the KMS-condition with respect to {ate}teN. P r o o f . Let X, Y E 79(t) 1 N 79(A). Using t.4 satisfies the KMS-condition with respect to the one-parameter group {@A}tcN of A, where ch~(A) z~it

. A--it

aAA/--3a ~ for A E A and t E N, and (2.7.1) ~ (2.7.4), it is shown similarly to the proof of Theorem 2.2.4 that there exists a function f x , Y in A(0, 1) such that

fx,y(t) = (A(Y~)IA(@~(Y~))), fx,y(t + i) = ( A ( @ ~ ( X ) ) I A ( Y ) ) for all t ~ ]~. Hence it is shown similarly to the proof of Theorem 2.2.4 that S ~ t : z~.~ for all t E R , which implies all of our assertions. This completes the proof. Let (3,t, 4, A) be a triple of a closed O*-algebra Ad on D in 7/, ( E 79 and a yon Neumann algebra A on 7/. Suppose A4( is dense in 7 / a n d A' c M~w . Then, since 79(A() = Ad and A((X) = X ( for all X E Ad, it follows that 79((A().4,) = .4' and (A~)A,(K) = K ( for all K E A'. Hence, the conditions (GS)3,(GS)4 and (GS)5 in Definition 2.7.3 become as follows: (GS)3 A ' ( is dense in 7/; (GS)4 A A it( D C 7:) for all t E ]~, where AA~ is the modular operator of the full left Hilbert Mgebra A~ in 7/ equipped with the multiplication (A~)(B~) = AB~ and the involution A~ --+ A*~; Air A-it (GS)5 ,--%4~M~--%4~ = A,1 for all t E ]t{ and (GS)0 holds always. Hence we have the following L e m m a 2.7.5. Let (A/I, 4,-4) be a triple of a closed O*-algebra 2t4 on 79 in 7-/, ~ E 79 and a yon Neumann algebra A on 7-/. Then (Ad, A~, .A) is a standard system if and only if the following statements (GS)I ~ (GS)5 hold: (GS) 1 A4~ is dense in 7-/;

(CS)2 A' c M'w; (GS)3 A'~ is dense in 7-(; (GS)4 z ~itm j ? = 79 for all t C ]I{; Air

AAA-it

(GS)5 "--
.a~ = A4 for all t E ]i{.

2.7 Generalized standard systems

109

D e f i n i t i o n 2.7.6. A triple (Ad, ~,,4) of a closed O*-algebra M on 79 in 7-t, ~ E 79 and a yon Neumann algebra .4 on ~ is said to be a standard system if the conditions (GS)~ ~ (GS)5 of Lernma 2.7.5 hold. To apply the unbounded Tomita-Takesaki theory to more exmnples, we weaken the conditions (GS)4 and (GS)s in Lemma 2.7.5 and define the notion of modular systems. D e f i n i t i o n 2.7.7. A triple ( M , 4, A) of a closed O*-algebra M on 79 in 7-/, ~ E 79 and a v o n Neumann algebra A on 7-/is said to be a modular system if the condition (GS)I, (GS)2 and (GS)3 in Lemma 2.7.5 and the following condition (GM) holds: (GM) There exists a subspace s of 79(cA, (A/I)) such that

(GM)I ~ c 8 (GM)2 eA,(.hd)g = g; it (GM)a AA~g = g for all t E It{. Let ( M , ~ , A ) be a modular system. We put 79M = U g' where F is the set of all subspaces g of 79(eA,(M)) satisfying (GM)I, (GM)2 and (GM)3 in Definition 2.7.7. Then 79M is a subspace of :/:)(cA' ( M ) ) containing M~, and so it is dense in 7-/. It is easily shown that the linear span of A'79M belongs to F . Since ~[:)M is maximum in F , we have A'79M = :DM, which implies that N(M,r

= {X E s

is affiliated with A}

is an O*-algebra on ~)M in ~ . Since 79M =

A 79(X) E $-, it follows XEU(~,~,A) from the maximum of 79M that U(AJ, ~, A) is closed. Hence U(Ad, ~, A) is a generalized yon Neumann algebra on ~)M over A and it is called a left generalized yon Neumann algebra of the modular system (Ad, 4, A). We have the following T h e o r e m 2.7.8. Suppose (A/t, ~, A) is a modular system. Then a system (N(~A, ~, A), ~, A) is standard. We may define the notion of modular systems (Ad, A, A) for generalized vectors A, but we omit it in this note. Notes It is well known that the Tomita-Takesaki theory plays an important rule for the study of yon Neumann algebras and for the applications of quantum physics (Brat.tell-Robinson [1,2]). Inoue [4, 5, 9, 10, 14] tried to develop the Tomita-Takesaki theory in O*-algebras, mainly in case of O*-algebras with cyclic and separating vector. To treat with such a study more systematically,

110

2. Standard systems and modular systems

Inoue-Karwowski [1] defined the notion of generalized vectors for O*-algebras which is a generalization of cyclic vectors, and using it they have developed such a study in (Inoue[15, 17, 18, 19], Antoine-Inoue-Ogi-Trapani [1] and Inoue-Karwowski [1]). Here we have introduced these studies. The works in Section 2.1, 2.2, 2.3 are due to Inoue-Karwowski [1]. The works A (resp. B and C, D) in Section 2.4 are due to Inoue-Karwowski [1] (resp. Inoue [19], Antoine-Inoue-Ogi-Trapani [1]). The results in Section 2.4, E are generalizations of those obtained in Inoue [9] for yon Neumann algebras with cyclic and separating vector. The works in Section 2.5, 2.6 are due to Inoue [17]. The standard system (Ad,~,A) consisting of an O*-algebra, a cyclic vector ~ and a v o n Neumann algebra `4 in Section 2.7 was introdcued in Inoue [14], and the general standard system (A4, A, .4) has been introduced here. A Tomita-Takesaki theory in partial O*-algebras have been studied in Antoine-Inoue-Ogi [1], Ekhaguere [1] and Inoue [13]

3. Standard w e i g h t s on O*-algebras

Weights on O*-algebras (that is, linear functionals that take positive, but not necessarily finite valued) appear naturally in the studies of the unbounded Tomita-Takesaki theory and the quantum physics. Thus it is significant to study weights on O*-algebras for the structure of O*-algebras and the physical applications. Further, the weights on O*-algebras occasion some pathological phenomena which don't occur for weights on C*- and W*-algebras. From this viewpoint we should study systematically weights on O*-algebr~. In Section 3.1 we define quasi-weights and weights on O*-algebras and give the fundamental examples. Let A4 be a closed O*-algebra on a dense subspace ~Din a Hilbert space 7-/. The algebraic positive cone P(2~I) and the operational positive cone A/l+ are defined and the corresponding weights are defined. The phenomenon arises for the GNS-eonstruction of ~ which is important for such a study : r ~ - {X E A/l; ~(X*X) < oo} is a left ideal of Ad in the bounded case, but it is not necessarily a left ideal of A/I. For example, the condition { ( I ) < oo doesn't necessarily imply { ( X t X ) < c~ for all X E A~I. So, using the left ideal ~ _= {X E A4 ; ~((AX)t(AX)) < o o f o r a l l A E ~/I} of A/I, we construct the GNS-representation 7r~ and the vector representation ~ on the similar method to positive linear functionals, that is, 7r~ is a *-homomorphism of A/I onto the O*-algebra ~ ( A / I ) on the dense subspace ~D(Tr~) in the Hilbert space ~ , and ~ is a linear map of .91~ into D(7~) satisfying )~(AX) = 7r~(A))~(X) for each A E ~4 and X E r However, there are non-zero weights ~ such that r ~ has many elements but r = {0} (Example 3.6.2) and so the GNS-construction for such a weight is meaningless. We don't treat with such a weight. We give two important examples of weights. For any ~ E ~D the positive linear functional w~ is defined by ~ ( X ) = (X~]~), X E M , but if ~ E ~-/\ ~D then the definition of the above a~ is impossible. We regard w~ as the map A --~ (At*~[~) of the positive cone P(cJI~) generated by the left ideal 9 I ~ --- {X E A/I;~ E ~D(X**) and Xt*~ E ~ } of ~ l into I~+ satisfying w~(A + B) = w~(A) + w~(B) and w~(aA) = aa~(A) for all A , B E P(91,~) and a >__ 0. So, we need to study such a map (called quasi-weight) which is strictly weaker than the notion of weights. We give another important (quasi-)weight constructed from a net {f~} of positive linear functionals on Ad. It is natural to consider whether supf~ is a (quasi-)weight on P ( ~ I ) . We show that if {f~} has a

112

3. Standard weights on O*-algebras

certain net property for 7)(Ad) (resp. 7)(~

then supf~ is a weight (resp.

a quasi-weight) on P(A4). In Section 3.2 we define and study the notions of regularity and singularity for (quasi-)weights ~ on 7)(M), and give the decomposition theorem of ~ into the regular part ~ and the singular part Ps. A quasi-weight %zon 7~(,~4) is said to be regular if ~ = supf~ on P(cY~)

c~

for some net {f~} of positive linear functionals on A/I, and %a is said to be singular if there doesn't exist any positive linear functional f on M such that f ( X t X ) <_ %a(XtX) for all X C r162 and f r 0 on P(cY~). Using the trio-commutant T(~)~ for ~, we characterize the regularity and the singularity of ~ and show that ~ is decomposed into the sum of the regular part ~ of %a and the singularity part %os of %c. In Section 3.3 we define and study an important class in regular (quasi-)weights which is possible to develop the Tomita-Takesaki theory in O*-algebras. Let %c be a faithful (quasi-)weight on 7)(M) such that. 7r~(A4)'wZ)(Tr~) C Z)(Tr~). Then, the map A~ : Try(X) --~ A~(X), X E r is a generalized vector for the O*-algebra ~r~(M), that is, it is a linear map of the left ideal z)(A~) - 7r~(fY~) into ~D(Tr~) satisfying A~(:r~(A):r~(X)) = 7c~(A)A~(Tc~(X)) for all A C M and X E cYSt. Using (quasi-)standard generalized vectors defined and studied in Section 2.2, we define the notion of (quasi-)standardness of ~ as follows: %z is said to be standard (resp. quasi-standard) if the generalized vector A~ is standard (resp. quasi-standard). And we obtain that if %a is standard, then the modular automorphism group {a~}tci1 ~ of r N fJl~ is defined and ~ is a {a~'}-KMS (quasi-)weight, and if ~ is quasi-standard, then it is extended to a standard quasi-weight ~ on the positive cone P(Tr~,(Ad)'w'r of the generalized von Neumann algebra 7r~(M)~. In Section 3.4 we generalize the Connes cocycle theorem for weights on O*-algebras. In Section 2.5 we generalized the Connes cocycle theorem for standard generalized vectors. As the notion of generalized vectors is spatial, such a generalization is possible to a certain extent, but the notion of (quasi-)weights is purely algebraic and the algebraic properties don't reflect to the topological properties in general, for example, ~r~,(3,1) is net necessarily a generalized von Neumann algebra when M is a generalized von Neumann algebra, and so such a generalization for (quasi-)weights have some difficult problems. We here need the notions of semifiniteness and a-weak continuity of (quasi-)weights. Let p and ~ be faithful, a-weakly continuous and semifinite (quasi-)weights on 7)(34) such that ~r~ and 7r~ are self adjoint. We consider the matrix algebra 34 | M ~ ( ~ ) on D |

{ X = (Xll X12"~ } \X2~ X22J ;X~j 9 M and a faithful, a-weakly continuous semifinite (quasi-)weight 0 on "/'(./%4 | M2 (C)) by

3. Standard weights on O*-algebras

o(xtx)

113

: (/~(X~IXll Jr- XtlX21) Jr- ~d(Xl2Xl2 -[- x t 2 2 2 2 ) ,

x = ( x ~ ) c M | M2(C).

C

C ,

We obtain the result that if A o ((l:)(Ao) A/?(Ao)) 2) is total in 7-/o then 7r~, and ~rw are unitarily equivalent, and then the cocycle [D~ : D~] associated with the quasi-weight ~ on 79(Tr~(A4)~c) with respect to the quasi-weight I! on P(Tr~(Ad)wc) is defined, but rc~,(Ad) is not a generalized yon Neumann algebra in general even if AJ is a generalized von Neumann algebra, and so the eocycle [D~ : D~] for the generalized von Neumann algebra n~(A4)" c does not necessarily induce the cocycle [D~ : DW] associated with the (quasi)weight ~ on 79(Ad) with respect to the (quasi-) weight W on 7~(A4). We show that if 34 is a generalized von Neumann algebra with strongly dense bounded part and W is strongly faithful, then rr~,(34) is spatially isomorphic to 3,t, and so it is a generalized yon Neumann algebra and the cocycle [D~ : DW] for the generalized von Neumann algebra M is well-defined. In Section 3.5 we study the Radon-Nikodym theorem for (quasi-)weights on O*-algebras. Let W be a (quasi-)weight on a closed O*-algebra M on 7? such that 34w/9' c /:). A (quasi-)weight ~ on ;D(34) is said to be w-absolutely continuous if r c 9"[w and the map: ) ~ ( X ) --* )~w(X), X E eff~, is closable from the dense subspace , k ~ , ( ~ ) in a Hilbert space "H~ to the Hilbert space "He, denoted by K~, w the closure of this map. Using the map K~+W,~, we characterize the w-absolute continuity of ~; in particular, in case that W is standard. Suppose W is standard. Then we show that ~ is a w-absolutely continuous, {a~}-KMS (quasi-)weight such that W+~ is standard if and only if there exists a positive self-adjoint operator H in ~ affiliated with (1r~o(34)~)' Mrr~(34)~ such that D(H) D X~,(cJI~) and ~b(XiX) = IIH,X~(X)II 2 for each X E r and if there exists a standard, {at~}-KMS (quasi-) weight r on 7)(34) such that W+~b _< r and r = r then ~ is w-absolutely continuous and {cr~}-invariant if and only if there exists a positive self-adjoint operator H ' in "]-/~ affiliated with % , ( 3 4 ) ~ - {C E rc~(34)~;a~(C) = C, vt E R } such that ke(cJ2~,) C D ( H ' ) and ~b(XIX) = I I H ' ~ ( N ) I I 2 for each X E r In Section 3.6 we give some concrete examples of regular (quasi-) weights, singular (quasi-)weights and standard (quasi-)weights. We first investigate the quasi-weights :v~ on 79(A/1) defined by elements { in the Hilbert space. When is co4 extended to a weight ~ on 79(34) such that r = r ? We show that if A4 is commutative and integrable then the above question is affirmative. Further, we investigate the regularity, the singularity and the standardness of the quasi-weights w{. We next investigate the regualarity and the standardness of quasi-weights defined by density matrices which are important for the quantum physics. We shall apply these results to the physical models, namely the unbounded CCR algebra, a class of interacting boson model in the Fock space and the BCS-Bogolubov model of superconductivity in Chapter IV. And we shall give regular quasi-weighs and standard quasi-weights for the relative models.

114

3. Standard weights on O*-algebras

3.1 Weights

and

quasi-weights

on O*-algebras

In this section we define the notions of quasi-weights and weights on O*algebras and give the fundamental examples. Throughout this section let A4 be a closed O*-algebra on :D in ?-/. For a subspace A / o f Y~4 we put n

7)(N') = { E X~X~; Xk E N" (k = l,2,... ,n), n E N} k=l

and call it the positive cone generated by Af. D e f i n i t i o n 3.1.1. A map ~ of 7)(3//) into 1~+ U {+oo} is said to be a

weight on P(AJ) if (i) ~ ( A + B ) = p ( A ) + ~ ( B ) , A, B E T ) ( M ) ; (ii) ~(~A) = a~(A), A E ~(A4), a > 0, where 0. (+oo) = 0. A map ~ of the positive cone P ( g ~ ) generated by a left ideal r of A4 into ]~+ is said to be a quasi-weight on :P(A4) if it satisfies the above conditions (i) and (ii) for P ( g ~ ) . Let ~ be a quasi-weight on P(A4). We denote by ~ ( ~ ) the subspace of A/I generated by { X t X ; X E gl~}. Since gl~ is a left ideal of A/I, we have ~D(~) = the linear span of {YtX; X, Y E 9"1~}, and so each

Ec~kY:Xk ( ak E C, Xk, Yk E g~) is represented as k

E ~ j Z J Z j for some ~j E C and Zj

~.

C

Then we can define a linear

J

functional on :D(~) by

EkY:Xk k

E ,j (Z Zj) j

and write it by the same F. It is easily shown that

I~(ytx)[ 2 ~ r 1 6 2

X,Y E ~.

(3.1.1)

We put

N~ = ( X e gt~; ~ ( X t X ) _- 0},

,~(X) = X + N~ E ~ / N ~ ,

X E gt~.

Then it follows from (3.1.1) that N~ is a left ideal of g ~ and A~(cJ~) g ~ / N ~ is a pre-Hilbert space with the inner product ( . ~ ( X ) [ Ar

-- ~ ( Y t X ) ,

X,Y E r

3.1 Weights and quasi-weights on O*-Mgebras

115

We denote by 7-/~ the Hilbert space obtained by the completion of the pre0 of M by Hilbert space A~(cYi~). We define a *-representation 7r~

7r~

= A~(AX),

A E .All, X E r

and denote by 7r~ the closure of 7r~ We call the triple (zr~, A~, 7-/~) the GNSconstruction for 7:. Let 7: be a weight on 9o(AA) and put

r162 = { X 9 M ; 7:((AX)t(AX)) < ~

for all A E M } .

Then r is a left ideal of ~/[ and the restriction 7:F P(r of 7: to the positive cone P(cY[~) is a quasi-weight on P(A4) and it is called the quasi-weight on P ( M ) generated by 7: and is denoted by 7:q. We denote by (Tr~,Av,7-/~) the GNS-construction for the quasi-weight pq generated by 7:. We remark that even if 7: ~ 0 the case of 7:q = 0 arises (Example 3.6.2), and so the GNS-construction for such a weight is meaningless. We don't treat with such a weight. We next define a weight by another positive cone A/l+ -- {X E

M; x_>0}. D e f i n i t i o n 3.1.2. A map 7: of A/l+ into ]~+ tA{+oc} is said to be a weight

on jk4+ if (i) 7:(X + Y) = 7:(X) + 7:(Y), X, Y E A/l+ (ii) 7:(aX) = aT:(X), X E A/l+, a _> 0. A map 7: of a hereditary positive subcone/)(7:)+ of A4+ into ~ + is said to be a quasi-weight on A4+ if it satisfies the above conditions (i) and (ii) for /)(7:)+. A positive subcone P of A/I+ is said to be hereditaw if any element X of A/I+ majorized by some element Y of P (that is, X _< Y) belongs to 7~. It is clear that if 7: is a weight on A/t+ then it is a weight on 7~(M). We 9denote by 7:[ P(A/[) the restriction of 7: to P(jk/[). Suppose 7: is a weight on J~4 4- We define the finite part 7:q of 7: by

/)(7:q)+ = {x ~ M+; 7:(x) < oo}, 7:q(~--~~ x k ) = ~ k T : ( x ~ ) , k

xk ~/)(7:q)§ ~ _> 0.

k

Then/)(7:q)+ is a hereditary positive subcone of J~4+ and 7:q is a quasi-weight on A/l+. Suppose 7: is a quasi-weight on Jk4+. We put

r

= { X E .A4 ; ( A X ) t ( A X ) E/)(7:)+ for all A c A4}.

Then r is a left ideal of ~/[ and the restriction of 7: to P ( c J ~ ) is a quasiweight on P ( M ) . In fact, for each X1, X2 E ~ and A E M we have

(X1 + X2)t AtA(X1 + X2) + (X1 - X2)t AtA(X1 - X2) ----2(X{AtAX1 + X~AfAX2) E/)(7:)+,

116

3. Standard weights on 0 -aloebras *

r

and since D(~)+ is a hereditary positive subcone of 3,l+, it follows that (X1 + X2)*AtA(X1 + X2) 9 D(~)+, that. is, X1 + X2 9 r It is clear that aX, AX 9 r for all e 9 C, A 9 M and X 9 r Thus, r is a left ideal of A4. Further, since P(cJI~) c D(g))+, the restriction of ~ to 7)(r is a quasi-weight on 7)(Ad). We denote by ~[ P ( M ) the quasi-weight p on 3A+ regarding it as the quasi-weight on 7)(3/1). The following diagram holds: weight on M+

~ ~(M) weight on P(.M)

quasi-weight on 34+

quasi-weight on 7)(34)

The above equality g)q[ 7)(3//) = (g,[ P(M))q follows from 9"{~F ~(M) = 92~ = 92~ = r162 ~(~) = r

~(~))~"

This means that the GNS-constructions of all these (quasi-)weights coincide. YVe give two kinds of important examples of weights and quasi-weights on ~~ or J~4+. We first give (quasi-)weights defined by vectors. Let ~ E 7-l\D. We put r162 = {X 9 A4; ~ 9 D(X t*) and Xt*~ 9 D},

=

It k~ll,

xk 9

k k Then w~ is a quasi-weight on P(Ad). The following question arises: Is w~ extended to a weight on 7~(Ad)? In general, this question is inalfirmativc, and so this is one of the reasons why we have to consider quasi-weights. In Section 3.6, we shall investigate such quasi-weights w4 in more details. We next give some (quaM-)weights defined by a net of positive linear functionals on A/I. Let {f~ } be a net of positive linear functionals on JkI. We put

supf~ : A E P(Ad)

, supf~(A) c [0, +cx~].

Then it is easily shown that max(sup f,~(XtX), sup f,~(yty))

(3.1.2)

<_sup f~(Xt X + y t y ) o~

<_sup f~(Xt X) + sup f~(YtY) Oe

Oe

(3.1.3)

3.1 Weights and quasi-weights on O*-algebras

117

for all X, Y E 34. We define the finite part of sup f~ by ~s0up

f~ = {X E A4: sup f~(xt x) < oo}.

oe

Since ( x + Y ) t ( x + Y) + ( x - Y ? ( X - r ) = 2 ( x t x + r t y ) 0

for each X, Y E ~

0

p f~, it follows that CJlsup f~ is a subspace of 34. But,,

o~

C~

(supf~)(xtx+yty) r supf~(xtx) + s u p f ~ ( y t y ) in general, and we have t3e

12,

~e

the following result.: L e m m a 3.1.3. Let N" be a subspace of r

0

p f.

The following state-

ments are equivalent. (1) (sup f~) (A + B) = (sup f~) (A) + (sup f~) (B) for all A, B E 7)(Af). (2) For each finite subset { X I , " " { a . } of {a} such that lim ~ - - + (X)

k= 1,2,.--,m.

Ge

m

c*

of ~ there exists a subsequence

f~(xtxk)=(supf~)(XtXk),

P r o o f . (1) ~ (supf~)

, Xm}

(2) Take an arbitrary {X1,... ,Xm} c At. By (3.1.2),

( Z X ~ X k ) < oo, and so there exists a subsequence { a ' } of {a} k=l

such that m

J"

~i2~ f~" (F_,x~x~) k=l

m

: (sup f ~ ) ( ~ ~

x~x~).

k=l

m

Since supn f~" ( X ~ X 1 ) ~ ( s u p

f ~ ) ( Z XtkXk) < oo, there exists a subsequence k=l

{a~} of { a ' } such that

l i m f~,{,(XIX 1) = sup f~-(XI~X1)

~ (/~(X{X1).

n

Since {a'n' } is a subsequence of { ,~}, we have m

m

~f~l i1m7 6~,,. (,__.,~-'X*kXk) = (sup f~) ( Z k=l

Furthermore, since

~#(X~X1).

supf~g(Xt2X2) < oo, there exists a subsequence {c~'} of n

{a~} such that

XkXk),t ~oolimfa,,(X{X1)~ =

k=l

118

cr 3. Standard weights on 0 * -aloebras

m

t

nlirn fc~,,,(E XkXk ) = k:l

m

(sup fa)( E c~

xtxk),

k:l

lim f,~,, (X~Xl) = ~ o ( X { X 1 ), lirn f~,(Xt2X2) = (sup f~)(Xt2X2) =--7:(X~X2). 7t

Repeating this argument, there exists a subsequence {aN} of {a} such that m

m

t

li~rnoof ~ " ( E X~Xk) = (sup f~)(EXkXk), k=l

k=l

k = 1,2,.-. ,m,

(3.1.4)

which implies by the assumption (1) that m

t

m

m

E~(XkXk) = lirno~E f t . (X~Xk) = lirno~f~. (EXIkXk) k=l

k=l

k

1

m

=

(supf~)(E(X~Xk)) c~

k=l

m

= E(sup

f,~)(XtkXk).

(3.1.5)

k:l

qo(X~Xk) <_ (supf~)(X~Xk), k = 1,2,.-. ,m, it follows from t that ~(X~Xk) -- (supf,~)(XkXk), k ---- 1,2,-.. ,m. Therefore, we

Since 0 < (3.1.4)

have by (3.1.3)

f,~.(X~Xk)=(supf~)(xtxk),

lim

k=l,2,...,m.

(2) r (1) Take an arbitrary subset {X1, X2,-.- , Xm} of A/. By the assumption (2) there exists a subsequence {a,~} of {a} such that

f~. (xtxk)

lira

= (sup f,~)(XtkXk),

k= 1,2,---,m.

The statement (1) follows from m

m

m

(supf~)(~-~X~Xk) <_E(sup fi~)(xt xk) : 2LGZfo (x x ) k=l

k=l

k=l m

:

f

t

( XkXk) k--1 m

___(sup fol(Zx k=l

xkl.

3.2 The regularity of quasi-weights and weights

119

When {f~} satisfies the condition of Lemma 3.1.3, (2), we say that {f~} has the net property for 7)(H) and then denote the restriction of the map supf~ to 7)(A/') by Sup f~[ 7)(A/). In particular, when {f~} has the net property for 7)(r176 f~ ), we simply say that {f=} has the net property and then denote the map supf~ by Sup f~. By Lemma 3.1.3 and (3.1.2) we have the following P r o p o s i t i o n 3.1.4. Let {f~} be a net of positive linear functionals on Ad. Suppose {f~} has the net property for 7)(5[), where 5[ is a left ideal of AJ which is contained in 9Is~ f . Then Sup f~ r 7)(Z) is a quasi-weight on 7)(M). Suppose {f~} has the net property. Then Supf~ is a weight on 7)(M). Let {f~} be a net of strongly positive linear functionals on M . A linear functional f on Ad is said to be strongly positive if f ( X ) > 0 for all X E A/t+. We put supf~ : X E M + , sup f~(X) E [0, +cx~], D(sup f~)+ = {X C Ad+;sup f~(X) < oo}. C~

{3t

Then D(supf~)+ is a hereditary positive subeone of 3,t+. Let 7) be a positive subcone of D(supf~)+. When {f~ } satisfies the condition of Lemma 3.1.3,(2) for 7), we say that {f~} has the net property for 7) and then denote the restriction of the map supf~ to 7) by Sup f~ r 7). In particular, when {f~} C~

has the net property for D(supf~)+, we simply say that {f~} has the net

property and then denote the map supf~ by Sup f~. Similarly to the proofs of Lemma 3.1.3 and Proposition 3.1.~I we can show the following result: P r o p o s i t o n 3.1.5. Let {f~} be a net of strongly positive linear functionals on JM and P a hereditary positive subcone of D(sup fa)+. Then {f~} has the net property for 7) if and only if Sup f~ I 7) is a quasi-weight on A/I+. Further, {f~} has the net property if and only if Supf~ is a weight on Ad+.

3.2 The regularity

of quasi-weights

and weights

In this section we define the notions of regularity and singularity of (quasi)weights, and give the decomposition theorem of (quasi-)weights into the regular part and the singular part. Let Ad be a closed O*-algebra on 79 in 7-/.

120

3. Standard weights on O*-algebras

D e f i n i t i o n 3.2.1. A quasi-weight ~ on 7)(2t4) is said to be regular if ~ = Sup f~[ P(r (= supf~ on P(.r by Lemma 3.1.3 ) .for some net {f~} of c~

positive linear functionals on M , and it is said to be singular if there doesn't exist any positive linear functional f on A4 such that f ( X ~ X ) < ~(X~X) for each X ~ r and f ~ 0 on 7)(r A weight ~ on 7)(2k4) is said to be regular if ~ = Sup f ~ ( = supf~ on 7)(A4) by Lemma 3.1.3 ) for some net c~

{f~} of positive linear functionals on M , and ~ is said to be quasi-regular if the quasi-weight ~q on P(AJ) defined by ~ is regular. If there doesn't exist any positive linear functional f on A4 such that f ( X i X ) <_ ~ ( X i X ) for all X ~ AJ and f ~ 0 o n P ( M ) , then ~ is said to be singular. We define trio-commutants T(V:)} and T(~)~c for a quasi-weight ~ which play an important rule for the regularity of p as follows : T(~)~ = { K = ( c , ~ , ~ ) ;

!

c c ~(M)w,~,~

*

c z)(%)

s.t. C)~(X) = 7r~,(X)( and

C*,~(X) = ~r;(X)71 for all X ~ g[~,}, T(~)'c = { K = (C, ~, ~1) c T(~)~; ~, 71 c z ) ( ~ ) } . For K = (C, (, 71) E T(~)} we put ~ ' ( K ) = C,

~ ' ( K ) = ~,

X'.(K) = 71.

We have the following L e m m a 3.2.2. (1) T(~)~ is a *-invariant vector space under the following operations and the involution : K1 + / ( 2 = (C1 + C2, ~1 + 42,771 + 712), c~K = (aC, ct~, ~71), K* = (c*,71,~)

for K1 = (C1, ~1,711),/(2 = (C2, ~2,712) and K = (C, ~, 71) in T(~:)~ and a E C. (2) T(~o)'~ is a *-invariant subspace of T(~o);. In particular, if 7r~,(Ad)'w~(Tr~) c Z)(Tr~), then T(~)'c is a *-algebra under the following multiplication : gl/~

2 z

( C 1 C 2 , C 1 ~ 2 ' C~711)

t and 7r; is a ,-homomorphism for K1 = (C1, ~1,711), K2 = (6"2, ~2,712) E T (V))c, of T(~)'c into the yon Neumann algebra 7r~(A4)'w and ,V is a linear map of T(~)~c into /)(Try) satisfying 7r'(K1),V(K2) = M(K1K2) for all K1,/(2 e

L e m m a 3.2.3. Let W be a quasi-weight on P(Ad). Suppose a linear functional f on r satisfies the following conditions (i) and (ii):

3.2 The regularity of quasi-weights and weights

121

(i) 0 _< f ( X t X ) < ~ ( X t X ) for each X E r (ii) For any A EAd there exists % > 0 such that If(AtX)l 2 <_% ~ ( X t X ) for each X E g~,. Then there exists an element K E T(~)} such that 0 _< 7d(K) < I and f ( X ) = (A~,(X)I~'(K)) for all X E r Conversely, for each K E T(~)~ with 0 <_ rr'(K) _< I we put

f(x) = (~(x)I~'(K)),

x c 9%.

Then f is a linear functional on 92~, satisfying the above (i) and (ii). P r o o f . Suppose f is a linear functional on r satisfying the conditions (i) and (ii). Similarly to the GNS-construction for quasi-weights, we can define the GNS-construction (TrS,AS, ~ i ) for f. By (i) there exists a bounded linear transform C from ~ , to ~ f such that Ct~o(X) = Af(X) for all X E r Further, we have

C*C r rr~,(Ad)'w and f ( Y t X ) = (C*Ct~,(X) I A~(Y))

Vx, Y E r (3.2.1)

It follows from (ii) and the Riesz theorem that there exists an element ~ of D(zr~) such that

f(x) = (~(x)

I~), v x ~ 9I~,

which implies by (3.2.1) that

I~;(x)() = f(X*Y) = (;ko(Y)tC*C),~(X))

(A~(Y)

for all X , Y ~ fits, and so C*C),~(X) = 7r$(X)( for all X e gl~. Hence, K = (C*C,(,() ~ T(~)~, 0 _< rr'(K) < I and f ( X ) = ( ) , v ( X ) I ,V(K)) for all X ~ r We next show the converse. Take an arbitrary K 6 T(~)~ such that 0 <_ 7r'(K) <_ I. Then it is clear that. f is a linear functional on 9l~ and further, since I(X*X)

=

f(A*X)

:

for all X E r (ii).

(A~o(X)Irr~o(X))~'(K)) = (.k~(X)Irr;(A).V(K))

(A~(X)I~'(K)A~(X))

,

and A E Ad, it follows that f satisfies the conditions (i) and

R e m a r k 3.2.4. For K E T(~)~ the linear functional w~,(/~) o 7r~ on 3,l defined by (~'(K)~

9

( %*( X ) A ' (K) I),'(K)),

X C 34

is not necessarily positive in case 7r~ is not a *-representation of 7~4. When K E T(~)' c and 0 <_ 7r'(K) _< I, 7r~,(K ) o ~r~ is a positive linear functional on f14 satisfying

122

3. Standard weights on O*-algebras

But, the above inequality does not hold for all X E jr4 because the equality r % ( X ) M ( K ) = rr'(K)A~,(X) holds for each X C ~II~, but this doesn't hold for X C fl/l \ .r in general. For the regularity and the singularity of quasi-weights we have the following T h e o r e m 3.2.5. Let ~ be a quasi-weight on ;0(2~4). (1) Consider the following statements: (i) There exists a net {K~} in T(~)~c such t h a t 0 < 7r'(K~) < I for each ct and 7ff(K~) --~ I strongly. (ii) ~ = Sup (a~o otto) I P(cJI~) for some net { ~ } in D(Tr~). (iii) ~ is regular. (iv) T h e r e exists a net {K~} in T(~)} such t h a t 0 < 7r'(K~) < I for each a and 7r'(K~) --~ I strongly. T h e n the implications (i) ~ (ii) ~ (iii) ~ (iv) hold. In particular, suppose 7r~, is self-adjoint, then the statements (i) ,-~ (iv) are equivalent. (2) Suppose n~ is self-adjoint. T h e n ~ is singular if and only if there doesn't exist any element K of T(~P)~c such t h a t 7if(K) > 0 and 7r'(K) ~ 0. P r o o f . (1) (i) ~ (ii) We put ~ = M(K~). Since

( ~ o ~)(xV x)

=11

~'(K~)~(X) II2

for each X C r and c~, and rr'(K~) ---, I strongly, it follows t h a t the net {,w4o orr~,} of positive linear functionals on Ad has the net property for P ( 9 ~ , ) and ~ = Sup (w~o o 7r~) [ P(fJI~,). (ii)=~ (iii) This is trivial. (iii) ~ ( i v ) This follows from L e m m a 3.2.3. Suppose ~r~ is self-adjoint. Then, T(~)} = T (7:)c, ' and so the implication (iv) ~ (i) and the statement (2) follow from L e m m a 3.2.3. Similarly we have the following result for the regularity of weights : T h e o r e m 3.2.6. Let go be a weight on 7~(Ad). Consider the following statements. (i) ~ = s u p ( w ~ o 7r~) for some net {~c~} in T)(Tr~). (ii) ~ is regular. (iii) ~ is quasi-regular. (iv) T h e r e exists a net { K a } in T(~)~ such t h a t 0 < n ' ( K ~ ) < I for each a and rrt(K~) ~ I strongly. T h e n the following implications (i) ~ (ii) ~ (iii) ~ ( i v ) h o l d .

3.2 The regularity of quasi-weights and weights Let ~ be a weight on 7)(A/I). It follows groin the definition of the equality

~A.X):~'(K)

:

~'(K):~(X),

(X

9

9l~,

K

123

T(~)~c that

9 T(~)'~)

holds, but it doesn't hold for all X 9 .M. For this reason, even if 7% is selfadjoint, the quasi-regularity of ~ doesn't necessarily imply the regularity of ~. So, we have defined the notions of normality and semifiniteness of ~ and investigated the equivalence of the regularity and the quasi-regularity (refer to Inoue-Ogi [1]) As the decomposition theorem of quasi-weights we have the following T h e o r e m 3.2.7. Suppose ~a is a quasi-weight on 7~(M) such that 7% is self-adjoint. Then ~a is decomposed into = ~ + ~s, where ~v~ is a regular quasi-weight on P(M) and ~v~ is a singular quasi-weight on P(A4) such that ~b. and ~r~8 are self-adjoint. P r o o f . We denote by P'~ the projection from 7-/v onto the closed subspace of 7-/~ generated by 7r (T(~)c)7-I ~. Then, P~ 9 7~(2VI)w and there exists a net {K~} in T(~)~c such that 0 _< 7r'(K,) < P~ for each c~ and 7/(K~) -~ P~ strongly. It is clear that the net {f~ -- w~,(K.) o 7r~} of positive linear functionals on M has the net property for 7~(~Jlv), and so it follows from Lemma 3.1.3 that ~a~ = Sup h [ 7)(9I~) is a regular quasi-weight on P(~/I) such that r = r and !

t

~r(x*x) :11 P~(x)

t

I

II2 for each X 9 r

(3.2.2)

We put Then ~v~ is a quasi-weight on ~(A/I) with c j ~ = r It follows from (3.2.2) that lr~r (resp. 7r~8) is unitarily equivalent to the induced representation (~r~)p5 (resp. (Tr~)i_ph) of 7r~, so that 7r~r and 7r~8 are self-adjoint. We show ~s is singular. Suppose there exists a positive linear functional f on A/I such that f ( X i X ) <__(f~(XtX) for all X C 9l~ and f(XtoXo) :fi 0 for some X0 E gl~. Since ~s _< 7% it follows from Lemma 3.2.3 that there exists an element K of T(W)'c such that 0 < ~d(K), 7d(K) # 0 and I(X) = (A,(X) ] A'(K)) for all X 9 ff[v. Then we have

IOr'(K):b(X) l :,~(Y))l 2 = If(YtX)l 2 _< % II(I - P'w)A:(X)II 2 for all X, Y E r

and so

124

3. Standard weights on O*-algebras '

X

= li'_,noJ(r/(K)A~(Xn) I A~(Y))] _< %, lira I[(I - P':)A:(X,)II z ~---* O0

=0 for all X, Y 9 r

where {X,~} is a sequence in 9"[~ such that limo A~(X~ ) =

!

P~,A~,(X). Hence, ~r'(K) = O, and so f(X~oXo) = O. This is a contradiction. Hence, ~ is singular. We investigate the relation of qu~i-weights and generalized vectors. Let be a quasi-weight on P ( M ) . Suppose rc~(X) --~ ) ~ ( X ) , X 9 ff{~ is a map and then put

A~,ffr~(X)) = ),~,(X),

X 9 9l~.

Then A~ is a generalized vector for the O*-algebra ~r~(Ad), and it is called the generalized vector induced by ~ . Then we have the following result for regularity of ~ and A~,: P r o p o s i t i o n 3.2.8. Let ~ be a quasi-weight on 79(M) such that 7r~, is self-adjoint. The following statements are equivalent: (i) ~ is regular and A~,(cJI~fJI~) is total in ~ , . (ii) A~ is well-defined and it is regular. P r o o f . Suppose A~(cJI~cJI~) is total in 7Y~,. Then it. is e ~ i l y shown that I ) ( A ~ ) = {Tr'(K); K 9 T @ ) ' } and A,(Tr , (K)) = ~ ' ( K ) for each K 9 T (~)~, ' which implies by Theorem 3.2.5 that the statements (i) and (ii) are equivalent. Conversely, we consider when a generalized vector A induces a quasiweight on P(AA). We put ' z ~ ( ~ ) = l)(~)

~:,(~_x~xk) = ~ II~(Xk)tl 2, y~ x~xk 9 z~(~). k

k

k

When A = Ar (~ 9 ~ ) , ~x~ is a quasi-weight on P(A4), and so the generalized vector ),r always induces the quasi-weight qax~ on P(Ad). But, for a general is not necessarily well-defined because E X ~ X k = k 0 ({Xk} C ~(A)) doesn't imply ~ IIX(Xk)ll 2 -- 0. For this problem we have k the following generalized vector ~ ~

~

P r o p o s i t i o n 3.2.9. Let A be a regular generalized vector for Ad. Then is a regular quasi-weight on P(A4) such that (%,~ (A/f), A ~ ) is unitarily

3.3 Standard weights

125

equivalent to (M~ - Ad[TP(A),X), that is, there exists a unitary transform U of ~ onto ~ such that UTP(,~) = 7P(A~,~), U~(X) = A~(Tr~,(X)), VX E :D(A) and ~r~(A) = U(A[~D:,)U*,VA E M . P r o o f . This follows from the equality: 7~

n

= 1~ ~ k=l

~ IlKoa(x~)II 2

k 1 n

= ~

~kll~(Xk)ll 2

k=l n

k 1

for each {Xk} C :D(A) and {ak} C ]~+, where {Ks} is a net in T)(AC)*NZ)(Ac) such that 0 <_ K~ _< I, v(~ and K s --~ I strongly. The ~

in Proposition 3.2.9 is said to be the quasi-weight induced by A.

3.3 Standard weights In this section we define and study an important class in regular (quasi)weights which is possible to develop the Tomita-Takesaki theory in O*algebras. Let fld be a closed O*-algebra on :D in 7-/. We need the notions of faithfulness and semifiniteness of (qu~i-)weights : D e f i n i t i o n 3.3.1. A E 2td implies A = 0, in cJl~ n r such that I, then ~a is said to be

Let ~ be a (quasi-)weight on P(Ad). If p(AtA) = 0, then ~a is said to be faithful. If there exists a net {U~} l] Us ]l_< 1 for each a and {U~} converges strongly to

semifinite.

Let ~ be a faithful semifinite (quasi-)weight on 7)(Ad). Then it is easily shown that %, is a *-isomorphism and the generalized vector A~ for the O*algebra %,(AA) is defined by

A~(~r~(x)) = .x~(x),

x E ~.

Suppose (S)l

~(.A'~)lw~)(Tr~, ) C ~)(71-~),

(S)2 A~((cY~ N r

is total in ~ . C

Then we can define a generalized vector A~ for the von Neumann algebra 7r~,( M ) ~wby

126

(r 3. Standard weights on O * -al~,ebras

c

/

s.t. K A y ( x ) = r ~ ( X ) ~ , c

VX 9 9I~ }

c

A~(K) = ~K, K 9 z~(A~). Further, suppose C C C (S)a A~((7)(A~) n 79(A~)) ) is total in 7-/~. .

2

cc

Then, the generalized vector A~ for the von N e u m a n n algebra (Tr~(3d)~w) ' is defined by i

~9(A~ cc) = { A 9 ( ~ ( M ) w! ) I ., 3 ~A 9 7% s.t. AA~(K) = K(A, VK 9 79(A~) } c

CC

c

CC

I, A~ (A) = ~A, A e I)(A~ ) and

A~ ((T)(A~) n z~(A~)) CC

CC

CC

,

2

) is total in ~ , . Hence, the maps ),~,(X) cc (A) c~) A~(Xt), X 9 r n r and A~ , A cc . ( A ,) , A 9 D(AC%, ,__~, n 7~(A~ are closable in 7-(~ and their closures are denoted by S~o and SA~, re1

spectively. Let S~o = J~Z]~ and SA~r = JA~A~A~r be the polar decompositions of S~ and SA~r respectively. T h e n we see t h a t S~ C SA~C, and by the T o m i t a fundamental theorem J A~~c( 7r~o(A d ). ') J a' ~ = rr~(M)'w and .

/

Z~or

t

.

.

.

~

-it

) ZS~A~r = (rr~(.h/l)~w) ' for all t ~ ~ . But, we d o n ' t know how it

the unitary group {Z~A~},ei R acts on the O*-algebra rRo(A/l ), and so we define a system which has the best properties : D e f i n i t i o n 3.3.2. A faithful semifinite (quasi-)weight ~ on 79(Jt4) is said to be quasi-standard if the above conditions (S)1, (S)2, (S)3 and the following condition (S)4 hold : it (S)4 Z~A~CT)(rr~o)C g)(rr~) for all t E ~ . Further, if it

-it

(S)5 Z~A~= c~( jk4 r ) Z~A cc

rr~(Ad) for all t C ]R,

t h e n q0 is said to be essentially standard, and in addition if

air

,,~t

-it = 7r~(cj~ n r

for all t c 1~,

then ~ is said to be standard. We remark t h a t a faithful semifinite (quasi-)weight qo is standard (resp. essentially standard, quasi-standard) if and only if the generalized vector A~, for rrr induced by ~ is standard (resp. essentially standard, quasistandard). Hence by T h e o r e m 2.2.4 and T h e o r e m 2.2.8 we have the following results for standard (quasi-)weights :

3.3 Standard weights

127

T h e o r e m 3.3.3. Suppose ~ is a faithful semifinite standard (quasi)weight on P ( M ) . Then the following statements hold : (1) S'~o= S'A~, and so d~ = JA~ and z ~ = AAron. (2) There exists a one-parameter group {cry}re R of *-automorphisms of it -it M such that 7~(~t~(X)) = for all X ~ ~4 and t ~ ]~.

A~(x)A

(3) ~ is a {a[}-KMS (quasi-)weight, that is, for any X, Y ~ r ~r there exists an element fx,Y of A(0, 1) such that fx,~(t) = ~ ( Y a [ ( X ) ) and f x y ( t + i ) = ~ ( a [ ( X ) Y ) for all t ~ ]~, where A(0, 1) is the set of all complexvalued functions, bounded and continuous on 0 < Imz < 1 and analytic in the interior. Using Theorem 2.2.8, we can show the following result for essentially standard quasi-weights: T h e o r e m 3.3.4. Suppose ~ is an essentially standard quasi-weight on P(Ad). We put

{

9~o = { x ~ M

; 3~x ~ ~ ( ~ ) s.t. ~ ( x ) A ~ ( K )

= E ,I k

k

tl

= K~x,

VK ~ ~(A~) },

Ex' x k

Then ~ is a standard quasi-weight on P(~4) such that ~ C ~r and ~ = 7r~ and S ~ = SA~ as unitary equivalence. Here ~ C ~ means that ~ r c r and ~ = ~ [ 7 ) ( ~ ) . We next consider quasi-standard (quasi-)weights. Let ~ be a quasistandard (quasi-) weight on P(A/I). We put { z)(A~) = { A c 7r~(M)'w'c ; ~ A e :D(Tr~) c c s.t. AA~(K) = K~A, VK E ~(A~) },

A~(A) = ~A, A ~ D(A~). By Theorem 2.3.2, A~ is a generalized vector for 7r~(A/t)~ c such that A~ c A~ and A~C = A~ C . We simply denote by ~ the quasi-weight ~-

on P ( ~ ( M ) ~ c ) induced by A~. Then we have the following

T h e o r e m 3.3.5. Suppose ~ is a quasi-standard (quasi-)weight on P ( M ) . Then the quasi-weight ~ on 7~(~(M)~c) induced by 7~ is standard, and so it is a {a~}tc~-KMS quasi-weight, where a~t(d) = ~..JA~Az.JA~r Air A-u , A E

~,(M)~rtl

tc ~ .

128

3. Standard weights on O -aloebras *

(r

Conversely we consider when a KMS (quasi-)weight is standard. T h e o r e m 3.3.6. Let {at}teIR be a one-parameter group of *- automorphisms of Ad. Suppose ~F is a {c~t}-KMS (quasi-)weight on 79(3,4) such that ),~((r n r 2) is total in 7-/~. Then the following statements hold: (1) The map A~(X) , k~(Xt), X c r N .cRy, is a closable conjugatelinear operator in "H~,. Let S~, be the closure of the above operator X~(X) ---+ X~(X t) and S~ = J ~ A ~ the polar decomposition of S~.

(2) A~),,o(x) it

= ~(~(x)),

v x E 9I~,

vt E IR.

(3) ~ is standard if and only if the following statements hold: (i) A~ is well-defined. c 2) is total in ~ . (ii) a ~c, ( (~( d ~ , c) , n :D(A~,)) (iii) J ~ A ) ( $ ( A ; ) c ; N D ( A ; ) ) c d ~c~(v(Av~) ~ . n 79(d~cc )). (iv) ( & , A ~ (A)IA ~ (g*)) > 0, v m E D ( A ~ ) 9 n 7;)(A~~ ). P r o o f . We put

vp,~(x) = ~ ( ~ ( x ) ) ,

x e 9l~.

Since ~ is {c~t}-KMS (quasi-)weight on 7)(M), for any X, Y 9 ~ there exists an element f x , g of A(0, 1) such that fx,y(t) = ~(at(X)Y)

and f x , g ( t + i) = qo(Yc~t(X)),

n r vt 9 I~.

\Ve now have lim II u r

- ~ ( x ) II2

t--+0

: }~{

~(~(x)*~(x))

- ~(~(x)*y)

-

~ ( x t ~ ( x ) ) + ~(x*x) }

= lim{ 2qz(XtX) - fx* x ( t ) - f x x t ( t + i)} t--~O

'

'

=0 for each X E r n r which implies that {Utt}tc]R is a strongly continuous one-parameter group of unitary operators on 7-/~,. Let {X~} be any sequence in r Nr such that li+mooA~o(Xn) = 0 and limo A~(Xn*) = ~. For any Y E 9'~ n r

we have

3.4 Generalized Connes cocycle theorem for weights

129

m

lim sup [fx~y(t) - (A~(Y)[Ut~)I n--+ oc t E R =

lira

sup

-

n ---+oc t E ] R

_ < l i r n l I A r ( Y ) II I I A ~ ( X ~ ) - ~ I I z0,

lim sup I f x ~ y ( t + i)l = O, n---~ ~

tE]~

a n d hence there exists an element f of A(0, 1) such t h a t f(t) = (;~(Y)IUt~) and f ( t + i) = 0 for all t E IR. Hence we have f = 0, and so ~ = 0. T h u s the s t a t e m e n t (1) holds. T h e s t a t e m e n t (2) is shown similarly to the proof of T h e o r e m 2.2.4. We show the s t a t e m e n t (3). It. is clear t h a t if q# is s t a n d a r d , then the s t a t e m e n t s (i) ~ (iv) hold. Conversely suppose the s t a t e m e n t s (i) (iv) hold. We put. CC

CC

,

CC

T A v (A) = J~A~ (A ), A E :D(A~)

,

n

~

(A~). CC

T h e n T is a well-defined from (iii) t h a t T is positive and T = J~SA~ = 1

J~JA~A~A~. V~Te put U = J~JA~. T h e n U is a u n i t a r y o p e r a t o r on 7-/~. Since :F* = S ~ ~ J~ and

~ c . n ~(A~)) J~A~(~(A~)

is a core for S ~ ,

it. follows

from (iv) t h a t T is a positive self-adjoint o p e r a t o r in 7-/~, and so U = I a n d J~ = JA~. Hence we have /A~ = / ~ A ~ , which implies by (2) t h a t ~ is s t a n d a r d . This completes the proof.

3.4 Generalized Connes cocycle t h e o r e m for weights In this section we generalize the Connes cocycle t h e o r e m for weights on O*algebras. In 2.5, 2.6 we studied to generalize the Connes cocycle t h e o r e m and the Pedersen-Takesaki R a d o n - N i k o d y m t h e o r e m to generalized von N e u m a n n algebras in case of s t a n d a r d generalized vectors. As the notion of generalized vectors is spatial, such a generalization is possible to a certain extent, but the notion of (quasi-)weights is purely algebraic and not spatial a n d the algebraic properties don't reflect to the topological properties in general (for example, rr~ (A/I) is not necessarily a generalized yon N e u m a n n algebra when J~4 is a generalized yon N e u m a n n algebra), and so such generalizations for (quasi-)weights have some difficult problems. We first need the notions of semifiniteness and a-weak continuity of (quasi-)weights. Let A// be a closed O*-algebra on 19 in 7-/. D e f i n i t i o n 3.4.1. For any X c ~

~x(A) = ~(XtAX),

we p u t

A e A/I.

130

3. Standard weights on O*-algebras

Then g:x is a positive linear functional on Ad. If Cx is or-weakly continuous for each X 9 g'l~, then ~ is said to be a-weakly continuous . L e m m a 3.4.2. Let ~a be a (quasi-)weight on P ( M ) . Then the following statements hold: (1) ~ is a-weakly continuous if and only if ~ x y is a a-weakly continuous linear functional on J~4 for each X, Y 9 .r where

~ x , y ( A ) = ~(Y*AX):

A 9 34.

(2) Suppose ~ is v-weakly continuous on D(~), then ~ is a-weakly continuous. (3) Suppose qo is faithful, a-weakly continuous and semifinite. Then A~ is a semifinite generalized vector for ~r~(M) such that A~,((:D(A~,)* n D ( A ~ ) ) 2) is total in ~ , . P r o o f . (1) This follows since any Px,Y is a linear combination of

{~xk; xk 9 9I~ }. (2) This is almost trivial. (3) Since ~ is semifinite, there exists a net {U~} in r N eff~ such that II g2~ I1< 1, v~ and {Us} converges strongly to I. Take an arbitrary X 9 r Since qox is a a-weakly continuous positive linear functional on the bounded part Adb of 3/l, it follows that ~ x can be extended to a a-weakly continuous positive linear functional qa~ on the yon Neumann algebra Mbb". Hence we h ave

rr~(U~) 9 ~(A~)* n / ) ( A . ) , II ~ A u ~ ) ~ C x )

II2

v~,

= ~x(U*~G)

= ~5~ ( ~ * ~ 7 ) <- Il G II~'~ x" ( ; ) -< II x ~ ( x ) i l l v~ and

II ~ ( G ) ~ ( x )

- ~(x)

112= ~ x ( ( U ~

- I)*(G

- I))

which implies that Ar is semifinite. Further, it follows that rcr (D(A~)* • D(A~)) 2 and

limA~Or~(U~U~X))~,~ = I ~ , ~ ( U ~ ) ~ r = ~(x), = A~Or~(x)),

which implies that

A~((I)(A~)* n/)(A~)) 2) is total

in 7/~.

c~ o,

c

3.4 Generalized C,onnes cocycle theorem for weights

131

Let ~ and ~p be faithful, a-weakly continuous semifinite (qu~i-)weights on P(A4) such that 7% and 7r~ are self-adjoint. Let M2(C) be the 2 x 2-matrix algebra on C and put.

0)

Ell =

0

, E12 =

(00/ 0

, E21 =

(000) 1

0

"

Every element X of A/I | M2(C) is represented as

X= (XnXl2) =Xu|174174174 We put = ~(X{1Xll ~- X2tlX21) -[- ~)(X{2X12 -~- x2t2x22),

o(xtx)

x = (x~j) c M | M 2 ( C )

Then we have the following L e m m a 3.4.3. (1) 0 is a faithful, a-weakly continuous, semifinite (quasi)weight on P(A//| M2(C)) such that 7r0 is self-adjoint and ~f~o = { X ~- ( X ) i j E .A/[ | M2(C); Nil , X21 E ~)'{~ and X12, X22 C r

(2) Aw(~JlwG 9{;) is dense in 7-/~ and Ar162

N ~}l~) is dense in 7-{r

P r o o f . (1) It is easily shown that 0 is a faithful, a-weakly continuous (quasi-)weight on P(A4 | M2(C)) such that 7r0 is self-adjoint and ~0 = {X = (X)i j E M | M2(C);Xll,X21

C

~

and X12,222 E ~ b } .

Let {Us} and {V~} be nets in r Nr and r Nr respectively such that ]1~ ]1< 1, va and II ~ I]< - 1, vfl and {Us} and {V~} converge strongly to I. Considering

0)

0 v~

c~ngI0,

we can show that 0 is semifinite. (2) Take an arbitrary X E r

II ~ ( v ~ x ) - ~,~(x)

v~,9,

VzX E 9l~ n ~Jlto, II~- ~x((V~ I)) We have

-

I)t(v~

-

for v~, and hence it follows from the a-weak continuity of ~ x that Ar ~ll~) is dense in ~ . Similarly, ),r162 N ~ ) is dense in 7-/r

n

By Lemma 3.4.2,(4) and Lemma 3.4.3,(1) we have ko((cjlto N r

2) is total in 7-/o. C

(3.4.11

Hence we can define the generalized vector A o for the yon Neumann algebra C 7ro(AzI| M2(C))~w, and so to decide it we first define the following map A~,~:

132

3. Standard weights on O -aloebras C

s.t. K ~ ( X ) C

= ~r,(Xb,

~ x e 9% },

C

A~,,(K) = rh K ~ ~(A~,,). Then we have the following C

C

L e m m a 3.4.4. A~,~, is a linear ,nap of ~9(A~,,~) into Z)(rcv) satisfying C C ~ C K (i) C K E z)(A~,,,) and A~,,,(CK) = 6A~,,,( ) for each C E 7r,(M) /w C and K E D(A~,r C C C (ii) C K ~ 79(A~o) and A~(CK) = c A ~ , , ( K ) for each C ~ lI(Tr~, ~r~) and C K ~ ~(A~,~). We here put

~1 = A0(91~, | E u ) , ~2 = Xo(N~, | E21), ~ 3 = ~ 0 ( 9 l , | E~2), ~ 4 = ~0(92, | E.)2)

and

U~A~(X)

:

Xo(X | Eu),

X E 92~,

V2),~(x) = ),o(X | ~2,), X E 9"[~, U3A~(X) = Ao(X | El2),

X

u 4 ) , , ( x ) = ~ o ( x | E22),

X E 92,.

E

r

Then {7-/i}i=1,..-,4 is a set of mutually orthogonal closed subspaces of 7-s such that 7-/o = 7-/1 (9 7-12 (9 7-13 (9 7-/4, and U1 and U2 (resp. Ua and U4) can be extended to the isometrics from 7-/~ (resp. ~ ) to ~1 and 7-/2 (resp. 7-/3 and 7-/4), and they are also denoted by Ut and (]2 (resp. U3 and U4). For {' X u X12 ) E Ad | M2(C), 7to(X) is given by the matrix: X = \ X21 X22

(

glTr~(X11)g { U17r~(X12)U ~ 0 0 ) U27r~o(X21)U; U27r~o(X22)U~ 0 0 0 U3~,(X11)g~ g37r~b(X12)g 2 " o 0 U4T,-,4,(X21)U ~ U4~,(X22)U~

We now have the following results for the yon Neumann algebras 7re(A4 | M2(C))~w and Oro(Ad | M2(C))~)' and the generalized vector A0: Lemma 3.4.5. ~o (.',4 | M2(C))'~

=

U1C1U~ 0 U1C2U~ 0 C1 E ~,(.M)w, 0 U2C1U~ 0 U2C2U~ C2 E Ir(~%,r%), U3CaU~ 0 U3C4U~ 0 ; C3 c II(%,,7r~), ' 0

U4C3U~

0

U4C4U 4

C4 ~ 5T,(.~)lw

3.4 Generalized Connes cocycle theorem for weights

133

(,~0(M | M~(C))'w)' 0 O0 (Aij, Bir ~ 91~,~0, U2A ~U~ b~A~U~ 0 U3B11U~ U3B19Ug ; (i,j = 1 , 2 ) 0 U~B~U~ U4B~eU2 /

'

where !

I

t

I

A ~ ( ~ ( ~ ) ~ . ) , B C @~(M)w), ~'~

= /

(A,B)

:

AC = C B

a n d A*C for all C C II(Tr,p, ~r~)

CB*

and

( ulc~u; z)(A~) =

c

0

=

u~C3U~ 0

o U2C~U~

0

uac4v~

U2 C2 U,~

o

o

; ca e ~(A~,~),

C2 E

;),

U4Ca U~

0

U4C4U~

C4 C "D(Ar

'

X~

U3 § u~:,;(c~)

c e Z)(A;).

] /

Ae((D(Ao) C

C

,

In b o u n d e d case nD(A~)) 2) is total in ~ e , but, in u n b o u n d e d case this fact doesn't necessarily hold even if (p and ~b are standard. We have the following result for this problem: T h e o r e m 3.4.6. Let ~ and ~b be faithful, a - w e a k l y continuous, semifinite (quasi-~eights on 7~(M) such t h a t :r~ and :r e are self-adjoint. Suppose c c , c 2 c c 2 A~,((z)(A~,) nl)(A~)) ) is total in ~ and A~((~(A~)* n~(A~)) ) is total in 7-/~. T h e following s t a t e m e n t s are equivalent: (i) 7c~ and :r~ are unitarily equivalent. (ii) ll(:r~,,:r>)*ll(%,,:rr and ]I(:rr ,:rw) are nondegenerate *subalgebras of the yon N e u m a n n algebra :r~, (Ad)'w and we (M)~w, respectively. (iii) is dense in H~, and A~,r162 is dense in H e .

A~,~,(~P(A~,~o))

(iv)

Ao((~(Ao)* n z)(A0)) 2) is total in He.

P r o o f . (i) r (ii) This follows from T h e o r e m 1.8.9. (i) - ~ , (iii) T h e r e exists a unitary t r a n s f o r m W of 7-/~, onto H ~ such t h a t WD(:r~) = D(:rr and :r~,(X) = W * ~ r r for all X e 3/1. T h e n we have

z)(A;,~) = ( w c

[ A;,r Hence,

; c 9 z)(A~,) ),

: WA;(c), C 9 z)(A;).

A~,v(Z)(A~,~) ) is dense in Uv. Similarly, A~,~(z)(A~,~)) is dense in

7-L~. (iii) :==* (iv) By L e m m a 3.4.5 we have C

C

Ao((Z)(Ao)

,

n z)(A0)) 2)

134

3. Standard weights on O*-algebras C1, D 1 9

=

Uec2A~(D4)] 9 | UzC3A~iD~)+ U3c4A~.o(D3)| \ U4c3A~,r + UaC4A~(D4)] [ U2cIA~ ~(D2) +

C2~

7~(A~) n ~(A~),

~(Ar

D 2 , C ~, , D 3, 9 9

c

,

c

c

.

c

(3.4.2) c

,

c

c

,

c

which impliessinceT)(A~) ~T)(A~)and D(A~) {~D(A~)are nondegenerate

~a((~(A0)* n/)(A~)) 2) is total in T/o. (iv) ~ (i) Since A~((D(Ao)* n D(A0)) 2) is total in 7-/0, it. follows t h a t cc A o is well-defined and that

~(A$ c) o(A~C), B~j 9 7)(A~~) (i, j = 1, 2) Aij 9

s.t.

A=

( U~,AllU~ UIAI2U~ 0 0 I U2A21U; U2A22U~ 0 0 0 0 U3BllU~U3B12U4 0 0 U41~21U~U4B22U~

B,,A~,r = C3A~(AI,), B2,A~,w(C3) k~3 ~ A~ ~cA ~ 21)~

A12A~.~(C~) = c2A~ (B12), A12A~,.~(C2) = c2A.~ (B2"~) for VC2 9 D(A~,~)

and VC39 D(A~,~p)

] )

}

u, AC~C(A11) "~

Ao~(A)

u2A~(A2~) ]

A 9 D(A~C).

=

u4Ar (B=) } T h e n we have

Ul~lU; SA~C =

oo u2s21u~ o oo I Ua&2U~ * 0 0 ' 0 0 U4&2U2

where Sij (i,j = 1,2) is a closed operator defined by cc

A~ ( A n ) ,

S= : Ar (B22)

cc

A , (B=),

812 : A,~cc(B12)

, A~cc(A12),

A~(A21)

Ar (B21).

$11 : A~ (All)

cc

S21 : 1

cc

cc

cc

.

,

Let Sij = Jij.A~ be the polar decomposition of S O (i, j -- 1, 2). T h e n we have

3.4 Generalized Conues cocycle theorem for weights

i

U2 A2~ U~ 0 0 U3//~ 12 V~

o

JA~r =

/U1J11U1~ 0

0 0

135

(3.4.3)

0

UnA~ug

0 0

0 0 U2J12U~ 0 U3J21U~ 0 0 0 o UnJ~2UU

I

Then it follows from Lemma 3.4.5 that 0

U1U~

u~u;

0

0 0

o o

(o

C=_J~ c

0

0

0

"~

0 ) j~c

o v s g 4~ v 4 u s~ o 0

UIJ11J12U~

0 0 = U3J21JI1U; 0 U4J22J2~U~ 9 7ro(M | M2(C))'w.

0 0 0

0

o)

UIJ12J22U~

0 0

Hence we have

Oro(x)c

= 7re(X),

VX 9 M | M2(C),

which implies that W - J22J21 is a unitary tranform of 7-(r onto ~ that Try(X) = W * z c ~ ( X ) W ,

such

vX 9 M.

This completes the proof. P r o p o s i t i o n 3.4.7. Let ~ and ~ be faithful, or-weakly continuous, semifinite (quasi4~reights on 7~(M) such taht 7c~ and 7ce are self-adjoint. The following statements are equivalent.: (i) ~ and r are quasi-standard (quasi-)weights which satisfy one of (i) ~-, (iv) in Theorem 3.4.6. (ii) 0 is quasi-standard. P r o o f . (i) ==~ (ii) By Theorem 3.4.6 there exists a unitary transform W of ~ onto 7-(~ such that WT)(~-~) = :D(Tr~) and Try(X) = W % r r for all X E A4. Since W* c lI(Trr it follows that 9A~,~ = { (A, W A W * ) ; A c (7c~(~4)'w)' }, which implies by Lemma 3.4.5 and (3.4.3) that for each Aij c (7c~o(M)tw)t ( i , j = 1,2) A~t A~ ~

(U~A~ ~A~U;

0

0

U2A21U~ U2A22U~ 0 0 0 0 U3WAllW*U~ U3WA12W*U2 0 0 U4WA21W*U~ U4WA22W.U~

)

-~,

Z2tAcc= --0

136

3. Standard weights on O*-Mgebras it

-it

// U 1 Z ~ l l A l l A 1 1 u2A21A All

o

*

o

U 1 UlZ21~tlA12A21'tU ~ 0 U 1 u2A21A22A21 U 2 it O u3A12WAllW ~t 0 U4A22WA21~V

* *

it

*

it

*

~t

0

A i 2 U3 u 3 A 1 2 w A 1 2 -~t

A12

w W

Ua u4A22WA22

) *

--it * A 2 2 U~

*

/~122 U 4

it

*

(~0(M | M2(C))')'. Hence we have by L e m m a 3.4.5 it

~PZ~llAllZ~ll

-it

W

W Ait

A

A-itT~z*

W Ait

~

A-itT~p*

W Ait

a

A -it

z---~21~22z-.~21

,

w

z--allZa12z--a21

=

A it Hz~

T~;*A

-it

(3.4.5)

A it ~rn

I;fz*A

-it

(3.4.6)

~ z...a22vv -~22vv

vv

(3.4.4)

A it Hzn Ixz*A -it Z - a l 2 V V ZlllVV z--al2 ,

z--~22 ,

~ z...a12vv ~ 1 2 v v *

~..a22 ,

it

,

-

L--a21~21~-~11 W ----Z~22WA21W A12*t.

(3.4.7)

-itl~7*A It follows from (3.4.4) and (3.4.5) that A~11 " ~ 1it2 "T~z , ~ 2A~- i t T "z v *~A2i t2 "I~v E 7r~o(2M)~w for all t E N, and hence it

it

z~12T)(Tr~b) :

c

--it

W/~ll(Z~ll

,

W

it

,

/~12W)W

~)(7i'~b)

z~(~)

for all t E ~ . Similary, it A~lZ)(~r~) c Z)(~r~), vt E R.

Hence we have it

Z3Ag~D(Tro) C Z)(~ro), vt E R . Therefore, 0 is quasi-standard. c c , (ii) ==~ (i) Since Ao((~(Ao)

n z~(Ao)) ~) is total in ~ 0 , it follows from (3.4.2) that A~ ((Z~(A~) nT)(A~)) ) and Ar162 nz)(A~)) ) are total c

in ~ ,

and ~ r

c

,

c

2

c

c

.

c

2

it respectively. Furthermore, since Z]A;~7?(rr0 ) C Z)0r0) for all it

it

t ~ ]~, it follows from (3.4.3) that Z~A~:D(~r~) C :D(~r~) and z21~T)(Tcr c :D(~r~) for all t E ]~. Therefore, p and ~/~ are quasi-standard. This completes the proof. T h e o r e m 3.4.8. ( G e n e r a l i z e d C o n n e s c o c y c l e t h e o r e m ) Suppose and r are faithful, a-weakly continuous, semifinite, quasi-standard (quasi)weights on P(Ad) which satisfy one of (i) ~ (iv) in Theorem 3.4.6. Then H there exists a strongly continuous map t C ~ ~ Ut E rc~,(Ad)wc, uniquely determined, such that (i) ~ is unitary, vt E li~ ; A cc

(ii) U~+t = Utat ~ (Us), Vs, t E ~ ; (iii) aA$C(WAWt) = WutoA~C(A)U[W*, VA E 7c~trM'",wc, vt E ~ , where W is a unitary transform of T/~ onto ~V~ such that WZ)(Tr~) = T)(Trr and 7rr = WTc~(X)W t for all X E M ;

3.4 Generalized Connes cocycle theorem for weights (iv) for any A ~ ( w t g l g W ) A 9 2 ~ a n d / 3 E an element. FA,B of A(0, 1) such that

ff~Tn ( w t g l ~ W )

137

there exists

FA,B(t) = ~( AUtc~A;~ (/3)), FA,B(t + i) = -~(atA~ ( W B W t ) W U t A W t ) , for all t E I~, where ~ and ~ are the qu~i-weights induced by ~ and ~, respectively. P r o o f , We put

z)(a~) = { ~ ( x ) ; x c 9l~ }, , A ~ ( ~ ( x ) ) = w*x~(x), Then it is easily shown that

A~

x c fits.

is a generalized vector for ~r~(Ad) such that

/ Z)((A~) c) = { W*KW ; K E 7)(A~) }, c

*

,

c

[ (A~) ( w K W ) = W A~(K),

c

K c T)(A~) ;

Z)((A~) cc) = { W*AW ;cA ~ v(A 7) }, { (A~)cc(W*AW) = w*A~ (A), A ~ 7)(Ar ) ; cc

S(A~)co = W*SA;~W. Hence we have ,

it

for all t E ]~, and so A~ is quasi-standard. By Theorem 3.3.4 A~ and A~ are standard generalized vectors for the generalized von Neumann algebra 7r~(Ad)~,~, and so it. follows from Theorem 2.5.6 that there exists a strongly continuous map t E 1~ ) Ut E 7r~(AA)~c satisfying the conditions (i) (iv) and it is identical with the Connes cocycle IDA; : DA~]t (= Ait ~ 2 ~ uA-it~J associated with A ; with respect to A~. This completes the proof. The map t E ]R --~ Ut E %,(Ad)~c, uniquely determined by the above theorem, is called the cocycle associated with the quasi-wei9ht ~ with respect to the quasi-wei9ht ~, and denoted by {D~ : D~]. It follows from (3.4.6) that the coeycle [D~ : D~]t associated with ~ with respect to ~p equals W[D~: D~]~W*. By (iii) and (iv) in Theorem 3.4.8 we have cc

(iii)' a A* ( ~ ( X ) ) A/t, vt E]R ;

Acc

= WIDe:

__

D~]ta t ~ ( ~ ( X ) ) [ D ~ : D~]'[W*, VX E

138

3. Standard weights on O*-algebras (iv)' for any X c r

Nr

and ]/ E ~r

Nr

there exists an element

Fx,r of A(0, 1) such that. A cc

Fx,y(t) = ~(Tr~(X)[D~ : D~lta t ~ (~r~(Y))), Fx,~(t + i) = ~(ar

: D~]~(~r~(X)),

for all t E ]~. C o r o l l a r y 3.4.9. Suppose ~ and ~, are faithful, a-weakly continuous, semifinite, standard (qu~i-)weights which satisfl" one of (i) ~ (iv) in T h e o r e m 3.4.6 and Try(M) is a generalized von Neumann algebra. T h e n there exists a strongly continuous map t C R ~ [De, : D~]t E A d , uniquely determined, such that (i) [ D e : D~]t is unitary, vt E R ; (ii) [D~:: D~]s+t = [D~b: Dp]tcr[([D~: D~z]~) ; (iii) (7~'(X) = I D a : D~]ta[(X)[D~b : D~]~,* V x E fl~l, vt E ]R ; (iv) for any X E r V/r and Y E r Nr there exists an element Fx,g of A(0, 1) such t h a t

rx,g(t) = p(X[D~b : D ~ ] t a ~ ( Y ) ) , Fx,y(t + i) = O(a[(Y)[Df~ : D~]tX) for all t E It{. This [ D e : D~] is called the cocycle associated with the (quasi-) weight with respect to the (quasi-)weight ~. As seen in Corollary 3.4.9, if Try(M) is a generalized yon Neumann algebra, then the generalized Connes cocycle theorem for weights on O*-algebras becomes the best form. Here we show that if 34 is a generalized von Neum a n n algebra with strongly dense bounded part and ~ is a strongly faithful, a-weakly continuous (qu~i-)weight on 79(3/l), then :r~(fl4) is spatially isomorphic to M , and so it is a generalized yon N e u m a n n algebra. L e m m a 3.4.10. Let 34 be a self-adjoint O*-algebra on ~D in 7-/such t h a t A4~~ = (A4~)' and ~ a a-weakly continuous (quasi-)weight on 79(34). Then there exists a normal *-homomorphism ~ of (M~w) ' onto (%,(M)~)' such t h a t ~ ( A ) = 7c~(A) for all A E Mb. P r o o f . Since ~x can be extended to a a-weakly continuous positive linear functional on 34~' for each X E ~r it follows that.

~ x ( A t A ) -
Mb,

3.4 Generalized Connes cocycle theorem for weights

139

~(Mb) C ~(M)b, II ~r~(A)II_
(3.4.8)

VA 9 3.tb.

We now have the following: If {An} is any uniformly bounded net in ~4b such that Am ~

A E B ( ~ ) weakly (resp. strongly, strongly*),

then {Tr~(A~)} converges weakly (resp.strongly, strongly*) to an element of B(7/~). In fact., for each X, Y c r

(3.4.9)

we have

lim ((7c~(A~) - 7c~(A~)).k~,(X) [ ~ , ( Y ) ) = lira ~x,y(A~ - A~)

0~ and so we put

B()~(X), ~ ( Y ) ) = lim (7c~,(A,~))~,(X) [ ~ ( Y ) )

X, Y E r

OL

By (3.4.8) g is a bounded sesquilinear form on ~,(r

x ~(r and so it can be extended to a bounded sesquilinear form on ~ x ~ , . It hence follows from the Riesz theorem that {Tr~(An)} converges weakly (resp. strongly, l 1 strongly*) to an element of B ( ~ ) . Since M~' = (Adw) , it follows from the Kaplansky density theorem that for each A E (M~w)' there exists a net {A~ } in M b such that II Am 1I---IIAII for all a and A~ , A strongly*, and so we put ~-~(A) = s* - lira 7c~(A~), c~

!

!

A 9 (YMw) .

By (3.4.9) ~ ( A ) is well-defined, i.e., it is independent for taking a net {A, } in 3rib, and ~ is a normal *-homomorphism of (~,t~)' to B ( ~ , ) . Hence, it follows that - -

I / 7r~((Yt4w) ) is a von Neumann algebra.

(3.4.10)

We finally show that

7r~(Mb)"

7r--~((Ad') !)

( ~(Ad)w ) .

(3.4.11)

In fact, take an arbitrary C 9 7r~,(Adb)'. Since X is affiliated with (3/l'w)' = Ad~! for each X 9 Ad, there exists a net {Am} in YMb which converges astrongly* to X. Hence we have lira II ~ ( A ~ ) X ~ ( Y ) - ~ ( X ) : ~ ( Y ) c~

II2

= lim ~y ((A~ - X ) t ( A ~ - X ) ) c~

=0

140

3. Standard weights on O*-algebras lim s

II ~ ( A ~ ) ~ ( Y ) - ~(XV)~,~(Y) II= 0

for each Y E .r162 and so

(C~(X)),~,(Y) I ),~(Z)) = lira ( C ~ ( A s ) , ~ ( Z ) o~

= lim c~

I ),~(Z))

(C)~(Y)I~(A~);~(Z))

= (C)~(z)[~,(xt)~(Z)) for all Y, Z E gl~,. Hence, C ~ rr~(M)~. T h u s we have ~r~,(.Adb)' C %,(M)~w, which implies by (3.4.10) that.

7r~(Mb)" C 7r~((M~w) ') C (Tr~(M)~w) ' C 7r~(Mb) 't. Therefore, the statement (3.4.11) holds. This completes the proof. As shown in L e m m a 3.4.2, if ~ is a faithful, semifinite (quasi-)weight on "P(M), then 7c~ is a *-isomorphism, but we d o n ' t know whether ~ is a *-isomorphism in general. For this we have the following L e m m a 3 . 4 . 1 1 . Suppose ~ is a faithful, or-weakly continuous (quasi)weight on 79(A4). T h e n the following s t a t e m e n t s are equivalent.: (i) ~-~ is a *-isomorphism. (ii) T h e m a p 7c~-1 from 7r~(Mb)[W~] to ( M I w ) ' [ v [ T ~ ] is closable. P r o o f . (i) ~

(ii) Let {As} be any net in A4b such t h a t r o s - l i m lr~,(As) =

0 and r~s - lim As = A E ( M w1 ) ! Iv 9 By L e m m a 3.4.10 we have s

~(~)

= ~s - l i m ~ ( B , ) ,

where {Bz} is a uniformly b o u n d e d net in Mb which converges a-strongly* to A. A n d we have lira II ~ ( A s ) A ~ ( X ) c~,r

- ~r~(Bz)Ar

II2

= lim p x ( ( A s - BS)t(A,~ - BZ) ) s,,3

=0 for all X E gl~. Hence we have

rr~(A))~,(X) = lim 7r~(B~)~(X) = lira rcr s

=0 for all X E r A=0.

and so ~-~(A) = 0. Since ~-~ is a *-isomorphism, we have

3.4 Generalized Connes cocycle theorem for weights

141

(ii) ~ (i) Suppose ~ ( A ) = 0, A E ( M r ) ' . T h e n there exists a net {A~} in Adb such that, ]] A~ ]]< r for all c~ and 7-~ - lim A~ = A. By L e m m a 3.4.10 we have ro~ - l i m ~ ( A ~ )

= ~(A)

= 0.

Hence, A = 0. This completes the proof. Definition 3.4.12. A a-weakly continuous (quasi-)weight ~a on P(AA) is said to be strongly faithful if ~ is faithful and one of the conditions (i) and (ii) of L e m m a 3.4.11 holds.

P r o p o s i t i o n 3.4.13. Let. M be a self-adjoint O*-algebra on D in ~ such t h a t 3A~~ = (3A~w)', and ~ a strongly faithful, a-weakly continuous (quasi)weight on 7 ) ( M ) such t h a t ~r~, is self-adjoint. Suppose (M'w)' and (~r~ (Ad)'w)' satisfy one of the following statements: (i) they are s t a n d a r d yon N e u m a n n algebras. (ii) Ad'w and 7c~,(2t4)'w are properly infinite and of countable type. (iii) 7-/and T/~, are separable and (M'~)' and (7c~(M)~w)' are yon N e u m a n n algebras of t y p e III. T h e n the O*-algebras Ad and 7c~(M) are spatially isomorphic. P r o o f . It follows from L e m m a 3.4.10, 3.4.11 and (Stratila-Zsido [1] w t h a t ~ is spatial, t h a t is, there exists a unitary transform U of 7-/onto 7-/~ such t h a t ~ ( A ) = UAU* for all A E (Ad~w)'. This implies t h a t U D = :D(Tr~,) and Try(X) = UXU*

(3.4.12)

for all X c 3,t. Take an arbitrary X E 3A. For each ~ E D and Y c r have

we

( ~ ( X * ) ; ~ ( Y ) I U~) -- lim (Tr~,(A~)A~(Y) I U() Cl

t * A~(Y) I U~) = lim (UA,~U

= (A~,(Y) IUX~), where {A~} is a net in Adb which converges a-strongly* to X. By the selfadjointness of 7r~ we have U~ c D(Tr~) and 7r~,(X)U~ = UX(,

(3.4.13)

and further (x*r

(ux*~ I ~) = ( ~ ( x t ) u ~ I ~) =

for all ~ E 7:) and 7] C :D(Tr~). Hence it follows from the self-adjointness of

142

3. Standard weights on O*-algebras

Ad that U*/)(%,) c I9, which implies that the statement 3.4.13 holds. This completes the proof. Throughout the rest of this section let 3// be a self-adjoint generalized yon Neumann algebra on I9 in 7-/ such that A4~~ = (A4")' and (Ad'w)' is a standard form. We denote by W~(M) the set of all strongly faithful, aweakly continuous, semifinite, quasi-standard quasi-weights p on T'(M) such that 7r~ are self-adjoint. Suppose ~; 9 W~(AJ). By Proposition 3.4.13 %,(did) is a generalized yon Neumann algebra on/9(7r~) in 7-/~, and so ~ is standard. By Theorem 3.3.2 we have the following C o r o l l a r y 3.4.14. For every ~ 9 W~(A/I) there exists a one-parameter group {%~}teR of *-antomorphisms of M such that (i) % ( a ~ ( X ) ) = A it% ( x ) A -it , X 9 M , t 9 R; (ii) ~ is a {a~'}-KMS quasi-weight on "P(3,t). Suppose % r 9 W~(A4). By Proposition 3.4.13 ~(M) and ~r are generalized von Neumann algebras, and ~ and ~ are standard. Hence, by Corollary 3.4.9 we have the following C o r o l l a r y 3.4.15. Suppose % r 9 W~(A4). Then, the cocycle [De : D ~ ] associated with the quasi-weight r with respect to the quasi-weight ~ is welldefined in Jk4, that is, t > [De : D~]t is a strongly continuous map of into Ad satisfying the conditions (i) ~ (iv) in Corollary 3.4.9. We generalize the Pedersen-Takesaki theorem (Pedersen-Takesaki [1], Stratila [1]) for standard weights on yon Neumann algebras to those on O*algebras. Let { E W~(M). Since M is a generalized von Neumann algebra, the quasi-weights {e and ~ on P(Ad) defined by Theorem 3.3.3 and Theorem 3.3.4 coincide, that is,

{

9 l ~ = { x 9 M ; 3~x 9 z ~ ( ~ ) s.t.

e(~-~x~xk) = ~ k

k

% ( x ) A ~ ( K ) = K~x, VK 9 :D(A~) },

II ~x~ II2, ~ x ~ x k

9 p(9~).

k

Using Theorem 2.6.2 and Theorem 2.6.6, we can show the following results: C o r o l l a r y 3.4.16. Suppose ~, r c W~(A/I). The following statements are equivalent: (i) ~ o c r ~ = ~ , vt CI[~. (ii) ~ o a ~ = ~ , v t E ~ . (iii) [De : D~]t ff Jk4~*, vt c R. (iv) [ D ~ b : D ~ I t E M ~ vt c ]~.

3.5 Radon-Nikodym theorem for weights

143

(v) { [Dr : DV]t }teR is a strongly continuous group of mfitary elements of A4. C o r o l l a r y 3.4.17. Suppose V, z~, E Ws(Ad). The following statements are equivalent: (i) r satisfies the KMS-condition with respect to {a[}te R. (ii) at~ = o t ~, vt E]R. (iii) There exists a positive self-adjoint operator A in 7i affiliated with (%,(Ad)~w) ' r~ ~r~,(Ad)~ such taht tp = ~ , where VA is the quasi-weight on /)(7r~,(M)~,c ) induced by the quasi-weight on 79(JUl) defined by

{ 92~2 = { X e M ; A~(X) E 79(A) },

2(xtx) --II AA~(x) I[2, x 3.5 Radon-Nikodym

theorem

fits.

for weights

In this section we extend the Radon-Nikodym theorem for positive linear functionals (Gudder [1], Inoue [8, 10]) to for weights on O*-algebras. Throughout this section, let A4 be a closed O*-algebra on 79 in 7/-/. D e f i n i t i o n 3.5.1. Let V and r be (quasi-)weights on P(A4). If r c r and r < "yv(X*X), VX E .r for some constant "~ > 0, then ~b is said to be v-dominated and denoted by r _< YV. If r c r w and the map K~,r : )~o(X) ---+)~r X Er is closable from the dense subspace )~(9'[~) in a Hilbert space 7/~ to the Hilbert space 7/r then r is said to be v-absolutely continuous. If .r c r162 and for any X E .r there exists a sequence {X,~} in r such t h a t = 0 and lirnccr - X ) ' (Xn - X)) = 0,

nlilnoov(X*nXn)

then r is said to be v-singular. If r each X E 92~,, then ~ is said to be an

c r and v(XtX) = r for extension of V and denoted by V c ~.

We have the Radon-Nikodym theorem and the Lebesque decomposition theorem for weights similarly proofs to Theorem 3.2, Theorem 3.3 in Inoue

N: T h e o r e m 3.5.2. ( R a d o n - N i k o d y m t h e o r e m ) Let V and ~ be (quasi-) weights on 7)(A4). Then the following statements hold: (1) g, is v-dominated if and only, if there exists a positive operator H ' in 7r~,(Ad)'w such that r = (H'A~(X)IA~(X)) for all X E r (2) Suppose cYSt, c r and :r~+~(Ad)'w is a v o n Neutnann algebra. Then the following statements are equivalent: (i) r is ~-absotutely continuous. (ii) There exists an increasing sequence {H;} of positive operators in

144

3. Standard weights on O*-algebras

lr~(M)~w such that lirnooH'A~(X ) exists in ~

for each X E r

and

~(X*X) = liInoo]IH'~)ho(X)]{2 for each X E r (iii) There exists a positive self-adjoint operator H' in ?/~o affiliated with (r%o(M){v)" such that D(H') D A~(cJI~) and ~(X*X) = I]H')ho(X)]l 2 for each X r r P r o o f . Here we simply state these proofs. (1) Suppose ga is Q-dominated and put H' = K~o,wK~,r Then H' ff rho(Ad )" and ~(X*X) = (H',X~(X)]t~(X)) for each X E r is trivial. (2) (i) ~ (ii) We put

The converse

K' = (K~+~, K~+w,~)89

1

K" = / , t - l ( 1 - t)dE(t), n E N, where K ~ =

~01tdE(t)

is the spectral resolution of K'. Let U denote the

isometry of 7-/~0into ~ , + e defined by UA~(X) = K'~+w(X), X E r put H88 = U *K~U, ' hEN.

and

Then we can show that {//In} satisfies our assertion in (ii). (ii) => (iii) We put /9(H;) = {~ E ~ ;

limooH;~ exists in ~ }

:

Then H~ is a positive operator in ?/~ such that 7)(H~) D s162 and H 0 is affiliated with (Tr~(A/l)~w)", so that the Friedrichs self-adjoint extension H' of H~ satisfies our assertion in (iii). (iii) ~ (ii) This is trivial. T h e o r e m 3.5.3. ( L e b e s g u e d e c o m p o s i t i o n t h e o r e m ) Let Q and r be (quasi-) weights on 5~ such that r C r and rr~+~(M)'w is a yon Neumann algebra. Then ~b is decomposed into the sum: ~OD ~b~'+ ~b~, where Cg is a Q-absolutely continuous quasi-weight on P(Jt4) with r = r and ~b~ is a Q-singular quasi-weight on T'(Ad) with r = r P r o o f . Let { H ' } be in the proof of (i) ~ (ii) in Theorem 3.5.2, and let P~+W,~, be the projection of ~ + ~ onto Ker K~+V,,~K~+w,~,. We here put

r

= lirn ]]H~A~(X)]I 2,

~ y ( x * x ) = I I f ~ + , < ~ ( x ) l l 2, x 9 9l~.

3.5 Radon-Nikodym theorem for weights Then ~b~ and r

145

imply our assertions. This completes the proof.

As seen in Example 3.5.15, the Lebesque decomposition of a (quasi-) weight is not unique in general. P r o p o s i t i o n 3.5.4. Let ~ and r be (quasi-)weights on ?)(M) such that r162 c r and 7r~,+W(A4)" is a v o n Neumann algebra. The following statements are equivalent: (i) r is ~-singular. (ii) ~bc~--- 0 (iii) If X is a (quasi-) weight on P(M) such that X _< ~ and X -< r then

X = 0 on 7)(14). P r o o f . (ii) =~ (i) This is trivial. (i) =~ (iii) Since r is ~-singular and X -< r it follows that 3~ is ~a-singular. On the other hand, since X <- ~, it follows that X is ~-absolutely continuous, which implies X = 0 on 79(r (iii) => (ii) By Theorem 3.5.2, 3.5.3 ~ is represented as

~ ( x * x ) = ~ i m IIH'n~(X)ll 2, X 9 ~ for some increasing sequence {H~} of positive operators in 7r~(M)~. For any n 9 N we put

~n(x*x) = tlHLA~(X)II 2, X 9 ~ . Then ~n is a quasi-weight on P(M) such that ~n _< CE -< r and ~an _< IIH'II2~. By the assumption (iii) we have ~n = 0 for each n C N. Hence it follows from r = edl~ that r = 0. This completes the proof. In Section 3.2 we have obtained tile generalized Pedersen-Takesaki RadonNikodym theorem for strongly faithful, normal, semifinite weights on generalized von Neumann algebras with strongly dense bounded part by using the generalized Connes cocyle theorem. Here we consider the generalized Pedersen-Takesaki Radon-Nikodym theorem in more general cases. T h e o r e m 3.5.5. Let ~ be a standard (quasi-) weight on 7)(Ad) and r a (quasi-) weight on T~(Ad). The following statements hold: (1) ~p is a ~-dominated, {a~}-KMS (quasi-) weight on ;o(Ad) if and only if there exists a positive operator H in (Tr~(3d)'w)' N 7r~(Ad)~w such that r = IIHA~o(X)II2 for each X E r (2) The following statements are equivalent: (i) r is a ~-absolutely continuous, {cr[}-KMS (quasi-) weight on ~ ( M ) such that ~ + r is standard. (ii) There exists an increasing sequence {H~} of positive operators in

146

3. Standard weights on O*-algebras

(Tr~(Ad)t) I A ~r~(M)' w such that

x e ~

and

r

=

li~n H~,A~o(X)

lira IIH;~A~(X)I?

exists in

for each 2 c

n--* OO

7~o for each

~.

(iii) There exists a positive self-adjoint operator H in ~ affiliated with (r%(2t4)')/ A rqo(A4)~ such that D ( H ) D k~(cJ2~,) and r = IIHA~,(X)II 2 for each X E r (3) Suppose r is a {cT/}-KMS (quasi-) weight on P(AA) such that. .r C r162 and ~ + g, is standard. Then ~b is decomposed into the sum: ga D ~c + g 4 , where ~b~ is a ~-absolutely continuous, {cr~}-KMS quasi-weight on P(Ad) with ff[r = r and ~b~ is a ~-singular, {cr[}-KMS quasi-weight on 79(A4) with .W/r = r P r o o f . (1) Suppose ~ is a ~-deminated, {crt~}-KMS (quasi-) weight on 7)(Ad). By Theorem 3.5.2 there exists a positive operator H in n-v(Ad)'w such that r = IIH;~(X)ll 2 for each X ~ 9l~. We put, r

.

f

cc

2

cc

t I H A ~ (A)II , if A E ~ ( A ~ )

/ oo

if otherwise.

Then ga" is a normal, semifinite weight on (rr~(Ad)'w)~, Take arbitrary A, B E cc D(A~C) * N D ( A ~ ). Since S~o = ~5'A~, there exist sequences {Xn} and {Yn} in t

~1l~, N r

such that lim A~(Xn) = n--+ oo

A~~ (A), lim A~(X*~) = ./1~o ~ (A), * n--+ oo

~im A~(Y~) = A ,~( B ) , l i r a ~ ( ~ * ) = A~~ ( ~ *) . Since r is a {a~~ (quasi) weight on P ( M ) , there exists a sequence {fx,,,v~ } in A(O, 1) such that 2

it

t

= (H A~o(Yn)IA,oA~(X,J),

fx~,v~(t) = r f x . , r ~ ( t + i) = r176

= (H zk~A~(X,~)I),~(Y~))

for all t E N and n E 1~1,which implies that 2

cc

it

cc

.

lira supffx~,r~(t) - (H A~ (B)IA~A~ (A ))l = 0,

n-ooo tEI~

lim s u p l f x ~ , v . ( t + i ) - (H 2 A~A~ it ~ (A)IA~~ ( B ), ) ] = 0. n--+oe tEg {

Hence there exists a function fA,B in A(0, 1) such that

fA,B(t) = (H 2 A~,c c ( B ) I A ~i tA ~c c ( A .) ) = ~ " ( ~ ( A ) ~ ) , fa,B(t + i) = (H 2 A,oA~o *t ~ (A)IA~oc ( B ), ) = r162 for all t ~ ]~, which means that ga" satisfies the KMS-condition with respect to {a~}, By Theorem 15.4 in Takesaki [1] we have H E (rr~,(Ad)')'~rr~(A/l)'w.

3.5 Radon-Nikodym theorem for weights

147

Conversely suppose H 9 (7r,(~4)~)' D ~(A4)'~. Then ~p" is a normal, semifinite weight on (Tr~(AJ)'w)~ which satisfies the KMS-condition with respect to {c~}. Since S , = SA~, we can show similarly to the above proof that. r satisfies the KMS-condition with respect to {g~}. (2) (i) ==> (ii) Let K', U and H~, n ~ N be in Theorem 3.5.2. By (1) we have/(' 9 (Tc~(M)~w)' D 7r~(M)~w. We show H~ 9 (~r~(M)~w)' for each n 9 !~. Take an arbitrary C 9 ~r~(M)tw. Since

(UCU*K%~+v(A)A~+~(X) I A~+r = (CTr~(A)~(X)]U*),~+~(Y)) .

t

= (CA~(X) IU ~ + r

)l~+f(Y)) t

= (UCU*K'A~+o(X) ITr~+o(A)A~+O(Y)) for each A 9 M and X, Y 9 ~ , which implies

it follows that UCU*K' 9 7r~+V(M)~,

rl

CH~I~(X) = CU*( / t-I(1 - t)dE(t))UA~(X) n

= U*(UCU*)K'(/1-t-I(1 - t)dE(t))l~o+~(X) = U*( l ~ t - l ( 1 - t)dE(t))(UCU*K')A~+r

: g'~c~(X) for each C 9 7%o(A/0~w, X E r and n 9 N. Hence we have H~ 9 (Tr~(A/l)~w)' for all n 9 N. (ii) ~ (iii) This is shown similarly to the proof of (ii) ~ (iii) in Theorem 3.5.2. (iii) ~ (i) It is clear that r is a w-absolutely continuous, {at~}-KMS (quasi-) weight on P(Az/) and

(~+r

= I[(I+ H2)89

(3.5.1)

2, X 9

We put I/P(( I +

H2

) 89A ~c~) = { A 9

], ((I + H2)89

~c

= (I + H2)89

A ~cc( A ) 9

2 ) 89)}

d 9 lP((I + g2)89

ThenA~

( I + H 2 ) !2 A~cc is a generalized vector for the von Neumann algebra (Tr~(J~4)~w)'. Since (I + H2)- 89 9 ~D(A)* N I)(A) for each A 9 /)(A~C) * A cc /)(A~ ), it follows that

A((~(A)* n z)(A)) 2) is total in 7-/~,

(3.5.2)

148

3. Standard weights on O*-Mgebras

and further it is easily shown that

{(I + H~)-89

K 9 v(A~)} c v ( A ' ) c v ( A ~ ) , ]

A'((I + H2)- 89 (I + H2)- 89

= A~(K),

VK 9 19(AC~), = A~(K), VK 9 z)(A*),

(3.5.3)

so that

A'((D(A')* n/p(A')) 2)

is total in H~.

(3.5.4)

By (3.5.2) and (3.5.3), A is a standard generalized vector for the von Neumann algebra (Tr~(M)~w)'. By (3.5.3) we have

X*w:A~(K)= A~o(K ~ , ) = A'((I + H 2 ) - 8 9 *) = S*AA'((I + H2)- 89K) ,

C

= SAA~(K) C

,

C

,

for each K E lP(A~) AlP(A~), and hence S~I7 c S A. Further, since D(A') c C

.

29(A~) by (3.5.3), it follows that S*A~ = SA, and so SA;~ = SA, which implies CC

by (3.5.1) and the standardness of A~ that q~+~b is a standard (quasi-) weight on P(M). (3) This follows from Theorem 3.5.2 and the statement (2). This completes the proof. We next study the Radon-Nikodym theorem for {cr~'}-invariant (qu~i-) weights on 79(M): T h e o r e m 3.5.6. Let ~ be a standard (quasi-)weight on P ( M ) and r a (quasi-)weight on 79(.A4). The following statements are equivalent: (i) ~ is ~-dominated and {crt~}-invariant. (ii) There exists a positive operator H' in 7r~(2t4)~ ~ such that ~(X* X) = I[H'A~(X)][ 2 for each X E r Further, suppose ~ and ~ are positive linear functionals. Then the above equivalent statements are equivalent to the following (iii): (iii) There exists a positive operator H in (Tr~(Ad)'w)'~ such that H ~ ( I ) c Z ) ( ~ ) and r = (~(X)H~(I)I~/~(I)) for each X ~ 9l~. P r o o f . (i) r (ii) This follows from Theorem 3.5.2. Suppose ~ and r are positive linear functionats on AJ. (ii) ~ (iii) We put H = J~,H'J~. Then it is easily shown that H is a positive operator in (G,(Ad)~) '~ such that HA~,(I) = H'k~(I), and hence HA~(I) c D ( ~ ) and r = (Tv~(X)HA~(I)IHA~(I)) for each X E r (iii) ~ (ii) This is proved similarly to the proof of (ii) ~ (iii).

3.5 Radon-Nikodym theorem for weights

149

T h e o r e m 3.5.7. Let (p be a standard (quasi-) weight on 7)(.Ad) and r a (quasi-) weight on 7)(A/I). Suppose there exists a standard, {at~}-KMS (quasi) weight r on 7)(M) such that ~ + r <_ r and 92, = 92~,. Then the following statements hold: (1) Suppose r is {cr~'}-invariant. Then r is decomposed into the sum: D r + ~/J~, where r is a ~-absolutely continuous, {cr~}-invariant qu~iweight on 7)(3,t) with 92~S = 92~ and ~ is a ~-singular, {a~'}-invariant quasi-weight on 7)(3.t) with 92r 92~. If r is ~o-absolutely continuous, then r D r and if r is ~-singular, then r D r (2) The following statements are equivalent: (i) r is ~-absolutely continuous and {crt~}-invariant. (ii) There exists a positive self-adjoint operator H' in ~ affiliated with %,(Ad)~ ~ such that ,\~(92~,) c ~D(H') and ~(X*X) = IIH'>,~(X)II 2 for each X E 92~,. Further, suppose ~ and r are positive linear functionals on M . Then the above equivalent statements (i) and (ii) are equivalent to the following (iii): (iii) There exists a positive self-adjoint operator H in H~ affiliated with (7r~,(M)~) ' ~ such that. X~,(I) E ~?(H), HAw(I ) E Z)(rr~) and ~(X) = (Tr~(X)H,k~(I)IHX~(I)) for each X E JVI. P r o o f . (1) By Theorem 3.3.6 we have it

z~ r )~,(X) : / ~ , ( c r / ( X ) ) ,

t

t

VX E 92. N 92~ : 92~ A 92~, vt E R.

(3.5.5)

Since ~ _< r and ~ _< r, there exist. R' and K ' in 7r,(Jk4)~w such that 0 < R', K ' _ < I a n d

~(X*X) = IIR%(X)II ~ and ~(X*X) = IlK%(X)ll ~ for each X E 92~ = 92~. Using (3.5.5) and the standardness of r, we can prove in the same way as in Theorem 3.5.5 that the generalized vector R'_/I~~ for the yon Neumann algebra (7r,(3.'l)~w)' satisfies the KMS-condition with respect to {0[}, so that R' E (Tr,(A/l)~w)'N 7r,(3/l)~w. Further, since ~ is {cr2}invariant, it follows from (3.5.5) that. K ' ~ 7c~-(3/l)~'. We denote by U the isometry of 7-t~ into ~ , defined by u A ~ ( x ) = R'A~(X), X E 92~ = 92~. We now put H~ = U * ( ~ where R' =

/0

~dE'(t))K'U, n ~ N,

tdE'(t) is the spectral resolution of R'. Since R' and K ~

commute, it follows that {H~} is an increasing sequence of positive operators in r r ~ ( M ) ? ~ and lirn H ~ ( X ) exists in ~ for each X E 92~,. We put

~ t (x* x ) -- ~ i m I I H ' ~ (X)It 2, ~ ( X * X ) = IIK'~'(0),Xr(X)II 2, X ~ 92, = 92~.

150

3. Standard weights on O*-Mgebras

Then it is easily shown that r is a u-absolutely continuous, {a[}-invariant quasi-weight on P(A4) with r = fits, ~ is a u-singular, {a~}-invariant quasi-weight on P(AJ) with r = r and ~b D r ~ + ~ . Suppose r is u-absolutely continuous. For any X E 9I~- there is a sequence { X , } in r such that l i r a A~(X,) = E'(O).k~-(X). Since r = .r C tile, we have lim A~(X~) = lim U*R'AT(X,~) = U*R'E'(O)AT(X) = O, n---~

n---*(x~

lira (~(X~)I~r

lira (K'~(X~)IK'~.(V)) = (K'E'(O)A,(X)IK'A~(Y))

for each Y ~ 92~. Further, since ~ _< r, we have lim Af(X,~) = 0, and so it rt~oo

follows from the u-absolute continuity of r that K'~E'(O)A,(X) = 0. Hence we have r = 0, and so ~ D ~ . Similarly, if r is Q-singular, then

r162 (2) (i) ** (ii) Using the statement. (1), this is proved similarly to the proof of Theorem 3.5.2. Suppose ~ and r are positive linear functionals on M. (ii) ~ (iii) We put H = J~H'J~,. Then H is a positive self-adjoint operator in 7-/~, affiliated with (Tr~,(Ad)'w)'~ such that X~(I) ~ :D(H), HA~,(I) = H';~,(I) ~ 5P(~r~) and O(X) = ffr~,(X)H,X~(I)IHA~,(I)) for each X ~ 3,t. (iii) ~ (ii) This is proved similarly the proof of (ii) ~ (iii). This completes the proof. It seems to be difficult to show directly the standardness of U + ~b in Theorem 3.5.5 and the existence of {a~}-KMS (quasi-) weight r with U + r _< r in Theorem 3.5.7, and so we consider when Theorem 3.5.5 and Theorem 3.5.7 hold without these assumptions. P r o p o s i t i o n 3.5.8. Let ~ be a standard (quasi-)weight on P(A/I) and r a (quasi-)weight on 7)(Ad). Suppose r is ~-absolutely continuous, 7r~+r (~4)~ is a von Neumann algebra and r

< ~/{u(XtX) + ~(XX*)},

VX e ~

Or

(3.5.6)

for some constant "y > 0. Then the following statements hold: (1) If r satisfies the KMS-condition with respet to { ~ }, then there exist a positive self-adjoint operator H in ~ affiliated with (lr~ ( A d ) ' ) ' N 7r~(A/I)~ such that A~(cJI~) c I)(H) and r = ]]HA~(X)]] 2 for each X E r (2) If r is {a~}-invariant, then there exists a positive self-adjoint operator H ' in 7-/~ affiliated with 7r~(Ad)~ ~ such that A~(cJI~) c :D(H') and r = ]]H'A~(X)]] 2 for each X E r P r o o f . (1) By Theorem 3.5.2 r is represented as

3.5 Radon-Nikodym theorem for weights

= ilH'~(X)ii ~, X 9 ~

r

151

(3.5.7)

for some positive self-adjoint operator H' in 7Y~ affiliated with %,(Ad)~ such that ),~(,r c 7P(H'). Since r is {cr~'}-invariant, we have it

I

IIH A~A~(X)II = IIHtA~(X)II, Vx 9 9 ~ .

(3.5.8)

cc

Take an arbitrary A 9 z)(A~) * n z)(A~ ). Since SAy = S~,, there exists a sequence

{Xn} in r A~ ( A ) . By

N~

such that

CC

lira A~(X~) = A~ (A) and

(3.5.6), (3.5.7) and (3.5.8) we have

A~ (A) 9 iD(H'),

lim H/A~(Xn)

it cc nlimooH t A ~i t ( X ~ ) = H I A~A~ (A),

,~ ,A), Vt

9 R,

(3.5.9)

which implies by (3.5.7) and (3.5.8) that t

it

I

it

cc

t

cc

IIH A~A~ (A)[I = IIH A~ (A)II,

(H A ~ ( X ) i H for each X E 9~t~ A r

I

it

cc

A A~ (A)) = (H t ~(X)IH'A~C(A))

and t 9 l~, so that

it ]IHI A~A~(x) - H I A i t A~c c (A)H = ]IH'~(N) - H'A~C(A)]I (3.5.10)

for each A ~ , _ ~ , A :D(A~ ) and X 9 r N 9I~. Since 0 is a {crt~}-KMS (quasi-) weight, it follows from (3.5.9) and (3.5.10) that the generalized vector I CC H A~o for the von Neumann algebra (Tr~(fl4)'w)t satisfies the KMS-condition with respect to {a~}. Hence there exists a positive self-adjoint operator H in 7-/~ affiliated with (Tr~(A4)~w)' N 7r~(A4)'~ such that cc

A~ (A)

9

~(H) and IIH'A~C(A)Ii = IIHA~C(A)Ii CC

for each A 9 /?(A~) * A ~D(A~ ), which implies by the polar decomposition theorem that !

CC

CC

CC

A~ (A) 9 Z)(H) and IIH A~ (A)[I = IIHA~ (A)I[ for each A E

cc

z)(A~ ).

Take an arbitrary X C r

__ polar decomposition of _X_ and IXI =

IXI. We put cc

j~000

__

Let X =

(3.5.11) m

UIXl be the

tdN(t) the spectral resolution of

X,~ = X E ( n ) , n 9 l~I. Similarly to (2.1.1) we can show X,~ 9 cc lira A~ (Xn) = , ' ~ ( X ) , and further by (3.5.11)

~(A~ ), n ~ N and

n ~T~ ---~ o o

152

3. Standard weights on O*-Mgebras cc

cc

,

lim IIHA~ ( x , d - H A w (xm)ll =

cc

lim IIH A~ ( x . ) - . .

=

rg,ACCf y

~,~..~,ll

lira [[(E(n) - E ( m ) ) H ' A ~ ( X ) I [ lz,fr~

oo

=0~

which implies A~,(X)

9

"D(H) and

lulq 9 H A ~(x

n~oo

~

n

) = HA ~ ( X ) " Hence we

have !

CC

f~(X*X) = IIH'A~(X)II 2 = lirno I]H A~ (x,,)l[ 2 9

= limllHA~

cc

(Xn)ll

2

= ilHA~(X)II 2 for each X 9 ~ . (2) This is proved similarly to the proof of (1). P r o p o s i t i o n 3.5.9. Let ~ be a standard positive linear functional on ~4. Suppose r is a v-absolutely continuous positive linear functional on M such that ~+f~(Ad)~ is a v o n Neumann algebra and

<

r

~ ( X Yk YkX), VX r M

(3.5.12)

k=l

for some finite subset {Y1, Y2,"" , Y~} of Ad. Then the following statements hold: (1) If r satisfies the KMS-condition w.r.t. { t }, then there exists a positive self-adjoint operator H in T/~ affiliated with (Tr~(M)~)' A 7r~(A.~)~ such that A~(I) 9 :D(H), HA~(I) 9 D(T:~) and r = (T~(X)HA~(I)IHA~(I)) for each X 9 A4. (2) If ~ is {a[}-invariant, then there exists a positive self-adjoint operator H in H~ affiliated with ( ~ ( M ) ~ ) '~ such that Aw(I) 9 D(H), HA~(I) 9 ~ ( ~ ) and r = (~(X)HA~(I)IH),~(I)) for each X 9 M . P r o o f . (1) By Theorem 3.5.2 r is represented as r

= IIH'A~(x)II 2, x 9 M

for some positive self-adjoint operator H ~ in 7-/~oaffiliated with rrr such that A~(3A) C :D(H'), and further by (3.5.12) we have :D(lr~,) C :D(H'), and so A~,(AJ) C TI(H'2). Hence, since f) is {~f}-invariant, it. follows that

H '2 A ~i tA A ~ ( I ) = A ~t H '2 AA~(I),

vA

, 9 ~r~(A4)w n ( ~ ( M ) w,) , ,

v t 9 ]~,

which implies since r is a {a[}-KMS positive linear functional on AA that H the normal positive linear fuctional ~H,2X~([) on the von Neumann algebra (~r~(~4)~)' satisfies the KMS-condition with respect to { t }, so that the

3.5 Radon-Nikodym theorem for weights

153

statement (1) is shown similarly to the proof of Proposition 3.5.7,(1). (2) We can show similarly to the proof of the above (1) and Proposition 3.5.8, (2) that there exists a positive self-adjoint operator H / affiliated with 7r~(34)~ ~ such that A~(M) c Z)(H I) and r = (~r~(X)HIA~(I)IHIX~(I)) for each X ~ M . We put H = J~HIJ~. Then H satisfies our assertion in (2). This completes the proof. We next consider the Radon-Nikodym theorem for positive linear functionals on O*-algebras with the von Neumann density type p r o p e r t y . Let A4 be a closed O*-algebra on ~D in 7-/such that Jt4'w~D c ~D. Suppose that the O*-algebra 34 has the von Neumann density type property, that is, = 34w~" Let ~o be a standard vector for J ~ and put 9~o = W~o. We denote by P(~4")'~0 the natural positive cone associated with the left Hilbert al~ is the closure of {AJ~;AJ~'o4o; A ~ (34~w)1}, P(Z4'w)'~o and denote P(z4,),~o~Z)~by P~0" Let 34+ be the set of all a-weakly continuous

/ /40, that is, gebra (Adw)

positive linear functionals on 34. Lemma

3.5.10. (1) Any element r of Jkd,+ is represented as r = a ~

for the unique vector 4r in 7'~0. (2) Let ~ be any element o f ~ 4 .+. Then (~r~,o+r ~ is unitarily equiv/ / alent to (2t4w) , and so 7r~,o+~(M)~w is a v o n Neumann algebra. P r o o f . (1) By L e m m a 5.2 in Inoue-Ueda-Yamauchi [1], r is represented as r = a~ for some 4 in ~D. Hence it follows from Theorem 10.25 in StratilaZsido [1] that (A414) = (A4r

VA c (3/I'w)'

(3.5.13)

for the unique vector (r in P ( ~ , ) , ( o . Take an arbitray X C M . Let IX[ =

/o

talE(t) be the spectral resolution of IxI and En =

Since Added C 7), it follows t h a t (3.5.13)

/o

dE(t), n E N.

E~, XE~ E (M'w)' for n E N, so that by

lim En(~ = 4r n ----+o o

lim ~Tt , n -----r O 0

[[XEm~r -XEn4~][ =

[]XEm~- XEn~[[ = O.

lim 7Tt , n ~

C2~

Hence, ~r c ~D and ~ = w(,. Suppose r = w41 = cu(2 for ~1, ~2 C ~Dgo. By ----JMwo and (3.5.13), we have ~1 = ~2. (2) By (1), Wo + ~ is represented as ~o + r = W~o+ ~

(3.5.14)

154

cr 3. Standard weights on O * -aloebras

for the unique vector ~ o + r 9 P~o' which implies by MT$~ = A / ~ that (7~o+~(M)~w) ! is unitarily equivalent to (A/Yw)'. This completes the proof. P r o p o s i t i o n 3.5.11. Let ~4 be a closed O*-algebra on T) in T/such that !

It A/~w:D c T) and A/~~ = A//w~ , ~0 a standard vector for A/t and r 9 ~ 4 .+. Then r is ~0-absolutely continuous if and only if there exists a positive selfadjoint operator H ! affiliated with M~w such that M~0 is a core for H ! and r -- ~H'~o" If this is true, such an operator H ! for r is unique and it is denoted by H~.

P r o o f . Suppose r is ~0-absolutely continuous. By Lemma 3.5.10, r = ~ for some ~ 9 P~o" We denote by K~o,~ the closure of a closable map: X~0 X ~ , X 9 M . Then it follows from M ~ = A//~" that (~4w)' '~o c T~(K~o,~) / ! and K~o,r = A~r for all A 9 ( M ~ ) , whmh implies that (M~)'~0 is a core for K~o,~ and K~o,~ is affiliated with M~w. We put H ' = (K~o,r189 Then, H ' is a positive self-adjoint operator affiliated with 2~4~w such that A/[~0 is a core for H' and r = WH'~o- The uniqueness of H ~ follows from that of the polar decomposition. The converse follows from Theorem 3.5.2. This completes the proof. P r o p o s i t i o n 3.5.12. Let M be a closed O*-Mgebra on :D in ~ such that , ~T~s ,, A/IwT) C :D and = Mw~,~0 a standard vector for ~4 and r E A/[ +. The following statements hold: (1) r satisfies the KMS-condition with respect to {at~~} if and only if r = WH~o for some positive self-adjoint operator H affiliated with ( M ~ ) ' A A/Vw such that ~o E T)(H) and H~0 E TL (2) r is {at~~ if and only if r : ~H(o for some positive selfadjoint operator H affiliated with (M'~) '~~ such that (0 E :D(H) and H ( o 9 :D. P r o o f . By Lemma 3.5.10, r : ~(r for some (r 9 P~o" Suppose r is a {crt~~

(resp. {at~~

positive linear functional on A/L Then,

since ~ s ,, it follows that the normal positive linear functional -- A/lw~, ~tt ~ on the von Neumann algebra (A/I~)' satifsies the KMS-condition with respect 'to (at~~} (resp. (at~~}-invariant), so that by Theorem 15.4 in Takesaki [1] (resp. Theorem 15.2 in Takesaki [1]) there exists a positive self-adjoint operator H affiliated with (A/t~)' N .M~ (resp. ( M ~ ) ' ~ ~ such that ~o

9

T)(H) and (A~J~r

: (dH~oJg~o), VA

9

(~4~)'.

(3.5.15)

We denote by U ~ the partial isometry of 7-/ defined by A ~ ---* AH~o, A 9 (Ad~)'. Then U' 9 A/Vw, and so H~o -- U'~r 9 :D. Further, since X is affiliated with (A/Vw)' for each X 9 J~4, it follows from (3.5.15) that r = wilco. This completes the proof.

3.5 Radon-Nikodym theorem for weights

155

We finally investigate the absolute continuity and the singularity of positive linear functionals on the O*-algebra generated by the differential operator, on the O*-algebra defined by the SchrSdinger representation and on the maximal O*-algebra s (,5(][{)) on the Schwatz space S ( ~ ) . E x a m p l e 3.5.13. We put D-

{~ E C~[O, 1]; ((")(0) = ((n)(1),n = 0 , 1 , 2 , - . . } , .d Xo = - ~ j / [ / ) ,

G0(t) = [exp{- e x p ( -

)}](5 - 4 cos 27rt) -1, t E [0, 1].

Then the polynomial algebra 7)(Xo) generated by X0 is an integrable, commutative O*-algebra on 7) and Go is a standard, strongly cyclic vector for ~(Xo) (Takesue [1]). We consider positive linear functionals on 7~(Xo) defined by ~b(p(Xo)) ---- (p(aXo + b)~ol~o), a # O, b E N. Then the following statements hold: (1) For any n # 0, m E Z, ~ ' ~ is W~o-absolutely continuous. (2) For any bounded subset B of R and a # 0, b E ]~ we define positive linear functionals on P(X0) by ( ~ 0 o xB)(p(xo)) = (xR(xo)p(Xo)~H~o), (~ba o Xs)(p(X0)) = (XB(Xo)p(aXo + b){0[{0). Then ~b o XB is (W~o o Xs)-singular for each a E ~ or b ~ 27rZ. We show the statement (1). By ~2"m is represented as ~.m(p(x0))

=

(p(Xo)U~olV~o)

for some U E s (T))i - {U E s (T)); U is an isometry }. We put ( ~ ' ~ ) " ( A ) -- (AU~o[U(o), A E (P(Xo)'~)'. I" I I Since (P(X0)w) is a commutative yon Neumann algebra and Go is a cyclic tracial vector for (T~(Xo)~w)', it is easily shown that ( ~ 2,m ) n is w~o-absolutely continuous. Hence we have

(AU~olU~o) : (AH'~oIH'~o), A E (T'(Xo)')' for some positive self-adjoint operator H' in L2[0, 1] affiliated with P ( X 0 ) ' , which implies H'(o E 29 and ~ ' m ( p ( X o ) ) =

(p(Xo)H'(oIH'{o).

156

3. Standard weights on O*-algebras

Hence (p2n~rmis aJ~0-absolutely continuous. We next show the statement. (2). For any polynomial p and n c N we define a polynomial p. by 2n+1

p~(t) = Z k

ak{(t + 2nTr)(t + 2 ( n - 1))~r)... (t + 27c)t(t - 27c)--. ( t - 2nTr)} k, 1

where {al, a2,--. , a2~+~} is the unique solution of the equation: p,~(2rnrca+b)

p(2m~ra+b), m=-n,...

,-1,0,1,..-,n

(the existence of the unique solution dues to a ~ 0). Since /3 is a bounded subset of It{, it follows that

o xB)(pn(Xo)'p,

(Xo))

= 0,

o x . ) ( ( p . ( X o ) - p( Xo) ) * (pn(Xo) - p( Xo) ) ) = o

for sufficient large all n E N. Hence, ~b o XB is 'W(o o XB-singular. -

-

m

m

Let L 2 | L 2, 8 | L 2, (8 | L2)+, 8+, ~'2{c~} ({OLn} E 8+) and { f n } be in Example 2.4.24. Let ~r be the self-adjoint representation of s (8(R)) in the Hilbert space L 2 | L 2 defined by 7c(X)T = XT, X E s

T r 8|

and 7/' and 7r' *-representations of B(L2(]I{)) on L z | L 2 defined by 7r"(A)T = A T and rr'(A)T = T A , A c B(L2(II{)), T C L 2 | ~ .

We here consider strongly positive linear functionals on s (8(N)) defined by Ax) = t r p 2 X = <

(X)plp >,

(3.5.16)

Since Y2{~.} is a standard, strongly cyclic vector for 71"(st(8"(R))) with /'~{~n}it 7"(trf~-2itt {~.}j~Tr'tf22itt {~,}j( ~ Vt C ~ ) for each {a,~} E s+ (Example 2.4.24,(3)), it follows from (3.5.16) that ~{~n} is a standard, strongly positive linear flmctional on s (8(1~)). Let 3.4 be a self-adjoint O*-algebra on 8(]1{) defined by the Schr6dinger representation of the canonical algebra for one degree of freedom. Since Y2{~-.,} is a standard, strongly cyclic vector for 7r(Ad) for each /3 > 0 (Example 2.4.24,(5)), it follows that ~{~ ~,~ is a standard, strongly positive linear functional on M . In next Example 3.5.14 we consider the qa{~ ,,)-absolute continuity, the ~{c_n~}-singularity and the :

{at~

"~} }-invariance of strongly positive linear functionals on Ad.

3.5 Radon-Nikodym theorem for weights

157

Example 3.5.14. Let p ~ (8 | L2)+, {c~n} r s+ and/3 > 0. (1) Suppose f2~-2~}p is densely defined. Then ~p is a p(~}-absolutely continuous, strongly positive linear functional on 3/I. (2) qz{~) is a {a[ (* ~'~ }-invariant, strongly positive linear functional on A4. Conversely, suppose K2-1{_~}p is densely defined and p2f-2?el_,~} r 8|

2

(in particular, ~ is ~{~ ~z}-dominated) and ~ is a t a t " "t-mvarmnt positive linear functional on 3A. Then ~ = ~ { ~ } for some {a,~} ~ s+. (3) Every {at~ ~ } - K M S positive linear functional ~ on AJ is represented as ~ -- YW{e-~) for some constant 7 > 0. The statement (1) follows since ~ is represented as

I~ (n{~.}p)ln(~.} >, x 9 M for a posiitve self-adjoint operator l~-'(~2~-~=}p)l affiliated with 7r'(B(L2(R))) such that ]7r'(~-2n}p)[~2{~} e 8 | ~ . We show the statement (2). It is clear that ~ { ~ } is a {a[ {~ }-invariant, strongly positive linear functional on Ad. We simply put ~2~ = ~2{~ ~} and ~

-- W{~-~}. Suppose s

is densely defined, p2f2~l C 8 | ~L and ~p is

qo~

{a t }-invariant. We put Ho = (~2~Ip)(~2~Ip) *. Then 7/(H0) is a positive self-adjoint operator in L2(~) affiliated with 7/(B(L2(]t{))). Since p2~2~1 r 8 | -L~ it follows that s

e T)(Tr'(H0)) andzr'(H0)t9 z = p2$2~1 E 8 | ~L,

(3.5.17)

and hence

7r(M)~2 z C 79(7r'(H0)), ~r'(H0)rr(X)f2# = 7r(X)rr'(H0)~2Z,

~(x)~'(Ho)n~ln~ >, v x c M.

~Ax) :< ~o~

.

~

Since qop is {a t }-mvanant, it follows that

~p(Y*a[ ~ (X)) =< 7r(Y*a[ ~ (X))Tr'(H0)~2~l~2z > = < rc'(Ho)A~erc(X)f2z]Tr(Y)f2 z > ~p(Y*a[ ~ (X) ) = ~p(a[" (Y')X) = < A~jc'(Ho)rc(X)n~lTr(Y)n z > for all X , Y C M T)(Tr'(H0)) that

and t E 1~, which implies since 7r'(/~(L2(~)))f2~ C

158

3. Standard weights on 0 -alc,ebras *

o"

= < A~'(Ho)Tr(X)f2~I~r"(A)F2Z =< ~T(X)~[Tr'(Ho)A~"(A)F2Z

> >

for all A 9 B(L2(]~)), X E ~4 and t E ]~. Hence it follows since ,4~z = ~r'(~2~2~)~"(~2~), vt 9 ]~ that ~r"(~22'~)~'(~2~)~'(g0)Tr"(d)~2 ~

=

7r'(Ho)Tr"(~'t)T~'(~;2i~)~"(A)~z

for all A 9 B(L2(~)) and t 9 ]~. We here put A = f~ | T~, n 9 N u {0}. Then, since {fk} c / ) ( H 0 ) , it follows that

e-2kZ~(Hof~lf,~)f,~=(fn

| -~--~H Jn) o ~2f~-2~t~~

= ( f n | -f -, , ) ~ ; ~- ~

goA

= ~-~nz~'(UoA IA)fn, which implies HoA = (HofnlA)A,

n ~ N u (0}.

By (3.5.17) we have

{c~n = e-'~Z(Hofnlf,~) 89} e s+ and : p

=

:{~,}.

Suppose ~op is ~o~-dominated. By Theorem 3.5.2, ~p is represented as ~p(X)

=<

T:(X)~'(Ho)F2z[cr'(Ho)F2~ > ,

X

c 3..4

for some positive self-adjoint operator Ho in B(L2(~)), and

~l~'(H0)~z

= H0 e Z3(L2(iI~)),

(Tr'(H0)~2Z)2~2~l

=

(Tr'(g0)~)H0 9 S | n 2.

Hence, taking the above p to 7rr(H0)S2Z, we can show that ~op= ~o~,(Ho)~ = ~ot~,t for some {a,~} 9 s+. The statement (3) follows from Theorem 30 in Gudder-Hudson [1]. We finally give concreate examples of ~o{e-,,,}-singular positive linear functionals on s (S(]~)) and of ~{e-,z}-absolutely continuous positive linear functionals on /:* (S(~)), and characterize {a~~ ~ linear functionals on Z:t (S(]~)).

}

Example 3.5.15. (1) We put for -- ~ rt=O

e-~J'~, fs =

2]o -

f~r

. . }-mvarmnt positive

3.5 Radon-Nikodym theorem for weights Then ~vf~|

and ~f,|

on s (8(]~)) and ~f~o|

159

are ~v{e-~z}-singular positive linear functionals + ~F|

is not a ~{e_~}-singular positive linear

functional on s (S(~)). (2) The ~{~ ~z}-absolutely continuous positive linear functional ~v{ _=~ } on s (8(1~)) dominates a positive linear functional r on s (S(]~)) which is not ~v{~-~z]-absolutely continuous. (3) The Lebesgue decomposition of ~ { _ ~ } is not unique. r

~P{e-n3}'~

.

9

9

(4) Every ~o{e-~}-absolutely continuous and t~rt kmvarmn~, strongly positive linear functional p on s (S(]~)) is represented as ~o : ~{a.} for some {~} ~ ~+. (5) Every {a[ {~-~}}-KMS strongly positive linear functional ~ on s (S(]i{)) is represented as ~ = %v{~-~} for some constant "7 > 0. We show the statement (1). We put ~2Z = $2{~-=~} and qvz = qV{e-.Z}. For any X ~ s (8(~)) we put m

X ~ - log1m kE=xlem~(Xfoo@-fk), m = 2 , 3 . . . . Then we have '

7r(Xm)(y~ |

--

log 1

m ~ k (Xf~

ml

= (m~-~E-$)~r(X)(fo~ 1ug,,~ k= 1,4

If,Q , --

@ foo), m = 2 , 3 , - . . .

Hence it follows that lim lr(Xm)(2~ = 0 and

lim 7r(Xm)(foo | -'f"~) = Tc(X)(f~ | f ~ )

for each X E s (S(]~)), which means that ~fooeT'~-~ is ~%-singular. Similarly, we can show that ~vf, | is ~%-singular. Since

e2~ ( f ~ | YZ)2 + ( f s | ] ' ) 2 -- e ~z - i (Lo | foo + f " 9 Y-~),

2e2~ ((Lo | fo~) ~ + ( f ~ | fh)2)(fo~ + i f ) -- e2~ _ 1(fo~ + f ' ) , ((f~@foc)2+(f~|

f ~')

2) ( f ~ r

2e2fl - f ' ~ ) -- (e2Z - 1)2 ( f ~

- fs

it follows that f ~ + f ~ = 2fo and f c r ] ~ are eigenvectors for ((fee | foe)2 + ( f ~ | ~-/~)2) with eigenvalues 2e2Z/e 2~ - 1 and 2e2~/(e 2z - 1) 2, respectively, which implies

160

3. Standard weights on O*-Mgebras 2e2Z ( ( f ~ | ~ - ) z + ( f ~ | ~w~)2) > _ _ e 2~ -- 1 (fo | /o).

Hence we have

(~s~|

+ ~s-|

)(x'x) = t~((f~ | f~)~ + (f" | ~)~)x* x 2e2~ >_ d , ~ _ ~tr(fo | To)Xt X 2eZZ -- e2~ _ l ~ f 0 |

for all X E s (S(R)), and hence (2e2Z/e 2z - 1)~So|

is a non-zero positive

linear functional on s (S(N)) which is dominated by both ~ and ( ~ I ~ | ~S_| so that by Proposition 3.5.4 that (~f~| + ~S2~| is not ~,singular. We next show the statement (2). Let 7-/1 be the closed subspaee of L2(]~) generated by {fa, f a , " " , f 2 n + l , " " } and P the projection of L2(]~) onto T/1. Since f2{e ~ } P = PY2 {e- ,~2} and it is a non-singular compact operator on 7-/1, it follows from Lemma 8.8 in Kosaki [1] that there exists a unitary operator U on T/1 such that

Range(n _~

n 5Range(n e_n~ P ) = {0}.

We here put p= s

r

,~}UY2{e_,~} , where U = U P + (1 - P),

= trpp*X,

X Es

Since

r

= II~(x)~

.~}un<~ .~}112

i e-~z

t

le

2 J

_<~{ _.}}(x'x) for all X E/:* ( $ ( ~ ) ) , it follows that r is ~{e-"~ }-dominated. Suppose r is ~z-absolutely continuous. By Theorem 3.5.11 r is represented as

r

i t =< ~(X)HJ2zIH~2Z >, X E s (S(~)).

Hence, the positive linear functional r r

on B(L2(]t{)) defined by

= < 7r"(A)H~FizIH~r

>, A E B(L2(~))

3.5 Radon-Nikodym theorem for weights

161

is faithful and ~@-absolutely continuous, so that by Corollary 7.3 in Kosaki [1] that ~r'(B(L2(N)))f2Z ~ rc'(t3(L2(N)))p is dense in L 2 | ~ .

(3.5.18)

Take an arbitrary T E rr'(B(L2(]~)))X2Z N rr'(B(L2(]t{)))p. Since T = rc'(A)p = ~r'(B)/2~ for A, B E B(L2(]t~))), we have

u ~ ~A~= s ~-~e ~ ,

v~

L~(R),

which implies P f 2 { _ ~ } B ~ = UPf2{_n ~}A~ E Range(PY2{e ~ ~ }) C~URange(PY2{~_~_~ }) = {0}. Hence we have

P T ~ = P f 2 ~ B ~ = f 2 ( _ ~ } P~2 {~ ~ 2 } B ~ = 0 for each ~ E L2(IR), and so Range(T) c (I - p)L2(]R), which contradicts (3.5.18). Hence r is not ~-absolut.ely continuous. We show the statement (3). The ~-absolutely continuous positive linear functional ~ { _ ~ } on s (8(1~)) is decomposed into

~_~) =~_~)+0 = {(~

_~ ~ - r + r

}+

r

where r is in (3). Since ~a{e ~ } - ~p < ~B and ~ps~z =~ 0, it follows that ((~{r ~ } - ~p) + r

is ~{e ~ I-dominated and ~-absolutely continuous,

and r is non-zero and ~a~-singular, which shows that the Lebesque decomposition of ~{ _~r is not unique. We show the statement (4). By Proposition 3.5.12,(2) ~ is represented as ~(X) --< rr(X)HY2aIHs z >, X c s for some positive self-adjoint operator H in L 2 | -L~ affiliated with rc"(B(L 2 ( N ) ) )o~e such that f2Z E D(H) and Hs E 8 | ~ . It is easily shown that rc"(B(L2(R))) "~' = {Tr"(A); A

~ n=0

Hence we have

anf,~ | K E B(L2(R))}.

162

3. Standard weights on O*-algebras C~

Hn = ~--~/3k(~)fk | ~ 9 B(L2(]~)), n 9 N and lirno~ rr" (H~)S2z = H/2Z, k:0

which implies (X3

lim ~3~")e-ko : ak, k : 0, 1, 2,..- and H O e : Z n---+oo

c~kfk | f7 9 8 | ~7.

k=0

Hence we have {~k} 9 s+ and ~ = ~(~k}- We finally show the statement (5). By Proposition 3.5.12 ~ is represented as

~(X) --< ~(X)HY2eIHS? e >, X e s for some positive self-adjoint operator H affiliated with 7r"(B(L2(]~))) VI rd(B(L2(l~))) such that ~ E 2)(H) and HY2Z 9 ,5 | L 2. It, is easily shown that 7r"(B(LZ(~))) N rd(B(L2(]~))) = C I , which implies ~ = "y~e for some constant "y > 0.

3.6 Standard

weights

by vectors

in Hilbert

spaces

We first investigate the regularity, the singularity and the standardness of the quasi-weights 02r defined by elements ~ of Hilbert space. Let Jr4 be a closed O*-algebra on 2) in ~H and put 2)*(M) =

D 2)(X*)and 2)**(A~)= XEM

D

D((X*[D*(M))*).

XCM

Suppose ~ C 2)**(M) and put

02~(x)

--

(x**~ I(),

x e m.

Then 02~ is a positive linear functional on M . If ( c D* (M)\2)** (M), then 02~ is a linear functional on M , but it is not necessarily positive. If ( ~ 2)*(M), then 02r is not defined, and so we regard 02r as the quasi-weight on 7~(AA) as follows:

{

~Yt~ = { X e M

~ e 2 ) ( X t*) and X t * ( 9

02~(XtX) -----IIX**r

II2,

X 9 ~,.

This coincides the quasi-weights qo~ on P ( M ) induced by the generalized vector Ar We here investigate such quasi-weigCs 02r on :P(A4) in detail. We first investigate when a quasi-weight o2( can be extended to a weight. P r o p o s i t i o n 3.6.1. Let A4 be a commutative integrable O*-algebra on 7) in 7-/and ~ ~ 7-/\ 2). We put

3.6 Standard weights by vectors in Hilbert spaces

{

163

(~-~XtkXk ~l~) if ~ E D(y~XktXk)

k

co

if otherwise.

Then ~ is a weight on ~(~A) which is an extension of ~ such that 9I ~0 =_ { x E M ; ~ ( X t X )

< ~ } = {X e M ; ~ E ~ ( X * X ) } ,

91~ = { X E . M ; A X E 9 1o~ , VA C M } 910d~ "

P r o o f . Since A/I is commutative and integrable, it follows that ~ E

I:)(~-~X~Xk) if and only if there exists a sequence {~n} in 19 such that k

--* ~ and both {XkG~} and {X~Xk~} are Cauchy sequences in 7-/ for

~

each k if and only if ~ E 19(X~Xk) = : D ( X ; ~ )

for each k, and then

~ _ x t x k ~ = y~X;Xk~, which implies that ~ is a weight on T'(M). It k

k

is easy to show that r

= 91~ and ~ is an extension of a;~.

E x a m p l e 3.6.2. Let H be a positive self-adjoint unbounded operator in T/,/9~176 =- N :D(Hn) and Ho - H[lg~(H). nEN

(1) The polynomial algebra M - P(H0) is a commutative integrable O*algebra on D~ in 7-/and the following statements hold: (i) If ~ r D(H2), then 9 1w~ o = C I and 91~ = 91~, = {0}. (ii) If ~ E :D(H 2n) \ :D(H 2n+2) (n E N), then 0

91~ = {p(Ho) ; p is a polynomial with the degree _< n}, 9 l ~ = 9I~, = {0}.

(2) Let A//be a commutative EW*-algebra on/9~ containing H0 and E 7-/\ :D~(H). Suppose ~ is a cyclic vector for (A/Yw)'. Then r t* ~ is dense in/:)~176 (H) [tr As seen in Example 3.6.2, (1), there are the following cases: (i) Even if ~ ~ 0, 91~e = {0}, that is, w~ = 0. (ii) Even i f ~ 0 , (~)q=~=0. It is meaningless to consider such (quasi-)weights. We investigate the regularity and the singularity of quasi-weights w~. For. the singularity of w4 we have the following

164

3. Standard weights on O*-algebras P r o p o s i t i o n 3.6.3. Let ~ r 79"(M). Suppose that M'w = C I , 91!. :D is

dense in 7-/and 9Jlt*~ is dense in :D[tz4]. Then w~ is singular. P r o o f . Since f J lt ~9 is dense in 7?Its], 7r~..~(M) is unitarily equivalent to AJ, that is, there exists a unitary operator U of ~ onto 7-/ such that U . ~ ( X ) = Xt*~ for all X 9 f 3 ~ and UTr~(A)U* = A for all A 9 M . Take an arbitrary K 9 T('~)'e. Then there is a constant a E C such that aXt*~ = x t * u x ' ( K ) for all X 9 ~ . Since r is dense in ~ , we have a~ = U)((K) 9 D*(.M), and so a = 0. Hence K = 0, which implies by Lemma 3.2.3 that, ~ is singular. Since s (79) satisfies all conditions of Proposition 3.6.3, we have the following C o r o l l a r y 3.6.4. For every ~ 9 7-i \ :D*(s weight on p ( g t (:D)).

w~ is a singular quasi-

P r o p o s i t i o n 3.6.5. Let M be a self-adjoint O*-algebra on 79 in 7-/and E 7-{ \ 79. Suppose r is dense in 79[t~] and put

F~ = Proj C~7-/, Then the following statements hold: (1) w~ c w(~ + w~s , co(. is a regular quasi-weight on P(Ad) and w~ is a singular quasi-weight on P(Ad). Further, 7r~. (resp. 7 r ~ ) is unitarily equivalent to 7r~) (resp. 7r~s,), where cz~~) is the regular part of w( and w~) is the singular part of w~ in Theorem 3.2.7. (2) w~ is singular if and only if Cg = {0} if and only if ~ = 0, and w~ is regular if and only if C~ is a nondegenerate *-subalgebra of M~w if and only if~

= O.

P r o o f . (1) By Theorem 3.2.7, the quasi-weight co~ on P(Ad) is decomposed into w( = co~~) + co~s), where a;~~) is a regular quasi-weight on 7~(M) and w~s) is a singular quasi-weight on 7~(fl/l) with 9l~.) = r

~) = r

defined by

w~")(XtX) =l[ P~XI*~ II2, co~)(Xtx) =ll (I - P~)xt*4 ]l2,

x 9 r

We have the relation that the quasi-weights ' ~ . and w~~) are equivalent (w~. a;~~) ), that is, 7r~e. and 7r~.) are unitarily equivalent. In fact, it is clear that

3.6 Standard weights by vectors in Hilbert spaces w~~) C w~, that is, ~ll ~ , ( = ~II~r c .r all X 9 r

165

and w ~ ) ( X t X ) = w4~(XtX) for

~/, and so 7r~} C 7r~. unitarily and 7r~/ is self-adjoint. Hence

7r~r is unitarily equivalent to 7 r ~ Similarly, we have wfr

~ w4. Thus wr is

a regular quasi-weight on 7)(M) and wr is a singular quasi-weight on 7)(Ad) and w~ = w~ + w~ on 7~(~5l~). (2) This follows from Theorem 3.2.5. Hence, we call ~ and ~ the regular part and the singular part of ~, respectively. C o r o l l a r y 3.6.6. Let H be a positive self-adjoint unbounded operator in ~ , / M an O*-algebra on Z)oe(H) containing { f ( H ) [7?oe(H); f is a measurable function on ]R+ such that ]f(t)] < p(t), t c l~+ for some polynomial p}, and ~ c 7-/\ D~ Suppose r t* is dense in 7-/. Then the following statemerits hold: (1) Suppose Jt/Vw = C I . Then w~ is singular. (2) Suppose 3/1 is commutative. Then w~ is regular. P r o o f . It is easily shown that Ad is self-adjoint and 91t*~ is dense in :Doe(H)[t~] using the spectral resolution of H. (1) Since M~w = C I , we have C~ = {0} and hence by Theorem 3.6.5 w~ is singular. (2) Let H =

/5

t dF,H(t) be the spectral resolution of/-/. Since 3,1 is

commutative, we have {EH (n); n E N} C C~, and hence C~ is a. nondegenerate *-subalgebra of .Ad~. Therefore, by Theorem 3.6.5 co~ is regular. E x a m p l e 3.6.7. Let .4 be the unbounded CCR-algebra for one degree of freedom and 7r0 the SchrSdinger representation of .4. Then 7r0(.4 ) is a self-adjoint O*-algebra on S ( ~ ) satisfying 7r0(.4)~w = C I . Let Ad be the O*-algebra on S(]I{) generated by 7r0(.4) and { f ( N ) ; f is a real-valued continuous function on ~ + such that ]f(t)l _< p(t) (t e ~ + ) for some polynomial p}, and ~ c L2(~) \ 8 ( ~ ) . Suppose f f [ ~ is dense in L2(~). Then 8(]~) = N :D(Nk), where N is the number operator, and jr4 and ~ satisfy k=l

all of the conditions of Corollary 3.6.6, (1). Hence cz~ is singular. We investigate the standardness of the quasi-weights w~. The standardness of the quasi-weight w~ and the generalized vector A4 is equivalent, and so by Proposition 2.4.1 we have the following

166

3. Standard weights on O*-algebras

P r o p o s i t i o n 3.6.8. Suppose (S)I

{YXt*~ 9 X , Y E ofit M r

(S)2

Cr is dense in 7-/.

is total in 7-/,

Then oJr is a faithful regular quasi-weight on ~ ( M ) and ~ is a cyclic and It separating vector for the von Neumann algebra ( M ~ ) ' and denote by z~r the modular operator for the left Hilbert algebra (M~)t~. Further, we have the following results: (1) ,~r is quasi-standard if and only if the above (S)x, (S)2 and the following condition (S)3 hold: --ltit

/ ~ r :D C / 9 for each t E ~ . (2) '~r is standard if and only if the above conditions (S)1, (S)2 and (S)3 and the following condition (S)4 hold: (8)3

(S)4

ttit

A ~MA

tl-it

~

=M

for e a c h t E l ~ .

We next give some examples of regular quasi-weights , singular quasiweights and standard quasi-weights defined in the Hilbert space of HilbertSchmidt operators, which are important for the quantum physics. Let A/I be a self-adjoint O*-algebra on :D in 7-/such that Ad'w = C I . We put

(

~2(Y~) = {T E 7 - / | ; TT-/C 19 and X T E 7"t| n ( X ) T = X T , X r M/I, T E G2(A/I).

V x E MM}, m

Then rr is a selfoadjoint representation of dvt on ~2(A//) in 7-I | 7-/ such that 7r(~d)~w -- ~r'(B(1-/)) and (Tr(M)~w)' = 7r"(B(7-/)), where ~r'(A)T = T A and r&(A)T = A T for A E B(7-/) and T E 7-/| 7-/ (Lemma 2.4.14). Let f2 E 7-/@ 7-I a n d / 2 > 0. We define the quasi-weight ~ on 5o(M) by

= ( x e M; f i x )

e

},

9 o , ( X t X ) = t r ( X t ~Q)*(Xt I2) = w, 0r(X)tTr(X)),

X Er

.

Then we have

~(xtx)

= ~(~(x)*~(x)),

x e 9/~.

Hence, by Proposition 3.6.5 we have the following P r o p o s i t i o n 3.6.9, Suppose r following statements hold:

t*

is dense in G2(M)[G(M)]. Then the

3.6 Standard weights by vectors in Hilbert spaces (1) ~

is singular if and only if

Ce -- {Tr'(K); K E B(7-/) and s (2) ~

167

f2K* E G 2 ( M ) } = {0}.

is regular if and only if C9 is a nondegenerate .-subalgebra of

(3) f2 is decomposed into f2 = ~ + Y2~, where f2~ is the regular part. of f2 and f2s is the singular part of ~. Hence, ~ is a regular quasi-weight on ;~ ~ is a singular quasi-weight on 7)(Ad) and ~ = ~ a + ~ on

By Theorem 2.4.18 we have the following P r o p o s i t i o n 3.6.10. Suppose there exists a dense subspace 8 in 79[tz4] such that (i) M D { ~ N ~ ; ~,r/Eg}, (ii) f2g C 79 and f28 is dense in 7-/. (iii) s - t is densely defined, (iv) f2it79 c 79 for each t E ]~. Then ~ n is a quasi-standard quasi-weight on P(Ad). Further, suppose ( V ) ~itM~-it = J ~ for each t E l~. Then ~ is a standard quasi-weight on P(Ad). Let f2 be a positive self-adjoint unbounded operator in 7-/. We define a quasi-weight on P ( A J ) by t*

{x =

x

e

By Theorem 2.4.23 we have the following P r o p o s i t i o n 3.6.11. Suppose there exists a subspace g of 79N79(Z?) such that (i) g is dense in 79[t~], (ii) M D {~ | 7; ~, r/E 8}, (iii) t2g C D and [28 is dense in 7-/, (iv) 52-1 is densely defined and 79 A 79(f2 -1) is a core for f2 -1, (v) Suppose s c 79 for all t E ]R. Then ~ is a quasi-standard quasi-weight on P(M). Further, suppose (vi) Suppose ~itj~-~-it = .1~ for all t E ]~. Then ~ a is a standard quasi-weight on P(Ad). Notes 3.1, 3.2. The works of these sections are due to Inoue-Ogi [1]. 3.3, 3.4. These are due to Inoue-Karwowski-Ogi [1].

168

3. Standard weights on O*-algebras

3.5. Definition 3.5.1 ~ Proposition 3.5.12 in the first part of this section are generalization of the results obtained in Inoue [10] for positive linear functionals to (quasi-)weights. Examples 3.5.13, 3.5.14, 3.5.15 in the larst part are due to w in Inoue [10]. 3.6. The works of this section are due to Inoue-Ogi [1].

4. Physical Applications

In this chapter we apply the Tomita-Takesaki theory in O*-algebras studied in Chapter II, III to quantum statistical mechanics and the Wightman quantum field theory. Given a physical system, the first task of quantum statistical mechanics is to try and construct equilibrium states of the system. In the traditional algebraic formulation, the system is characterized by the algebra A of its observables, usually taken as an algebra of bounded operators. The latter in turn may be obtained by applying the well-known GNS construction defined by a state on some abstract *-algebra. Then the standard treatment of the basic problem consists in applying to A the Tomita-Takesaki theory of modular automorphisms, which yields states on A that satisfy the KMS condition. The latter is a characteristic of equilibrium: Gibbs states satisfy the KMS condition. For finite systems, the converse is also true, whereas, for infinite systems, the KMS condition characterizes only the local thermodynamieal stability. For many models, the equality between the sets of KMS states and Gibbs equilibrium states persists also after the thermodynamical limit. This fact suggests the general interpretation of KMS states as equilibrium states in the Gibbs formulation, at least if the system is described as a C*- or W*-dynamical system. However, there are systems for which the standard approach fails, typically spin systems with long range interactions such as the BCS model of superconductivity and its relatives. For such systems, indeed, the thermodynamic limit does not converge in any norm topology. An elegant way of circumventing the difficulty consists in taking for observable algebra, namely an O*-algebra on some dense invariant domain IP in the Hilbert space ~. The same technique may be applied when unbounded observables are considered, such as position and momentum in the CCR algebra (then 7-{ = L2(]l~ 3) and IP is Schwartz space S(1~3)). Since the examples presented in this chapter are of that nature, we will adopt the O*-approaeh. This means that the observables of the system (either local or in the thermodynamical limit) are represented by the elements of an O*-algebra Azl. Thus we are facing the same question as before: how does one construct KMS states on an O*-algebra? It is important to consider a Tomita-Takesaki theory in O*-algebras. The Tomita-Takesaki theory may be derived for }~I if, among other conditions, A/I possesses a strongly cyclic vector Go E IP. In that case,

170

4. Physical Applications

one obtains states on A// (in the usual sense) that satisfy the KMS condition. However, the existence of the cyclic vector is a rather restrictive condition as seen in late physical examples, that we want to avoid. An interesting possibility is to consider it as a generalized vector. The main advantage of this interpretation is that generalized vectors (vectors in ~ \ D are only the simplest case) are also closely related to the concept of weights and quasi-weights on O*-algebras. For a system whose observable algebra is assumed to be an O*-algebra A/l, we will be able to show the existence of quasi-weights on J~4 satisfying the KMS condition. In view of the discussion above, in a generalized setup where physical states of the system would be represented by quasi-weights on the algebra of obsevables, it is plausible that these KMS quasi-weights would represent equilibrium states. This chapter is organized as follows: In Section 4.1 we consider the quantum moment problem for states on an O*-algebras. Many, important examples of states f in quantum physics are trace functionals, that is, they are of the form f ( X ) = t r T X with a certain trace operator T E G1 (A/l). Hence we study the following quantum moment problem: Under what conditions is every strongly positive linear functional on an O*-algebra a trace functional7 The results of this section are applied to Section 4.2, 4.3 and 4.4. In Section 4.2 we extend some results obtained for states in Section 4.1 to (quasi-) weights. In Section 4.3 we first study standard systems in unbounded CCRalgebras in one degree of freedom. Let A be a *-algebra generated by identity i and hermitian elements p and q satisfying the Heisenberg commutation relation: [p, q] = - i l . For a self-adjoint representation rr of A satisfying rr(p) and rr(q) are essentially self-adjoint, Von Neumann [1] had the result that the strongly continous unitary groups U(s) = eis~(p) and V(t) = e it~(q) satisfy the Weyl commutation relation: U(s)V(t) = eistV(t)U(s) for each s, t E I~ if and only if 7r is a direct sum @ rr~ of ,-representations 7r~ which ~E10

are unitary equivalent to the SehrSdinger representation rr0. We call such a ,-representation the Weyl representation of the cardinal I0. Powers [1] defined the notion of strong positivity of self-adjoint representations of A and using it he characterized the Weyl representations. Here we introduce the Powers results. Further, we show that a Weyl representation of countable cardinal is unitarily equivalent to the self-adjoint representation rr| of .4 defined by D(rc| = 8(][{) | L2(1~) and rr| = rco(a)T for a E A and OO

T E 7)(rr|

and ~n = ~ e

_,~

= f,~|

03 > 0) is a standard vector for

•--0

rr|

where {fn},~=0,1,... is an ONB in L2(I~) consisting of the normalized

Hermite functions. We consider more general g2{~} =- ~

c~nfn | ~

(an >

n=0

0, n = 0, 1, 2 , - . . ) . Let ~4 be an O*-algebra on 8(]~) generated by rr0(A)

4.1 Quantum moment problem I

171

m

and f0 | f0. Then the positive self-adjoint operator ~ { ~ } defines a quasistandard generalized vector Ag(~,~ for the self-adjoint O*-algebra 7c(~4) on S(][~) | L2(][~) defined by 7c(X)T = X T for X 9 jk4 and T 9 8(]~) | L2(]~). Further As~o~) is standard if and only if an = r 9 N U {0} for some 9 ]~. We next give a standard generalized vector and a modular generalized vector in an interacting Boson model. In Section 4.4 we study standard systems in the BCS-Bogoluvov model. In case of the BCS model, a rigorous algebraic description, in the quasi-spin formulation, was given long ago by Thirring-Wehrl [1,2]. Using this formulation Lassner [2,3] solved the problem of the thermodynamicM limit discussed above by constructing a rather complicated topological quasi *-algebra. We show here that the existence of KMS quasi-weights may be obtained with a much simpler O*-Mgebra, provided one uses appropriate generalized vectors, as described in Chapter II, III. In Section 4.5 we study standard systems in the Wightman quantum field theory. The general theory of quantum fields has been developed along two main lines: One is based on the Wightman axioms and makes use of unbounded field operators, and the other is the theory of local nets of bounded observables initiated by Haag-Kastler [1] and Araki [1]. Here we characterize the passage from a Wightman field to a local net of von Neumann algebras by the existence of standard systems obtained from the right wedge-region in Minkowski space.

4.1 Q u a n t u m m o m e n t problem I In this section we consider under what conditions every strongly positive linear functional on an O*-Mgebra is a trace functional. This problem is closely related to the so-called problem of moments, and so we call it the quantum moment problem. Throughout this section let Yk4 be a closed O*-algebra on :D in T/ with the identity operator I. We denote by ~ ( ~ ) the *-invariant subspace of B ( ~ ) consisting of finite dimensional operators on ~ , and by A/t~:(7~) the linear span of M and ~-(T/)IT) regarded as operators on :D. We first investigate under which conditions a continuous linear functional f on A/l is a trace functional, that is, f ( X ) = tr X T , X 9 .M for some T 9 ~1(./~). We prepare the following lemma: L e m m a 4.1.1. Suppose f is a strongly positive linear functional on .s Then there exists an element T of 1 G ( M ) + such that f ( A ) = tr T A for all A 9 if(?-/). P r o o f . Since x | 5 _< []x][2I for each x E H, it follows from the strong positivity o f f that f(x| <_ [[x[]2f(I) for all x 9 ~ , so that by the Schwartz inequality

172

4. Physical Applications

If(z | 9) 12 _< f(i)f((x | 9)t(x | 9)) = f ( I ) l l x J l 2 f @ | 9)

S f(I)211<1211yll 2 for all x, y E ~ . Hence, (x, y) --~ f ( x | 9) is a continuous positive sesquilinear form on 7-{ @ ~ , and so there exists a positive b o u n d e d o p e r a t o r T on 7-{ such that

f ( x 6) 9) = (Tzly)

(4.1.1)

for all x, y E H. Since

I(X(ITy)I 2

= I/(X(~ | <_ f ( X X t ) f ( ( (

@ 9 ) t ( ( | .Y))

<_ f(xx*)lJ~ll2f(y | 9) < f(XXt)f(I)]lyjl211~]l 2 for all X E 2k//,~ E 2) and y E ~ , we have T ~ C 79"(A//), and so it follows from the closed graph theorem t h a t X * T E B ( ~ ) for all X E 3/l. We show t h a t X * T E G1(7-/) for all X E AJ. Let {x~}~eF be an o r t h o n o r m a l subset in 7-I a n d {il, i 2 , - " , in} any finite subset F . For each k E {1, 2 , - - - , n} there exists ~*k E 79 such t h a t 1

(4.1.2)

Ilx~k - ~11 < 2kllX,ZllFurther, we take a n u m b e r ak E C such t h a t ~ k ( x * r ~ i k l 4 i k ) . T h e n we have

n

s

II ~--} ~ k ( ~ k | k=l

~ k ( x ~ | ~2d)ll <--

-

I~kl --

1 and

I(x*r~l~)l

=

s II,~k | ( ~ - ~-r | ~7;dll k=l

k=l

-< s

II(,k - x,,~ll(ll,~,~ II + 1)

k=l

1

< <

k=l

1

2kllX,Ti I (2 + 2klIX*TII) 7

- 3IIX*TII ' a n d since {xik } is an o r t h o n o r m a l system, it follows t h a t

k=l

k=l

7 < (--+1).

311X*TII

(4.1.3)

4.1 Quantum moment problem I

173

We put n

A = ~ ak~i~ | Xt~ik. k=l

By (4.1.1) and

(4.1.3)

we have

n

Eak(X*Z~kl~/k)

f(A)

=

= If(A) l

k=l n

= If((~--~. c~kCik| ~T~)X)I k=l n

~q,

< f((~-~, ak~ik | ~ ) ( Z k=l

ak~ik | ~)t)l/2f(XtX) 1/2

k=l

n

-< II~ ~k~k

|

~ikllf(I)l/2f(xfx) 1/2

k=l

7

<- (SllX*TII + 1)f(I)l/2f(x*x)l/2 Hence we have

~-~ l(X*Tx~lxi~)l k=l

= ~

I(x*r(xik - ~ ) l x i k ) + (x*r~k[xi~ - ~ ) + (X*Tr162

k=l n

1

= '~.(~

+ 2-Vzf_ 1 + [(X*f~k[~)l)

k=l

_< 3 + ~-:~ ak(X*T~ I~k) k=l

7

< 3 + (3llx*zll

+

1)f(I)l/2f(XtX)l/2'

which implies

~_~ 7 + 1)f(I)l/2f(XtX)l/2" ~F I(Xr* x~lxdl ~ 3 + (311X*TII Hence we have X*T E G1 (7-/). Thus we have T c l ~ ( A d ) , and by (4.1.1) = tr AT for each A c .T(7-/). This completes the proof.

f(A)

We s t u d y the trace representation of continuous linear functionals on fld.

174

4. Physical Applications

T h e o r e m 4.1.2. Suppose f is a continuous linear functional on Ad[wr Then there exists an element T of GI(A4) such that f ( X ) = tr X T for all X E A/I. P r o o f . Since f is re-continuous, there is a relatively compact subset ~It of ~D[t~] such that If(X)I _< sup I(X~I~)I,

X E M.

(4.1.4)

Without loss of generality we may assume that ~Y~[t~] is a compact Hausdorff space. Let C ( ~ ) be the C*-algebra of all continuous functions on the compact Hausdorff space ~ . For each X E M we define a continuous function ~ x on ~ [ t ~ ] by ~ x ( ~ ) = (X~l~),

~ E 9Jr

and denote by .7- the subspace of C ( ~ ) generated by {~x; X E M } . By (4.1.4) we can define a continuous linear functional F on .7- by

F(~x) = f(X),

X E Ad.

(4.1.5)

By the Hahn-Banach theorem F can be extended to a continuous linear functional on the C*-algebra C ( ~ ) and also denoted by F, and F can be written as a linear combination F = F1 - F2 + i(F3 - F4) of positive linear functionals F I , . . . , F4 on C(~Y~). For any B = E xk | ~ E .7-(7-/) we define k a c o n t i n u o u s f u n c t i o n ~ B o n ~}rj~ by

a n d s o {~B; B E 9v(~)} C C(~CSt). Hence we have {~A; A E M~-(~)} C C(~[)~). Therefore Fj (j = 1 , . . . , 4) induces a strongly positive linear functional fj on A/t~-(~) by

f j ( A ) = Fj(~A),

A E My(x).

By the Riesz representation theorem there exists a positive Borel m e a s u r e p j on the compact space ~ such that Fj(~) = j,~ ~(~)dpj(~) for all ~ E C ( ~ ) . In particular, for the functions ~ A , A E M~-(7-t), it means that

f j ( d ) = Fj(~A) = f 'qd9~

(d~l~)dpj(~).

(4.1.6)

4.1 Quantum moment problem I On the other hand, by Lemma 4.1.1 there is an element Tj of that

f~(B) = tr TjB,

B 9 5r(7-/).

Take arbitrary U 9 79(X*) and z 9 7-{. Since ~x| (4.1.6) and (4.1.7) that

175

1~(./~)+ such (4.1.7)

9 C(ffJ~), it follows from

I(TjxlX%)l = I tr Tj(x | X*r/)l = Ifj(z | x*~)l = I f,(XCl~)(zlOdmj(()l

<_ II~ltllz[I _/i~ IIx~llll~lld#~(~)

which implies Tj'H c 79. Therefore, Tj 9 GI(A4)+. We show that f j ( X ) = tr T j X for all X 9 AJ. Let Tj = E

tn(n | "~ be a canonical representation

n

of Tj. Since Tj 9 GI(Ad)+, we have t,~ _> 0 for each n 9 N and {(~} c 79, and further since 9J[ is a compact subset of 79[t~], there exists an orthonormal set { ~ } in 7-/such that {~,~} c 79 and 9JI U {(n} c closed linear span of {~n}. Since OG

I(X~:l~,d(~,~lOI _< IIX~[lll~ll _< ((I + xtx),,l~) n:l

and ~gi+XtX(~) : ((I + XtX)~[~) is pj-integrable, it follows from the Lebesgue theorem that

176

4. Physical Applications oo

trTjX=

trXTj

~(Tj~IXt~.) n=l

tr Td(~n @ Xf~n)

: k n

1 oo

= ~'~fj(~n@Xf~.n)

(by 4.1.7)

n=l

(by 4.1.6) n

1

n=l oo

n

1

= f.~ (X~l~)d~,~(~)

(by 4.1.6)

= fj(X). Thus we have rd E

~l(.Ad)+ and f j ( X ) = tr T j X = tr XTj,

X r .M.

(4.1.8)

Here we put T = T1 - T 2 + i ( T 3 - T 4 ) . Then it follows from (4.1.8) and (4.1.5) that T r G I ( M ) and f ( X ) = tr T X = tr X T for each X r This completes the proof. C o r o l l a r y 4.1.3. Suppose Z)[tM] is a Fr~chet space and f is a linear functional on Ad. The following statements are equivalent: (i) f is continuous on Ad[Tc]. (ii) f = fT for some T E G I ( A J ) . P r o o f . This follows from Proposition 1.9.11 and Theorem 4.1.2. C o r o l l a r y 4.1.4. Suppose D[tz4] is a semi-Montel space and f is a continuous linear functional on Ad[r~,]. Then f = fT for some T E O 1 ( M ) . P r o o f . In a complete semi-Montel space, a set is precompact if and only if it is bounded. Hence we have r~ = re, and so by Theorem 4.1.2 f = fT for some T E ~ 1 ( . / ~ ) . Example 4.1.5. Let A4 be the O*-algebra generated by the position and momentum operators Qj, Pj (3 = i, 2,... ,n) on the Schwartz space 8(]~n). Then every linear functional f on A4 is a trace functional, that is, f -- fr

4.1 Quantum moment problem I

177

for some T E GI(Ad). In fact, since the graph topology tz4 is the usual topology of the space $(~{n) and the Schwartz space is a Fr6chet Montel space, it follows that 79[t~4] is a Frfichet Montel space. Further, it follows from Example 4.5.7 in Schmiidgen [21] that the uniform topology r~ on Ad equals the strongest locally convex topology "rst on jkd, which implies that every linear funtional f on 3/l is r~-continuous, so that by Corollary 4.1.4 f = f T for some T E @l(2Ul). We consider the quantum moment problem: Under what conditions is every strongly positive linear functional f on an O*-algebra Ad a trace functional f T , T E G I ( A d ) + ? If this is true, then an O*-algebra jkl is said to be QMP-solvable .

We first consider an ordered *-vector space 12(79,79*). Let 79 be a dense subspace of a Hilbert space 7-/. We denote by 79t (or 1;(79)) the algebraic conjugate dual of 79, that is, the set of all conjugate linear functionals on 79. The set 79* is a vector space under the following operations:

where < v,~ > is the value of v E 79t at ~ E 79. Tomita [2] has called an element of 79t an u n b o u n d e d v e c t o r in 7-(. We denote by 12(79, 79t) the set of all linear maps from 79 to 79t. Then 12(79,79t) is a ,-vector space under the usual operations: S + T, AT and the involution T --+ T t (< Tt~, r/ > = < Tr/.~ >, ~, r/ E 79). Furthermore, 12(79,79*)h - {T C 12(79, 79*); T* = T} is an ordered set under the order S _< T (< S~,~ >_<< T~,~ >, v~ E 79). We remark that any linear operator X defined on 79 is regarded as an element of 12(79,79*) by < X~,r/ > = (X~lrl) , ~,r/ E 79. In particular, 12'(79) and B ( ~ ) are regarded as ordered *-subspaces of 12(79,79t). For any element X , Y c 12*(79) and A E B(?-t) we define a multiplication y t o A o X by < ( y t o A o X)~, n > = (AX~In),

~, ~ e D.

Then we have y t o A o X e 12(79,Dr), VA E B('H), Vx, Y E s and X t o A o X < I I A I I X t X , V A * = A ~ g(J-t),Vx e s

(4.1.9)

T h e o r e m 4.1.6. Suppose d~4 is self-adjoint and there exists an element C of d~4 such that (I + C ' C ) -1 is a compact operator on 7-/. Then 3// is QMP-solvable. P r o o f . Let f be a strongly positive linear functional on A4. Let 120 denote the subspace of s generated by {X t o A o X; X E d~'l, A E /3(/-t)+}. By (4.1.9) 3// is cofinal in the ordered ,-vector space s ~ ~4 + s and so

178

4. Physical Applications

the functional f can be extended to a strongly positive linear functional on s and denote it by the same f . By L e m m a 4.1.1 and the self-adjointness of AJ there exists an element T of ~ I ( A J ) + such that

f(d) = tr TA = tr AT

(4.1.10)

for all A E 5c(7-/). We show that f(X) = fT(X) for all X E Ad. Let {An} be the eigenvalues of the compact operator (I+C* C) - 1 and { ~ } an orthonormal basis of the corresponding eigenveetors. Without loss of generality we m a y assume that An > I~+1 for all n E N. Take an arbitrary X E 3,l. We put n

X n

: E((I r @ d)X, k-1

";%E N .

Then Xn is a finite dimensional operator, but not bounded in general. Hence, we regard X~ ~ an element of s Since

x : ~ ~(~k | C~)(I + c+c)x, k:l

In = s

~k,((k @ ~ ) ( I

-4-C t C ) X ,

k=l we have

I((X -x~)(l()l

--

I E

Ik((I + CtC)X~l~k)(~kl~)l

k=n+l

( sup Ak){ Y~ I((I 4- C*C)X~l~k)ll(~klOI}

k=n+l oo (x? ~n-I-1 {~--~ I(( I -I- c*c)x~l~k)12}& {~--~I(~klOI2}89 k:l k=l k>n+l

~,,+~11(I + C*C)X~llll~ll _< ~n+,{ll(I + CtC)X~II 2 + I1~112}

--

= )'n+l((I

+

Xt(I + ctc)2x)~J~)

for each ~ E 79 and n E N, which implies that (X - X . ) I < I,~+1(I + Xt(I + CtC)2X),

(X -- Xn)2 <_ )~n+l(I 4- Xt(I + CtC)2X), where ( X - X~)I and (X - Xn)2 are the real part and the imaginary part of ( X - X n ) E s respectively. By the strong positivity of f we have

If((X - x , d J l _< A,~+lf(I + Xt(I + CtC)2X), [ f ( ( X - X,~)2)I _< An+lf(I + Xt(I + CtC)2X),

4.1 Quantum moment problem I

179

so that

If(X) - f(Xn)l <_ x/2A,~+af(I + x t ( I + c t c ) 2 X )

(4.1.11)

for each n E N. Since T E GI(A/[)+, it follows from Lemma 1.9.7 that fT is a strongly positive linear functional on 34. Hence, fT can similarly be extended to a strongly positive linear functional on s and denoted by the same fT; and so similarly to (4.1.11) we have IfT(X) -- fT(Xn)l _< V'2X,~+lfT(I + x t ( I + c t c ) U x )

(4.1.12)

for each n E N. By (4.1.10), (4.1.11) and (4.1.12) we have

If(X) - fT(X)[ < v"2A.+l{f(I + x t ( I + c t c ) 2 x ) + fT(I + x t ( I + c t c ) 2 x ) } for each n E N. Since lim An+l = 0, we have f ( X ) = fT(X). This completes the proof. T h e o r e m 4.1.7. Suppose T)[t~] is a Fr6chet Montel space. Then A/[ is QMP-solvable and every strongly positive linear functional is continuous with respect to the topology T~. P r o o f . Let f be a strongly positive linear functional on Ad. Since T)[t~] is a Fr@chet space, it follows from the closed graph theorem that t ~ = tLt(~) ), which implies .A/[ h is a cofinal in /:t(T))h. Hence f can be extended to a strongly positive linear functional on s and denote it by the same f . It follows from Corollary 4.4.3 in Schmiidgen [21] that Tu = " l o r d on ~t(~)), which implies that f is continuous on s By Corollary 4.1.4 there exists an element T of G I ( / : t ( D ) ) such that f ( X ) -- tr X T for all X E s Then since 0 <_ f(~ | ~) = tr (~ | ~)T -- (T~I~) for all ~ E D, we have T > 0. This completes the proof. C o r o l l a r y 4.1.8. Suppose A4 is a countably generated closed O*-algebra containing the restriction N of the inverse of a compact operator. Then 34 is QMP-solvable. P r o o f . Since D [ t ~ ] is a Fr@chet Montel space, it follows from Theorem 4.1.7 that ]~4 is QMP-solvable. E x a m p l e 4.1.9. The O*-algebra A/[ generated by the position and momentum operators Qk, Pk, k = 1, 2,-.- , n, on the Schwartz space S(]~ n) is QMP-solvable. In fact, the graph topology t ~ is the usual topology of the space 8 ( ~ n) and the Schwartz space S ( ~ n) is a Fr@chet Montel space. Hence it follows from Corollary 4.1.8 that A/I is QMP-solvable.

180

4. Physical Applications

We remark that every positive linear functionM on 54 is not a trace functional. For example, the positive linear functional f in Example 1.9.2 is not a trace functional because O(3

tr((AtA - I ) ( A t A - 2I)T) = ~ ( n

- 1)(n - 2 ) ( ~ l r ~ )

___ 0

n=0

for each T E G l ( f l d ) + . T h e o r e m 4.1.10. Suppose :D[tct(~)] is a Fr6chet space. Then the following statements are equivalent: (i) 29[tct(~)] is a Montel space. (ii) s is QMP-solvable. (iii) Every O*-algebra Ad on 29 with t ~ = tct(z~) is QMP-solvable. P r o o f . (i) ~ (iii) This follows from Theorem 4.1.7. (iii) ~ (ii) This is trivial. (ii) ~ (i) Assume that 29[tLt(z~)] is not a Montel space. Then there exists a bounded set ~ in 29[tot(z))] which is not relatively compact in 29[tLt(z~)]. Hence ~l~ contains a sequence {~n}~eN which has no cluster point in 29[tot(z))]. Since {Cn} is bounded in the Hilbert space norm, it has a weakly convergent subsequence in 7-{. For simplicity suppose that {~n} is weakly convergent to ~0 E "H. The set K - { ~ ; n E N} endowed with the induced topology by 29[tzt(z~)] is a Tychonoff space. Hence there exists the Stone-Czech compactfication ~ ( K ) of K (Kelly [1], p.153). For A E s the function gA(~) = (A~l~) is a continuous bounded function on the topological s p a c e / ( , and so it can be extended uniquely to a continuous function on ~ ( K ) and denote it by the same gA. Since K is not compact, there is an element ~0 of /3(K) \ K. We define a linear functional f on s by f(A)

=

gA(~O),

A E

s

Since K is dense in/3(K), ~0 is a cluster point of K and since gA is continuous on ~ ( K ) , we have f ( A ) = limoo(A~njC,~),

A Es

Hence f is a strongly positive linear functional on s QMP-solvable, there exists an element T of G I ( s f ( A ) = tr AT, By (4.1.13) and (4.1.14) we have

A Es

(4.1.13) Since s such that

is

(4.1.14)

4.1 Quantum moment problem I (T~lrl)=

trT(~|

181

f((| =

lim ( ( ~ | ?~---~ o ~

=

lim (Cnlrl)(~l~n)

-- ((r

| C0)~l~)

for all ~,g E ~D, and so T = ~0 N~00. Since T E G I ( s Further, we have lim (ACtiOn) = f ( A ) = tr A(r

n ---* o(3

we have C0 E Z).

| Co)

= (Ar162

for all A E Z2t (Z?), which implies t h a t lira IIAr

- Ar

= lirnoo{(AtACn]r = (AtAr162

- (r162

- (r162

-

(AfAr162 + IlAr 2}

- (AtAr162

+

IIAr

=0 for each A E s This m e a n s t h a t Co is a cluster point in ~D[tL~(z))]. This is a contradiction. This completes the proof. We finally consider trace representation of positive linear functionals. For the relation of the trace representability and the a - w e a k continuity of positive linear functionals on 3,4 we have the following P r o p o s i t i o n 4 . 1 . 1 1 . Let f be a positive linear functional on 3/l. Consider the following statements: (i) f is a trace functional on 3,t, that. is, f = fT for some T E G I ( A d ) + . oo

(ii) f = ~ w ~

for some {~n} E ~D~176

nzl

(iii) f is a - w e a k l y continuous. T h e n the following implications hold: (i) ~

(ii)

(iii). P r o o f . T h e equivalence of (i) and (ii) follows from L e m m a 1.9.4. T h e implication (ii) =~ (iii) is trivial. To consider when the above implication (iii) ~ (ii) holds, we first define an O*-algebra [34] of diagonal operators by

182

4. Physical Applications [x]{~k} = {x~k},

x 9 M,{~k} 9 ~ ( M ) ;

[M] = {IX]; X 9 M } Then

we have the following

L e m m a 4.1.12. [.At] is an O*-algebra on ~ ( ~ 4 ) in 7-/~176 satisfying (i) [M/I] is closed if and only if A/I is closed; (ii) [A/l]'w is a v o n Neumann algebra if and only if A t " is a von Neumann algebra; (iii) [A/I]'w:D~ = ~P~(A/I) if and only if AJw:D' = ~; (iv) [M/I] is self-adjoint if and only if A4 is self-adjoint; (v) the topology T~ (resp. 7~, 7-~) on []~4] is the topology 7-~w (resp. 7-o~,~-*~) on .~.

T h e o r e m 4.1.13. Let M be a closed O*-algebra on T) in T/ such that .,Mw~D C ~D. Then the following statements hold: I

(1) Suppose 3d '-~ = .A,lw~ , . " Then f is a weakly continuous positive linear n

functional on .A4 if and only if f = E ~ k

for some {~k}k=l,...,n in ~D.

k=l

(2) Suppose M~-J~ = .M~,. Then f is a a-weakly continuous positive lin(2,O

ear functional on .M if and only if f = E a ~

for some {(k} 9 ~D~(-A4).

k=l

P r o o f . We first prove the statement (2). Suppose f is a a-weakly continuous positive linear functional on A/L Then there exists an element {~?k} of T ~ ( A / 0 such that

If(x)l _< I ~ ( x ~ l , 7 ~ ) l ,

XEA4,

k=l

so that

II~i(X)l[ ~ II[X](~}[I,

(4.1.15)

x e M.

We put

Co[X]{~} = ~ ( x ) ,

x 9 M. m

By (4.1.15) Co can be extended to a continuous linear map Co of [M]{~?k} into ?-/f. Since [M]'wD~(M) c lP~(A/I) and [M-]~: = [ M ] ~ by Lemma 4.1.12, it follows from Proposition 1.2.3 that [A/l]{~k} = ([M]'w)'{7?k}, which implies that the projection P from ~/~ to [A/I]{~?k} belongs to [M]'w. We now show

4.1 Quantum moment problem I m

m

183

!

C - (CoP)*(CoP) 9

[.M],,,.

(4.1.16)

Take arbitrary A 9 (M~w) ' and x, y 9 ~oo. Since [(-~w), /] = ([-M]'.,) lc

[MV' ....

--T*

= [All

",

there exists a net {X~ } in J~4 such that 1 ~ ~--~ I I X , A k - Aqkll 2 --- tim k=l

I l X ~ ( k - A*qkll 2 = 0 k=l

for each {(k} 9 D ~ ( A J ) . Since g x , P y 9 [M]{Ok}, there exists sequences {Yn} and {Zn} in M such that lira [Yn]{~k} = P r and lira [Zn]{~k} = Py. Then we have

(C[A]xly) = (CoP[A]z]CoPy) (Co[A]PxICoPy) = lim (Co[A][Y,,]{~k}ICo[Zn]{~Tk}) n---*oo =

-- n~L%l~(~[XoY,,]{,?k}lU00tZ,,]{,~k}) = n--,~liml~(,~(X,#r,)l~.~(Zn)) = lira lim(Af(Yn)lAf(X~Zn)) = lim lim(Co[Y,,]{~k}]Co[X~]t[Z,,]{~k}) n--+oo

c~

= lira (Co[Yn]{~k}lCo[d]*[Z,,]{~Tk}) n--40o

= (CoPx]Co[A]*Py) (Cxl[d]*y).

=

I oo Hence C 9 [ ( M ~ ) ' ] ' -- [M]~. Since [M]w:D (Ad) c T)~176 it follows that C 1/2 9 [M]~w and {~k} - C1/2{~k} 9 D~176 which implies that

f(X) =

(At-(X)lA.t-(I))

=

(C[X]{r/k}l{r/k})

= ([XlCl/2{rlk}lC~/2{rlk})

-- ~ ( x r

I~k)

k=l

for all X E M . The converse is trivial. The proof of the statement (1) is similar to (2). This completes the proof. We next consider when a positive linear functional f on M is represented asf=wr for s o m e r

184

4. Physical Applications Theorem

4.1.14. Suppose M is a closed O*-algebra on I9 in 7-f such

that 2tdw19 ' " . Then the following statements hold: C 19 and M % * = Mwo (1) If (Ad'w)' has a separating vector, then every a-weakly continuous positive linear functional f on 2t4 is represented as f = co~ for some ~ E 19. (2) If there is a vector ~0 in 19 which is strongly cyclic for ~4 and separating for (3/Vw)' , then every a-weakly continuous positive linear functional f on M is represented as f = w~s for a unique element ~f o f P g - P(M,),{oN19, # where P(M'w)'~o # is the natural positive cone associated with the left Hilbert algebra (Ad'w)'{0 (Bratteli-Robinson[1]). Further, f = ~/~e0 for some positive selfadjoint operator H affiliated with (M~w) ~ such that {0 C 19(H) and H{0 E 19. P r o o f . (1) By Proposition 4.1.13, (2) there exists an element {~k} of

19~176 such that f = ~-~aa~k. We put k=l

E1 = proj[Ad]{~k},

E2 =

,

It is proved in the same way as the projection P in the proof of Proposition 4.1.13 belongs to [M]'w that E1 E [M]'w. It is clear that E2 E [M]'w and Z(E1) <_ Z(E2) = I, where Z(Ei) is the central support of Ei, i = 1, 2. Further, we have {~k} C E17Y~ and ([

]w)

l{~k}

[Ad]{~k}

E I ~ ~176

Let % be a separating vector for (M~w) ' and put

Then

we have

~o C E27Y ~ and E2[MI~E2r~0 =

= E27-/~

By Dixmier [2] (Part III, Chap. I, Lemma 4) there exists an operator [A/[]~ such that V*V = El and VV* < E2. Then we have

V in

VV*V{~k } z V E l ( ~ k } ~_ V(~k}, and so V{~k} E E 2 ~ ~ . Hence it follows from [M]w19 !

oo

(M) C 19~(M) that

4.1 Quantum moment problem I

185

Thus we have f ( X ) = ([X]{~k}l{4k})= ( V * V [ X ] { 4 k } [ { 4 k } ) = =

for all X 9 M . (2) By (1) f = co~ for some { 9 29. It follows from L e m m a 10.9 in StratilaZsido [1] that (A~[~) = (A~/l~y),

A 9 (~/l')'

(4.1.17)

for a unique vector ~f in P #(M'~)'~o" Take an arbitrary X 9 J~/l. Let IX] =

/5

t d E ( t ) be the spectral resolution of

IXI

and En =

/0

d E ( t ) , n 9 N.

Since Ad~29 c 29, it follows that En, X E ~ 9 (Ad'w)' for n 9 N, so t h a t by (4.1.17/

lim E n ~ / = ~/, n--4 oo

lira

mTn----+oo

I I X E m [ f - XEn~/I] =

lim

m~n----+oo

I l X E m [ - XEn~]] = O.

Hence, ~f E 29 and f = w[s" Suppose f = wfl = w Q for [1,~2 c J~# Then ~0" it follows from [A/I]T; = [d~/I]'o and (4.1.17) that ~1 = [2, which shows the uniqueness of ~f. We put HoC(o = C ( / ,

C 9 M ' w.

Since ( / 9 P #(M')'(o, it follows that H0 is a positive operator whose closure Ho is affiliated with (Ad~w)'. The Fiedrich extension H of H0 fulfills our assertions. This completes the proof. Applying the Randon-Nikodym theorem in Section 3.5 to O*-algebras, we have the following P r o p o s i t i o n 4.1.15. Let M be a closed O*-algebra on 29 in ~ such that ,A,4"29 c 29 and ~o a strongly cyclic vector for A4. Suppose f is a positive linear functional on AJ such t h a t r r / + ~ ~ (,&l)'w is a v o n Neumann algebra. Then the following statements hold: (1) f is co~0-absolutely continuous if and only if f = ~H'r for some positive self-adjoint operator H ' affiliated with Ad'w such t h a t 29(H') D M~o.

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4. Physical Applications

(2) Further, suppose ~0 is a separating vector for (M~w) ' and f is ~ o absolutely continuous. Then f = wr for a unique element ( f of P ~ . Further, f = ~H~o for some positive self-adjoint operator H affiliated with (A&w)~ such that ~0 9 79(H) and H~o 9 79. P r o o f . (1) This follows from T h e o r e m 3.5.2.. (2) This follows from (1) and Proposition 4.1.14. R e m a r k 4.1.16. We may change the condition " AJw79 ~ c 79" in Proposition 4.1.13, 4.1.14, 4.1.15 to a weaker condition "M,v is a yon Neumann algebra". For, when A//~ is a v o n N e u m a n n algebra, by Theorem 1.4.2 there exists a closed O*-algebra M on ~ in 7-/ which is an extension of A/[ satisfying (M)'w ~ Ad'w and (M)'~79 C :D. Hence we have only to consider the O*-algebra M instead of the O*-Mgebra Ad. A

4.2 Quantum

moment

problem

II

Let f be a trace functional on a closed O*-algebra ~4, that is, f(X) = trXT = trTX,

X 9 M

for some T 9 ~ I ( M ) + . Then, by L e m m a 1.9.13 f2 - T t/2 is a positive Hilbert-Schmidt operator on 7-/ such that t97-/ c 79 and Xt2 is a HilbertSchmidt operator for all X 9 M , and the f is represented as f(XtX)

-- t r ( X t 2 ) * ( X f 2 ) ,

X 9 2t4,

(4.2.1)

and so the GNS-representation 7~f for f is unitarily equivalent to a *subrepresentation of the *-representation 7r of A/[ on the Hilbert space 7-/| of Hilbert-Schmidt operators on 7-/. This result is useful for the unbounded Tomita-Takesaki theory and applicable to the quantum physics in later sections. In this section we consider under what conditions a weight on M + is represented as the similar form to (4.2.1); that is, the weight ~ is of the form ~(xtx)

-- t r ( X t * f 2 ) * X t * t ? whenever X e r176 v = {X 9 J~4; ~ ( X t X ) < oo} (4.2.2)

for some positive self-adjoint operator I2 in 7-/. In this case we can define the generalized vector AM for the O*-algebra 7r(~4) in 7i | 7-/ and develop the Tomita-Takesaki theory in the O*-algebra 7r(M) using the theory of generalized vectors in Section 2. Let M be an O*-algebra on 79 in 7-/ with identity operator I. For T c 6 1 (A/l)+ we define a strongly positive linear functional f T on A/I by f T ( X ) ----t r T X = t r X T ,

X r M.

4.2 Quantum moment problem II

187

Then we have

fT(XtX) = tr(Tll2xt)*T1/2X t = trXTXt,

X E ~4.

(4.2.3)

In fact, given T E 6 1 ( M ) + and X E M , we consider orthonormal sequences {~n} and {z/~} in 79 such that { ~ } contains a complete set of eigenvectors with non-zero eigenvalues of T and that the closed linear span of {~7,~} in 7-/ contains XTT-I U TT-t. Then, the equation (4.2.3) follows from

fT(XtX)

= ~-~.(XT~n[xtsJ n

?2

m

(Xbl. ITXhT.J

=

rn

=

ItT

/2Xt,mll 2

~rt

=

(XTXtvmlVr ). m

T h e o r e m 4.2.1. Suppose f14 is a QMP-solvable O*-algebra on D in 7-/ 0 and ~ is a regular weight on M + such that (91~)tD is total in 7-/. Suppose further ~-(D) = linear span of {~ | 7; ~, 7/ E l)} C M or ~ is sequentially m-regular, that is, ~ = sup fn for some increasing sequence {f,~} of strongly t~

positive linear functionals on M . Then there exists a positive self-adjoint operator F2 in 7-/such that 91o

= { x c M ; r xt 9 | c {X r M ; Xt*f2 E 7"/| 7-l},

~(xtx)

= tr(~2Xt)*~2Xt = tr(Xt*F2)*Xt*g2,

X E 9l ~

For the proof of Theorem 4.2.1 we prepare some notations and one lemma. Suppose AJ is a QMP-solvable O*-algebra on D in 7-/and ~ is a regular weight on M + , then there exists a net {T~} in G I ( M ) + such that ~(X)

= ~ { T ~ } ( X ) -- suptrXT~,

X E M+.

(4.2.4)

Hence we first investigate when such a weight ~{T~} is of the form (4.2.2). ~,Ve denote by T ( 3 4 ) the set of all nets {T~} in G I ( A J ) + for which the formula ~{T,~}(X) = suptrXT=,

X E M+

defines a weight on A4+, and denote by Ti(A4) (resp. Tic(A/l)) the set of all increasing (resp. increasing and mutually commuting) nets in GI(A4)+.

188

4. Physical Applications

Then, T~(Ad) C T/(Ad) C T(Ad). For trace representation of the weight ~{T~} we have the following L e m m a 4.2.2. Let Ad be an | on :D in 7-/. Let {T~} 9 T(A4) be given such that (92~ T~} )t:D is total in 7-{. Suppose {T~} 9 T~(A//) or that Y(:D) c A//. Then the following statements hold: (1) There exists a positive self-adjoint operator I? in 7-/such that 920

%o{T~ }

9 ~ | ~}

= {x 9 M; ~x*

c{X 9

Xt*{2 9174

and

%ro} ( x t x ) = tr (~?xt)*a'2X t = tr(Xt*y2)*Xt*Y2,

o

X E 92~'{r~}"

(2) If {T~} E Tic(Ad), then there exist a set {PZ}ZEB of positive numbers and an orthonormal system {(~}Z~B in 7-/ contained i n / 9 such that

flEB

sothat ~=

E

pflI/2 ~ | -~Z is a positive self-adjoint operator in T/and

flcB

920~O{Ta } = { x 9 3,l; x** f2 9 7{ | ~ } = { x 9 M ; ~2x* 9 ~ | ~},

qo{%}(XtX) = tr(Xt*f2)*Xt*f2,

X 9

0 92~{To}"

Proof. (i) W e define a form @ by

:Do = {( 9

sup (T~(I() < oo}, C~

0(~, ~) = sup (r~q{),

~ 9 :D0.

We show that :Do is a dense subspace of 7-i and that 0 can be extended to a closable sesquilinear form on :Do x :Do. Since ( T ~ ( ( + r/)l( + V) _< 2{(r~glg) + ( r ~ n b ) } for each (, r / 9 :Do, it follows that :Do is a subspaee of 7-(. Since sup (T~Xt {IX'r {) <__sup tr X T ~ X * O~

Og

= sup tr T ~ X * X Ol

= ~{To}(xtx)

(by 4.2.3)

4.2 Quantum moment problem II

189

for each X E 9~ ~ and ~ E 79, we have ( ~{T~})t79C :Do, and so 79o is dense in 7-/. If {T~} is an increasing net, 0 extends to a positive sesquilinear form on 79o x 79o (also denoted by the same 0) by 0((,~) = l i m ( T ~ l v ) :

1

s

+ ikrl)]~ + ikT1),

~,r] E 79o.

k=0

Suppose that 5c(79) c .M. Then ( E 79o if and only if {{T~}(( | ~) < oo. There exists a positive linear functional ~ defined on the linear span of {X E Ad+; ~ ( X ) < oo} such that %(X) = ~(X) for each X E A4+ with ~(X) < 0o. Consequently, 0 extends to a positive sesquilinear form on 79o x 79o defined by 0((, 7/) = ~(~ | ~)

= 1 ~:~jkr 4

+ i~v)

|

(( + ikv)),

~, ~ ~ 790.

k=0

We show that 0 is closable. In fact, take an arbitrary {~n} in 790 such that lim~n=0and lira 0 ( ~ n - ( m , ~ - ~ m ) = 0 . For any e > 0 there exists n ----* o o

n , r n ---4. c ~

a natural number Ne such that ]]~nll<e and O((n--(m , ( n - - ( r n )

f o r a l l n , m_>h 89

<~"

and so (T~(~n - ~,0 ] ~n - ~m) < e

foralln, m>_N~anda.

Taking rn ~ 0o, we have ( T ~ n ]~n) < e

for all n > ArE and ~,

and sup ( T ~ I ~ , ~ ) <_ e for all n > N~. Hence, lim 0(~n, ~n) = 0. Thus, C~

7Z - - - 4 0 C

0 is closable. Let 0 be the closure of 0. By Theorem 2.1 in Faris [1] there exists a positive self-adjoint operator /2 in ~ with 79(f2) = 79~ and (f2~ I f2r/) = 0(~,U) for each ~,7/ E Z~0. We next show r

~0 {Tc~ }

= {X E

0 Let S = {{(~}; {(~} Ad; f2Xt E 7-/| ~ } . Suppose that X E 9"[~,{To}. is an nonempty orthonormal set in 7-/contained in 79}. It follows that ~{~} ( x t x )

= sup tr T ~ X t X = sup tr XT~Xt

(by 4.2.3)

= sup sup ~-'}(Tc,Xt(~ ] X t ( z ) --

sup

y~llnXt@ll

{@}Es = tr (f2xt)* s

2

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4. Physical Applications

which implies that $2Xt is a Hilbert-Schmidt operator. Conversely, if ~2Xt is a Hilbert-Schmidt operator, we can write down the same equations with S' = {{~/Z} ; {7/~} is a nonempty orthonormal set in 2-{ contained in :D(tgxt)} 0 0 in place of S. This implies X E r Finally, given X E r ~ E T~ and 7/E :D(~2), we have

(xtr ]/2~2) : (f2xtr I ~) -_ (,~ I (f2xt)*n) -- (r x t * ~ = (y2xt) * E 7-/| ~{v~} (X t X) ----tr( Y?xt) *t2Xt ----tr ( x t * Y2)*Xt* Y2. (2) Let {T~} E Tic(2td). Then there exist an orthonormal system {~Z}ZEB in 7-I and nonnegative n u m b e r s / ~ , ~ such that T~ = E p ~ , z ~ Z | ~Z

for each c~.

(4.2.5)

~EB

In fact, using Zorn's lemma, one finds a maximal system {Pi}icx of nonzero finite rank orthogonal projections in 2-/such that (i) PiPj = O, i # j; (ii) for a E A and i E I there exists Ao,i such that T~P~ = Ao,iPi; (iii) for i E I there exists a E A such that T~Pi ~ O. We

show that

(EPi)Tc~

= ]'~

(4.2.6)

for each ~ E A.

iEI

Suppose

that this is not true. Then,

for some

s0 E A iEI

Let T~ o

X--'A(~~

(~~ be the spectral decomposition (A(~~ > 0). Then for

n

some n, p(~o) is not a subprojection of E P i .

This means that

iEl

q = p(~o)_ p ( ~ o ) ( E p i ) = p(~o)_

E

Pi ~ o.

Since the ,-algebra generated by {T,~Q} is a commutative C*-algebra of finite dimension, we can find a non-zero subprojection P of Q such that T,~P = A,~P for each a, which implies that {Pi, P} satisfies the conditions (i) ~ (iii). This contradicts the maximality of {Pi} and thus the statement (4.2.6) holds. Furthermore, we can choose in each Pi (~'/) an orthonormal basis and collect all these basis vectors together. By (4.2.4) we have

4.2 Quantum moment problem II {~Z} C :D and ~{T~}(X) : E p z ( X ( z I ( Z )

191

, X 9 A4+,

(4.2.7)

where #Z -= sup p~,z for each ~ E B. Now suppose #eo :

oo for some

190 E B. T h e n X{~ o = 0 for each X E r

= 0 for each

and so ({~o I x t ( )

X 9 ~ 0~{Ta} and ( E :D. Since ( ~ 0 { T ~ } ) t ~ is total in ~ , we have (Zo = 0. This is a contradiction. Hence we have

Pz < oo,

vZ 9 B.

(4.2.8)

1

Hence,/2 - E #~Z| is a positive self-adjoint o p e r a t o r in 7-{. Let X E A4. ~6B Since x t * ~ ? -- 0 on { ~ } • and

I(Xt*~r

I~)1 ~ = I ~ - ~ ( x ~

rEF

I v)l ~

~'EF

< (~--~. IA~ 12) ~--~'~l(X~z rEF

1,7)12

~EB

< II~--~,~ll~ll~ll~(~,~llX~zll ~) rEF

I~6B

-_ II~A~ll~ll~ll~{~o~(XtX) rEF

for each ~/E T{ and each finite subset F of B, it follows t h a t ( f 2 x t ) * = Xt*Y2 are b o u n d e d and

tr((Xt*~)*X**~2 = ~

IlXt*~ll ~

~EB

= y~.~zllX~zll ~ /3EB

: ~{~o}(xtx), which implies t h a t X E r 0

if and only if Xt* 52 is a Hilbert-Schmidt

o p e r a t o r on 7-{ and t h a t this is the case if and only if ~2Xt is a HilbertSchmidt o p e r a t o r on T/. This completes the proof of L e m m a 4.2.2. P r o o f o f T h e o r e m 4.2.1. : Suppose ~ is sequentially m-regular, t h a t is, ~ = s u p f n for some increasing sequence {fn} of strongly positive linear functionals on A4. In this case it m a y be represented also as ~ -- E g n ' n

where gl ~ f l and gn+l --- f n + l - f,~ are strongly positive. This implies t h a t = ~{T~} for some {T,~} E T ( A 4 ) . Hence our assertions follow from (4.2.4) and L e m m a 4.2.2.

192

4. Physical Applications

R e m a r k 4.2.3. In Lemma 4.2.2 we have showed { x 9 34; s ? x t e ~ | 1 4 9

xt*/2 9174

Do the above two sets coincide? Suppose X t*/2 6 "H| ~ a n d / 2 X t is densely defined. Then /2Xt 9 ?-/| ~ and /2X* = (X**/2)*. Hence, in the following cases two sets coincide: (i) D C 29(/2), that is, sup (T~(I~) < oo for each ~ 9 29. c~

(ii) /2 is bounded, that is, sup trT~ < oo. By Theorem 4.2.1 and Remark 4.2.3 we have the following C o r o l l a r y 4.2.4. Suppose A/I is a QMP-solvable O*-algebra on 29 in 7~. For every regular weight 7: on AA+ satisfying 7:(I) < oo there exists a positive Hilbert-Schmidt operator [2 on 7-/such that 9I ~

= (x

M; nxt

c

|

= {X E M ; Xt*/2 E 7-/| 7 : ( x t x ) = tr(/2xt)*/2Xt = tr(Xt*/2)*Xt*/2,

X 9 r

R e m a r k 4.2.5. In bounded case the condition 7:(1) < oo in Corollary 4.2.4 implies 7: is finite, that is, 7:(X) < o~ for each X 9 )A+. But., in unbounded case this does not necessarily hold as seen in next example. Let A4 be an O*-algebra on the Schwartz space S(]~) generated by the momentum operator P and the position operator Q and {/n}n=0,1,.-. C 8(]~) an ONB in L 2(~) consisting of the Hermite functions. For m C l%I u {0} we define a regular weight 7:,~ on 34+ by oo

7:re(X) = ~

1

(n + 1) 2m (Xfnlf, O,

X 9 34+.

n=l

Then the following cases arise: (i) I f m = 0 , t h e n 9 1 ~ = { 0 } . (ii) I f m # 0, then { I , N , . . . , N

TM}c

9lo

but N k ~ 9 I ~

for k _> r e + l ,

OG

where N = E ( n

+ 1)f, |

is the number operator.

n=0

We next consider trace representation of weights without the assumption of regularity. We generalize the Schmtidgen result (Theorem 4.1.6) for strongly positive linear functionals to weights. The proof is according to that of Theorem 4.1.6. T h e o r e m 4.2.6. Let jZ4 be an O*-algebra on 29 in ~ and 7: a weight on A/l+. Suppose that there exists an elemen N of 91o which has a positive

4.2 Quantum moment problem II

193

self-adjoint extension N such that. /~-1 is a b o u n d e d compact operator on ?-l. T h e n there exists a positive trace class operator T on T / s u c h that. (i) T1/2Xt is a Hilbert-Schmidt operator and y t * T X t is a trace class operator on 7-/for all X, Y 9 r176 (ii) ~ ( x t x ) = tr (T1/2Xt)*T1/2Xt

= tr X t * ~ for a l l X 9 1 4 9 (iii) p ( X ) = tr T X for each positive operator X in r176

~

P r o o f . Note first t h a t I 9 r ~ since N 9 r ~ and II]V-1/2lI2(N~l~) ___ II~ll 2, We put D(CO) = linear span of {X 9 2kd+; ~ ( X ) < oo}, n

n

T h e n D(CO) is a *-vector space with I E D(CO) and it is not difficult to show t h a t COis a strongly positive linear functional on ~D(CO). Since ( X + Y ) t ( x + Y) <_2 ( x t x + y t y ) ,

X=~1 ~

ik(X + ikI)t(x + ikI),

k=0 it follows t h a t r ~ is a subspace of D(CO). Let s

denote the subspace of

s T)t) generated by {X t o A o X; X r r ~ A E B ( H ) + } . By (4.1.9) D(CO) is eofinal in the ordered *-vector space s - Z)(CO)+ s and so there exists a strongly positive extension of COto s denoted also by CO. Since .T(7-/) - linear span of {x | ~[:D; x r 7-/} c s it follows from (4.1.1) t h a t there exists a unique positive trace operator T on 7-/such t h a t

Co(A) = tr TA,

A e ~-(~).

(4.2.9)

We show the s t a t e m e n t (i). Let X, Y E r Since a linear functional COx on 3c(7-/) defined by COx(A) = CO(Xt o A o X), A r )r(?-[) is strongly positive, it follows also from (4.1.1) t h a t

COx(A) = t r r x A ,

A 9 Y(~)

(4.2.10)

for a unique positive trace class operator Tx on 1-{. By (4.2.9) and (4.2.10) we have

194

4. Physical Applications

(TXt (]Xtu) : tr TXt ([ | 7)X : t r T x (~ | 7)

= (Vx(I,) for each (, ~/E l), and so

(4.2.11)

T x = x t * T x t = (T1/2xt)*TI/zx t.

Using the polarization formula, we get

(yt*TXt(b) = (TXt(IYt.) : ( ~ 1 ~ikTx+iky(l~ ) k:0 for each (, 77 9 1), which implies that yt*TXt is a trace class operator and in particular, yt*T is a trace class operator and (Yt*v)* = Vy{. Thus the statement (i) holds. We next show the statement. (ii). Let X 9 A4 such that

N X 9 glo. L e t / ~ - 1 - E I~[~ |

where {[~} is an orthonormal basis of

n=l

7-/ and A,~ ~ O. We put

E~=~&|

~9

k=l

Since

(NX)tNX

=

XtN2X > [IN - a l l - 2 x t x ,

we have X E 9I ~ Furthermore, since

< ( X t N ) o ( ~ - ~ ( I - E~)s

o (NXK,,

>

: ((s - E~)X~IX~)

= < (XtX-X

toEnoX)(,~>

for each (~ 7/E 1)e it follows from (4.1.9) that

0 < x t x - X t o E . o X : ( X t N ) o ( / V - I ( I - En)/V -1) o (NX) <_IIN-2(I- En)[I(NX)tNX, which implies o

<_ ~(xtx

- x t o E~ o x)

_ [IN-2(I - E~)lho(XtN2X)

<_A2+a~o(XtN2X).

(4.2.12)

4.2 Quantum moment problem II

195

Hence, it follows from (4.2.10) and (4.2.11) that ~(xtx)

= lira ~b(X t o E , ~ o X ) n---~oo n

= n -lim E(Tx[kl~k) --* oo k=l oo

k:l OO

= ~

IlZl/2Xi4kll 2

k=l

= trXt*TXi = tr (T1/2X?)*T1/2X t. We finally show the statement (iii). Suppose now that X E fl[t ( N X may belong to 9I ~ or not). Then, given r > 0 and k E 7/,, we have

0 <_ (r

+ l i k ( I - E , ) ) t o (eX + l i k ( I - E , ) )

1 = e 2 X I X + i k X t o (I - En) + i - k ( I -- En) o X + ~ ( I - En), which implies

7rt,~(ikX t o (I - En) + i - k ( I - En) o X ) <_ r

1 ) + ~-~b(I - En). (4.2.13)

Since N I = N c flio, it follows from (4.2.12) that n -lim q b ( I - En) = 0, which -*OO implies by (4.2.13) that I i r n ~b((I - En) o X) = 0. Moreover, it follows from (i), (4.2.10) and (4.2.11) that 3

lim ~b(E, o X) = lim ~b(1 E ~ -"* OO

n --~ O 0

i k ( X + ikI) t o En o ( X + ikI))

4

k=0

1

3

= lirn ~ E

iktr ( ( X + i k I ) i * T ( X + ikI)tEn)

k=0

= trXt*T. This completes the proof. R e m a r k 4.2.7. The statement (ii) in Theorem 4.2.6 does not hold for every X c flio as seen next. Let flJ, {fn} and N be as in Remark 4.2.5. We define a weight ~ on 34+ by

196

4. Physical Applications 1

~ ( X ) = limo~ - ~ ( X f,~[f~),

X C M+.

Then ~(I) = 0 and ~ ( N 2) = 1, and further the pair (A/[,~o) satisfies the assumptions of Theorem 49 Suppose now that there exists a positive trace class operator T on 7-/ such that ~ ( X t X ) = tr ( T 1 / 2 X t ) * T I / 2 X t for every X E .oR~ Since ~(I) = 0, it follows that T = 0, which implies that 1 =

~ ( N 2) = tr ( T 1 / 2 N ) * T 1 / 2 N = 0. This is a contradiction.

4.3 U n b o u n d e d C C R - a l g e b r a s A. Unbounded CCR-algebras in finite degrees of freedom Let .A be a *-algebra consisting of all polynomials in {Pi, qi; i = 1, 2, - 99 , s} satisfying

p~=;~,

q;=q~,

[Pi, qj] = Piqj -- qjPi = - i S i j 1, [Pi,Pj] = [qi, qj] = 0 ( i , j = 1 , 2 , - . . ,s), and it is called the canonical algebra for s-degree of freedom. We treat with only the canonical algebra of one degree of freedom, that is, .4 is a ,-algebra generated by identity 1 and two hermitian elements p and q satisfying the Heisenberg commutation relation : [P, q] = Pq - UP = - i l .

(4.3.1)

All of results obtained here hold for the canonical algebra for s-degree of freedom 9 It is not possible to find hermitian matrices P and Q satifsying the Heisenberg commutation relation (4.3.1), however Heisenberg found a solution in the form of infinite matrices:

/i -v~

-i P= ~

0

1L o oo o

ov~o

Q=~

0 0. 9149

o v~ 0 0 0 -v~ o 45oo o -v~ 0 v~O 0 0 -vq 0 o o

oo

o v~ o v~o o ov~o

'

(4.3.2)

4.3 Unbounded CCR-algebras

197

Schrhdinger found another solution to the Heisenberg solution in his formulation of quantum mechanics. The operators P0 and Q0 in L2(]~) defined by d

(Po()(t) = - i ~ ( ( t ) , (Qo~)(t) = t((t),

~ c S(R)

are essentially self-adjoint operators satisfying the Heisenberg commutation relation. Schrhdinger showed that his formulation agreed with the Heisenberg formulation, that is, P0 and Q0 have the same matirix elements as given in equation (4.3.2) with respect to an orthonormal basis for L 2 (N) defined by 1

1

d

n

t2

f~(t) = 7 r - Z ( 2 ' ~ n ! ) - ~ ( t - ~ ) e 2 ,

n C N U {0}.

We define a ,-representation 7ro of A on L2(R) with Z)(rr0) = S(IR) by 7r0(p) = Po and 7r0(q) = Qo, and it is called the Schr5dinger representation of A. We note some well-known results for the SchrSdinger representation. Let a + = 2- 89 - ip) and a - = 2- 89 + ip). Since a - a + - a+a - = 1, A is generated by {1, a +, a - } . We have the following L e m m a 4.3.1. (1) 7ro(a+)fn = V~ + lf,~+1,

7ro(a_)fn__{Ov~fn

-1

n = O, 1,.., n=0 ,

n>l.

(2) 7ro(a-a +) is essentially self-adjoint and 7ro(a-a +) = ~--~(n+ 1 ) f , ~ | n=0

(3) T)(rr0) = S(]~) = N :D(Tr~ and the graph topology t~ o = nEN the Schwartz space topology on S(]~) = the Fr6chet topology defined by the sequence {ll [In;n C N U {0}} of seminorms I[~l[n ~ 1lTro(a-a+)n~]l,~ E S(]~). (4) fo is a strongly cyclic vector for 7r0. (5) 7r0 is a faithful, self-adjoint representation of A such that 7r0(A)'w =

CI. This raised the question as to whether there were other self-adjoint representations 7r of A such that 7r(p) and ~(q) are essentially selfoadjoint. Weyl [1] proposed the question should be formulated in terms of the strongly continuous unitary groups U(s) - exp isTr(p) and V(t) - exp itTr(q). Suppose U(s):D(Tr) C :D(~r) for all s E JR. Then we have 9

.

(is)

e~S'~(P)Tr(q)e-~S~(P)~ = 7r(q)~ + is[re(p), 7r(q)]~ + T = (re(q) + sI)~

[Tr(p), [Tr(p), 7r(q)]]~ + - -

9

(4.3.3)

198

4. Physical Applications

for all ~ E T)(Tr) and s c N, and so rr is not bounded. Exponentiating both sides of equation (4.3.3) we have the Weyl formulation of the commutation relations:

U(s)V(t) = eistv(t)U(s),

Vs, t E ~.

(4.3.4)

For the SchrSdinger representation 7r0 we have (U(s)~)(x) = ~(z + s),

(v(t)~)(x) = e " ~ ( z ) for each ( E L2(]~) and s , t c 1~, and so { U ( s ) , V ( t ) ; s , t E 1~} satisfies the Weyl commutation relations. Von Neumann [1] characterized self-adjoint representations 7r satisfying the Weyl formulation as follows: T h e o r e m 4.3.2. Let ~- be a self-adjoint representation of .4 such that r(p) and 7r(q) are essentially self-adjoint. Then the following statements are equivalent: (i) {U(s), V(t); s, t C ]~} satisfies the Weyl commutation relations. (ii) 7r is a direct sum of ~i0~a of ,-representations ~ which are unitarily equivalent to the SchrSdinger representation. We call such a ,-representation the Weyl representation of the cardinal Io. (iii) U(s):D(Tr) C T)(~r) for all s E ]~. Fugledge [1] constructed a self-adjoint representation but not a Weyl representation. E x a m p l e 4.3.3. Let 7r be a *-representation of A on L 2 ( ~ ) defined by :D(Tr) = linear span of {x'~e -r~2+cx ; n = O, 1, 2 , . . . , r > O, c c C } , (~(p)~)(z) = -i~'(x) + e - ' / ~ ( x ) , 0r(q)~)(x) = x~(x) + ~(x + i v Y ) , ~ ~ T)(Tr). Then 7r(p) and 7r(q) are essentially adjoint and 7r** is not a Weyl representation of .4. The following problem arised again: under what conditions a *-representation 7r of .4 is a Weyl representation? Dixmier [3] obtained the following T h e o r e m 4.3.4. Let 7r be a ,-representation of ~4. Then the following statements axe equivalent: (i) 7r is a Weyl representation. (ii) 7r(p2 + q2) is essentially self-adjoint. Powers [1] defined the notion of strong positivity of self-adjoint representations of .A and using it he characterized the Weyl representations.

4.3 Unbounded CCR-algebras

199

D e f i n i t i o n 4.3.5. Let w be a positive linear functional on .4. If uo(a) >_ 0 implies a~(a) ~ 0, then w is said to be strongly positive . A , - r e p r e s e n t a t i o n ~r of .4 is said to be strongly positive if ~r0(a) > 0 implies ~(a) > 0. Lemma

4.3.6. Let 7r be a Weyl representation of .4. Suppose 711 is a

self-adjoint subrepresentation Proof. and T/--

Since ~r is a Weyl (~ ~o,

~EIo 1

1

where

of ~. Then

~I is a Weyl

representation

of .4.

representation

7r~ = ~o and ~

of .4 on 7-/, we have ~r -- G ~ ~Elo -- L 2 (]~) for all & E I0. The vector

2

fo(t) - ~ - ~ e - ~ t is a strongly cyclic vector for ~o- It is easily shown that. 7ro(ip + q)fo = 0 and E :D(u) and 7ro(ip + q)~ = 0 implies 77 = a f o for some a E C .

(4.3.5)

O

Let N o be a closed linear s p a n of {f~; c~ E I o ) in the Hilbert space 7-/, where O

f~ = fo and f ~ = (5~zfz)~eso for each (~ E Io. T h e n we have

ffr~o c v ( . ) and

(4.3.6) E ~o

iff ( E /)(r~) and u(ip + q)( = O.

In fact, let ~ E N 0 . T h e r e exists a sequence {~n} in 7-/such t h a t (,~ is a finite O

linear combination E

O

c,~ ((~) f ~ of f ~ and lim ~,~ = 4- T h e n since n'--~OO

o, E I o

Hrr(a)(~n - ~m)l] 2 = ][ ~

(cn(c0 - c m ( a ) ) r o ( a ) f ~ I]2

&EIo O

= ~

Ic~(~)

-

cm(~)1211~o(a) f ~ II2

aEIo

= II~n -- &ll211"0(a)f011 ~ 0 (n, m -+ o0) for all a E .4, it follows from the closedness of 7c t h a t ~ E 7)(~r). F u r t h e r m o r e , since r(ip+q)~,~ = 0 for all n E N, we have 7r(ip+q)~ -- lim ~ ( i p + q ) ~ n = O. Conversely, suppose ~ = (~)~eSo E/:)(~) and u(ip + q)~ = O. T h e n we have uo(ip + q ) ~ = 0 for each c~ E I0. B y (4.3.5), ~ = c(&)fo for some c(~) E C . Hence we have ~ E ~)~o. Since r l is a self-adjoint s u b r e p r e s e n t a t i o n of ~-, it follows from T h e o r e m 1.3.3 t h a t ~rl = ~rE for some projection E in 7r(.4)~w. Let r --- E~)~0. By (4.3.5) we have 7r(ip + q)E~ = ETr(ip + q)~ = 0

200

4. Physical Applications

for each ~ E 9J[0, and so E ( E ~)J~0. Hence we have r

= Eg:I[0 C ~)I0.

(4.3.7)

Let {~?Z;fl e I1} be an o r t h o n o r m a l basis for r and put r = 7r(A)r/Z. By (4.3.7) we have r]Z,r/~, E 92t0 and so rlz = (C(a)f~)~EI0 and ~Z' = (c'(a)f~)~Ei o. T h e n since (Tc(a)rlzlrlz,) = ~

c(~176

aElo

= (rl~Jrl~,)Oro(a)folfo) = 6~,Oro(a)folfo) for all a E "4, it follows t h a t 7rz ~ 7r~z is unitarily equivalent to ~ro, so t h a t 7r2 - ~ 7rz is a self-adjoint subrepresentation of 7r. By T h e o r e m 1.3.3, ~Elo

7r2 = 7rF for some projection F in rr("4)~. We show 7rl = 7r2. In fact, we have r = EgJ[0 c E/9(lr) = /9(7cl), and so rlfl E /9(7rl) and ef[~ = 7r("4)rlfl = ~rl("4)r]z C /9(rrl). Hence, ~r2 C 7r~, and so F < E. Take an arbitrary ~ E /9(7rl). Since 7r = O7r~ and f0 is a strongly cyclic vector for 7r~ = 7to, c~El0

it follows t h a t for any e > 0 there exists a sequence {as} in .4 such t h a t II~ -

7r(ak) f k I[ < e. T h e n we have k=l 7l

II~-

o

n

o

Zrr(a~) E f~ II = I I E ~ - E ~ r r ( a ~ ) f~ II k:l

k

1 o

-< I[~ - ~

7r(ak) f~ II

k=l

< e.

(4.3.8)

o

Further, since E f ~ E EffJ~0 = ~ for each a E I0 and rtZ E :D(~r2) C F ~

for

o

each fl E I1, we have E f ~ E FT-/ for each a E I0, which implies by (4.3.8) t h a t ~ EFT-/. Hence, E G F . Thus we have ~r~ = ~r2. This completes the proof. T h e o r e m 4 . 3 . 7 . Let 7c be a .-representation of .4 with a strongly cyclic vector ~0. T h e n the following statements are equivalent: (i) 7r is a Weyl representation of at most countable cardinal. (ii) 7r is self-adjoint and coco o rr is strongly positive. P r o o f . (i) ~ (ii) This is trivial. (ii) ~ (i) By E x a m p l e 4.1.9 there exists a sequence { ~ } in S(]~) such that

4.3 Unbounded CCR-algebras

201

oo

(7r(a)~01~0)

=

E(TCo(a)~nl~n)

(4.3.9)

n:l

for all a E .A. Let 7rl = ( t ~ 0 and ~ - {~-},~H- By (4.3.9) we have 4 6 "~(Trl) and (Tr(aK0140) = @rl(a)ClC),

x 6 A.

(4.3.10)

We p u t ~UI = 7q(A)~

and

7r2

z

(71-1)_~jl"

T h e n it follows from (4.3.10) t h a t 7r2 is u n i t a r y equivalent to ~r, and so 7r2 is a self-adjoint subrepresentation of 7rl. By L e m m a 4.3.6 7r2 is a Weyl representation. Hence 7c is a Weyl representation of at most countable cardinal. This completes the proof. C o r o l l a r y 4.3.8. Let w be a positive linear flmctional on A. T h e n the following s t a t e m e n t s are equivalent: (i) a~ is strongly positive and 7c~ is self-adjoint. (ii) 7r~ is a Weyl representation of at most countable cardinal. Further, we show t h a t a Weyl representation of countable cardinal is unitarily equivalent to the self-adjoint representation 7r| of ~4 on the Hilbert space of Hilbert -Schmidt operators on L2(]l{) defined by Z)(Tr| 7r|

= S(R) | L2(~), = 7ro(a)T ,

a E .4, T 6 Z)(rr|

For a n y / ~ > 0 we put _n ~ n

0

G u d d e r and Hudson [1] obtained the following Lemma

4.3.9. ~?Z is a strongly cyclic vector for 7r|

T h e o r e m 4 . 3 . 1 0 . Let 7r be a *-representation of A. T h e n the following s t a t e m e n t s are equivalent: (i) 7r is a Weyl representation of countable cardinal. (ii) 7c is unitarily equivalent to 7r| P r o o f . We put

V{~n} = i

~n @fn,

rt:O

202

4. Physical Applications

Then it is easily shown that U is a unitary transformation of ~ L2(]~) onto nz0

L2(I~) | L2(R).

oo

(i) =v (ii) Let 7r = G 7r~, rcn = 7r0 (n = 0, 1, 2,--- ). Since n=0 oo

Ib-|

oo - -

2 = II~|

2

(,~ | All2 = ~ n=O

s

II~ro(a)(nll 2

n=O oo

for each {~,~} 9 7P(~=~), we have U Z ) ( ~ = ~ ) C S ( ~ ) | L2(~). Take arbi(N3

trary ~ 9 S(]t{) and r / 9 L2(]I~). Since r / = ~

c~f~ for some {c~n} c C, we

n~0

have oo

r~O

II~0(a)~ll 2 = ( ~ b~l~)ll~0(a)~ll ~, n=O so that {Ws

n=O

9 :D(~__[,~) and ~|

(9O

9 UT)(G__[~). Hence we have

U:D(G=~rn) O0

= 8(]t~) | L2(][~). Further, since oo

7r|

= ~

oo

TCo(a)~n | ~n = U{Tro(a)~n} = U ( 2 ~ n ) ( a ) { ~ n }

redo O0

for each~ a C .4 and { ~ } E :D(n~[,~), it follows that ~r| is unitarily equivalent to ~Trn. n~0 (ii) ~ (i) By Lemma 4.3.9 f2~ is a strongly cyclic vector for 7r| and so a~n~ o ~r| is strongly positive. Hence it. follows from Theorem 4.3.7 that 7r| is a Weyl representation of at most countable cardinal. Further, since U is a unitary transformation of ~ L2(]~) onto L2(]~) @ L2(][~), it follows that ~r| nz0 is a Weyl representation of countable cardinal. This completes the proof. By Theorem 4.3.10 it is useful to consider the self-adjoint O*-algebra 7r| on S(I~) | L2(~) for the study of Tomita-Takesaki theory for CCRalgebras, and apply the results of Section 2.3, D to it. We put oo

_n ~

~e]R.

n=O

Then f2Z is a non-singular positive self-adjoint operator in L 2 (]~) and if ~ > 0 then ~2~ E S(]~) | L2(]~).

4.3 Unbounded CCR-algebras L e m m a 4.3.11.

f2i~trro(A)f2Sit =

203

7r0(A) for each t E ]~.

P r o o f . For each ~ E S(]R) and t E 1~ we have o~

rr~

t~ --

E

e ~n~ (glfn)rro(a+)f~

n=O oo

= ~-~e ~ ' ~ v ~ +

l(glf, Ofn+s

n:O

= E

e Bn~ (flrr0(a-)fn+l)f~+l

n=0

= e~ ~

e -~("+1)~

(~lrro(a-)fn+l)fn+l

n=O

= e~

~

e - z , ~= ({[rr0(a)fn)fn

n=l

= eZ~ ~ e -;~n~ (rro(a+)glf,Ofn n=O

=

o5%(a+1

.

Similarly, we have

rro(a-m)rro(a+n)f2i~t~ = e-(m-n)~ /2~trr0(a-m)rr0(a+n)~ for each m, n E NU{0}, ~ E S(]~) and t E ~ . Hence, we have f2~tTr0(fl.)f2~/t = rr0 (,4) for each t E ]~. T h e o r e m 4.3.12. Let/3 > 0. Then the following statements hold: (1) O~ is a standard vector for 7r| (2) rr| c N 79(Z]~) and it is a core for each z ~ . ~EC (3) There exists a complex one-parameter group {Z~(~); c~ E C} of automorphisms of A such that Z]~ rre(X)f2,3 = rr| for all x E A and a E C. P r o o f . (1) This follows from Lemma 4.3.9, 4.3.10 and Theorem 2.4.18. (2) By Theorem 2.4.18 we have

and

so

for each c~ E C

204

4. Physical Applications

~'(~;~)~"(~p)~|

+

2c~

/t

n=0

n=0

k--0 OG

e-~''~ Y } ~-*'~ ~| n

| K

0

e-Z~7~| (a +) #2#,

(4.3.11)

and (4.3.12) Similarly we have

A ~ Tc|

= e-(J-k)#~Tr|

for j , k = 0 , 1 , 2 , . . . , and so since 7r|

C T ) ( z ~ , ) and (I + A ~ e~)

~re(A) #2#(= ~ro(A)tg~) is dense in L2(]~) | L2(]~), it follows that ~r| is a core for A ~ . (3) By (4.3.11) and (4.3.12) we have

z~,,|

= ~-|

where p~(~) = cosh ( f 3 a ) p - i sinh 03a)q, qa'(~) = i sinh 03a)p + cosh 03a)q. Since p~,,(,~)qA,(~) _ qAp(~)p.~(~) = - i l , there is a unique automorphism A]#(a) of ~4 under which p~(~) and qaZ(~) are the images of p and q, respectively, and A~S|

~ = ~|

x c X , ~ c C.

Further, it is shown that {A#(a); c~ E C} is a one-parameter group of automorphisms of .4. This completes the proof. Theorem 4.3.12 shows that ~| is an unbounded Tomita algebra in L2(]R) | L2(R), that is, it has the properties of a Tomita algebra (Takesaki [1]), with the exception of the continuity of left multiplication.

4.3 Unbounded CCR-algebras

205

We consider generalized vectors ~2Z (/3 < 0). Let A/I be an O*-algebra on S(]~) generated by ~ro(A ) and fo | Too. Since 1

= nx/~.Wm.vTro(a+~)(fo| fo)zco(a -m)

f~ | ~

for n, m 6 H U {0}, we have {f~ | f-~; n, m 6 H U {0}} C M . Let a , ~ > 0 , n = 0 , 1 , 2 , . . ,

(4.3.13)

and put

n=O

Then f2{~} is a non-singular positive self-adjoint operator in L2(]~). Let ~r be a self-adjoint representation of the O*-algebra M defined by ~D(Tr) = $(]~) | L2(~) and ~r(X)T = XT,

X E J~, T E S(]~) | L2(~).

This has been defined in Section 2.4, D. T h e o r e m 4.3.13. Let A/I be an O*-algebra on S(]~) generated by ~0(A) and f0 | f0 and let ( ~ > 0, n = 0, 1,---. Then An(=~} is a quasi-standard generalized vector for 7r(A//). Further, An(o~} is standard if and only if an -e 3 n , n E N U { 0 } for s o m e 3 E ] ~ . P r o o f . By (4.3.13) it is easily shown that the conditions (i) and (ii) in Theorem 2.4.23 hold. We show only (iii) in Theorem 2.4.23. Let N1 =- {n E N;an > 1}U{0}andi~2={nEN;~,~ < i } For e a c h n E l ~ u { 0 } w e p u t n

--

I

{~--n~ n C l~1 '

r n ----

,

rt ~ 1~1,

n

II

"

{~-n~OLn

rn :

,n ~N2,

r n = r ~ -+- r/I

k----0

Then, {rn} E s+, Am E M N (S(]~) | L2(R)) and

kEN10zk

c $ ( ~ ) | L2(R). Further, we have

~Tt E ~ 2

kEN2

206

4. Physical Applications

A,~,AmXAnE~(An{~})tn~)(An{~,}), A2mXAnS2{~n} -~7m-A

n,m E NO{0},

A2X~{r~}

for each X E M , which implies by the non-singularity of A that In{~} ((T)(An{~}) t n T)(An{~})) 2) is total in L2(]~)@L2(]~). Since f2~t },S(][~) c S ( ~ ) for each t E ]~, it follows from Theorem 2.4.23 that ln{o~} is a quasistandard generalized vector for 7r(M). Suppose In{o~} is standard. Since

7~| ra+~2it ~ {~.}A = a ~ v ~ + l A + l for each n E N U {0} and t E ]~, it follows that a ~ / a ~ + l = constant for n = 0, 1, 2 , . . - , which implies C~n = e ~ , n = 0, 1, 2,. 99 , for some /3 E ]~. The converse follows from L e m m a 4.3.11. This completes the proof.

B. Dynamics of an interacting Boson model We consider standard generalized vectors and standard quasi-weights in a class of interacting Boson models in Fock space. Let 7-/ be a separable Hilbert space, and let ~ n be the n-fold tensor product of ~ . We define an operator S,~ on ~ n by

s~(/i | A |

| A) = (~!)-'~

L~ | L~ |

| L~,

7r

where the sum is over all permutations. We put

~=0(n) : C, Yn(~) = s~n ~, and o~

~- is called the Bose-Fock space. Let A be a self-adjoint operator in ~ . We put

dFo(A) = O, dFn(A) = A | 1 7 4 1 7 4 + I|174174174 +...+I@I|174174 (n _> 1), and

oo

dF(A) = @ dFn(A). n=O

Then dF(A) is a self-adjoint operator in .T. We denote by -To the subspace in f f spanned by vectors ~ = {~(n)}n~__0 E ~ such that ~(~) = 0 for all but finitely m a n y n. The subspace ~-o is dense in .T. For each f E 7-/there exists a closed linear operator a(f) in .T such that

4.3 Unbounded CCR-algebras

207

a(f)4 = {0,-.- , 0, (a(f)~) (k-l), 0,--. }, where k

1| (a(f))(k-1) = ~ 1 jE(flfj),_,ck_l(f 1

| fi-1 | fj+l |

| fk)

for ~ = { 0 , . . . ,O,~(k),O,...}, where ~(k) = Sk(fl @ ' ' " | fk),

( f l , ' ' " , fk E 7/).

The domain of a(f) contains 5to and a(f) leaves -To invariant. For each f E 7-{ there exists a closed linear operator a*(f) in 9r such that

(a*(f)~)(n) =

Sn(f|

n >_ 1

for ~ = {~(n) }or E ~-0. The domain of a* (f) leaves -To invariant. The closed operators a(f) and a* (f) are called the annihilation operator and the creation operator, respectively. We define a number operator N in 5 by

D(N) = {.(; { = t , sc(n)t~ s,=o,

~n211((n)ll2

<

+oo}

n 0

N~ and put D~

= {n~(") }n~_o

= f l D(Nk). For each f C 7-/ the domains of a(f) and k=O

a*(f) contain D~176 and a(f) and a*(f) leave :D~176 invariant. Let 9.1ccR be the self-adjoint O*-algebra on D~176 generated by the identity I and {a(f), a*(f); f E 7/} and it is a CCR-algebra over 7-/, that is, (a)

a*(f) is linear in f ,

(b)

a(f) + = a * ( f ) ,

(c)

f e 7-/,

[a(f),a*(g)] = (fig)I, [a* (f), a* (g)] = [a(f), a(g)] = 0

for f, g e 7-/.

Let A be the Laplacian operator in L2(~3). We put = L2 (]~3),

hz

--/~.

A two-body potential is a real function 9 over 1~3 • R 3 whose values 4~(xl, x2) represent the potential energy of interaction between a particle at the point xl and a second particle at the point x2. Thus the total interaction energy of n particles at the points xl, x2,... , Xn is given by

208

4. Physical Applications

u(n)(xl,'"

,Xn) =

~ ~(Xi,XJ)" l
Note that the s y m m e t r y of U (n) is reflected by the s y m m e t r y property

~(x~, ~ ) = r

~).

We assume that @ is bounded, that is, there exists a constant c such that t~( x, Y)I < c for all x, y 9 ]R3. The interaction operator V (n) is defined by

( v ( n ) f ) ( x l , ' ' ' ,Xn) = u ( n ) ( x l , "'" ,Zn)f(xl,"" ,Xn) (f 9 J:n(~)). We put Hn = dFn(h) +

v (~),

and o~

H =OHn. redo

Then H is a self-adjoint operator in .P. Let AJ be the O*-algebra on 79~176 (N) generated by 9dccR and { f ( N ) ; f is a continuous function on [0, oo) and there exists a polynomial p such that If(x)l _< p(x) on [0, oo)}. Then Ad is a self-adjoint O*-algebra such that J~l'w = C I and t| = e - ~ is a positive self-adjoint operator in ~ . We have the following T h e o r e m 4.3.14. AM is a quasi-standard generalized vector for r~(Ad) on 2P~(N) | S in F | 9~. In particular, if the two-body potential ~ is constant, that is, r y) = constant for all x, y 9 ]~3 then )~g~is a standard generalized vector for rr(A//). P r o o f . ~ is a positive self-adjoint operator in 5~ satisfying defined. We put

f~-I is densely

ffp (7-/) = {~ = {~('~)}n~__o 9 .F; there exists a positive integer no s.t. ~(n) = 0 for all n _> no}. Since ~ - F ( T / ) n T ) ( ~ -1) is included i n / ) n / ) ( f 2 - 1 ) , it follows that 79AT)(~2 -1) is a core for ~2-1. We put

f s ( ~ ) = {~ = {r

9 7p(~);

each ~(n) is a finite linear combination of simple tensors}. Let 7~ be the set of all rank one projections constructed from ~s(7-/), and

4.4 Standard systems in the BCS-Bogolubov model

209

let N" be the linear span of {R]~4R2; R1,/~2 C ~ } . Since 7~ C A/l, we have AK C A4. And we have

~(~r

c

z>(~.),

~(9),

.tCt.p c

and the linear span of j ~ t / p is a core for ~2. Since Fs(7-/) is a subspace of D M/P(~2) such that { { | {,~/ c 5~s(7~)} c M and Y s ( ~ ) is a core for $2, by L e m m a 2.4.22, we have ~ 9 ( ( D ( ~ 9 ) A D(A~)t) 2) is total in ~- | F . Furthermore, since we have ~2itZ) C l? for all t E ~ , it follows from Theorem 2.4.23, An is a quasi-standard generalized vector for 7r(J~4). By the way, if the two-body potential 9 is constant, then we have

~2uM$2 -it = M , v t

E

R.

Hence, by Theorem 2.4.23, An is a standard generalized vector for 7r(A/l). This complets the proof. Using Theorem 2.2.4 and Proposition 3.2.9, we can construct from An I! the quasi-weight ~ on P(;r(A~)wc) , which satisfies the KMS condition. As in the previous case, we may interpret it as an equilibrium state of the interacting Boson model.

4.4 Standard systems in the B C S - B o g o l u b o v model Let A be a finite region of a lattice and IAI the number of points in A. The x y local C*-algebra ~A is generated by the Pauli operators ap = (ap, ap, ap) at every point p E A. The O-p are copies of the Pauli matrices

: (10)01 is isomorphic to the C*-algebra of all 2 IAI • 21AI-matrices on the 2 IAIdimensional complex Hilbert space T/A = ( ~ Cp2, where C 2 is the 2peA dimensional complex Hilbert space at p. If A c A' and AA E ~ A , then AA ~ A A, = AA | ( (~ Ip) defines the natural imbedding of 9.1A into ~ A ' ' pcA'\A Let n = (g~,gv,gz) be a unit vector in ~ 3 , and put

~A

(an) = gxa x + ~yaY + ~za z. Then, denoting as Sp((Tn) the spectrum of an, we have

Sp(,~n) = { 1 , - 1 } .

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4. Physical Applications

Let Ini be a unit eigenvector associated with 1, and let {n} = {nl, n 2 , . " } be an infinite sequence of unit vectors in 1~3. Then [{n}} = ~)In;} is a unit p vector in the infinite tensor product H~o = ~)Cv2- We put P 9~=

UgJ.A A

and And we denote the closure of 2 ~ } in 7-/o0 by 7-/{~}. Let (n, n 1, n 2) be an orthonormal basis of ]Ra. We put

n' = ~(n 1 - i n 2) and

Im,~) = Then

(an')ml n) (m = 0,1).

we have

(o-n)]m,n) = (-1)mira, n} ( m = 0 , 1 ) . Thus {I{m},{n}} = ~ l m p , np}; m v = 0,1,

~mp

P

< co} forms an or-

P

thonormal basis in ~ { , } . In this space we define the unbounded self-adjoint operator M by

MI{-~}, {n}) = (y]~-~p)l{.~}, {n}). P

M counts the number of the flipped spins in ]{m}, {n}) with respect to ]{n}}. Now we put --

N k

and let n{n} : 9~ +/:t(:D{~}) be the natural realization of 9.1 on D{n}, i.e.

={~} (o~) I {m}, {n}) -- % I mp,~p | ( ~ , # |

I me,%,))

(i = x,>z).

The BCS-Hamiltonian in the quasi-spin formulation is given by 2g

IAI --

p

where

1

IAI

--

O'p O'q

pq, --1 P~q

4.4 Standard systems in the BCS-Bogolubov model

211

Note that for the limit of the dynamics,

at(A)

eiHAtAe-i~At (A 9 9.1)

= lim A ---* o o m

fails to exist on the C*-algebra ~[. Let N be a unit vector in N3, and let {N} be a sequence {r

e2N,r

(%= lor

- 1 for eachp)

such that 1

k

k-~lim~ E

cv = 77r 0.

p=l

Then the work of Bogo[ubov and Haxg shows that the total Hamiltonian of the BCS-model is given by a self-adjoint operator HB in ~{N} O(3

z.

= a E{

p -

p=l

where a is a constant_ Then we have the following T h e o r e m 4.4.1. Let rid be the O*-algebra generated by Zr{N}(~) and {f(M); f is a continuous function on [0, oo) and there exists a polynomial p such that [f(x)[ _< p(x) on [0, oo)} with domain 7){N}. We put f2 = e - -~~

Then A~ is a standard generalized vector for 71"{N} ( . / ~ ) and the quasi-weight co~ on P(Ad) constructed by A~ satisfies the KMS condition. This quasiweight may be thought of as an equilibrium state for the BCS-Bogolubov model. P r o o f . A,l is a self-adjoint O*-algebra such that ~4'w = C1 and Y2 is a positive self-adjoint operator in 7-/{N} such that Y2-1 is densely defined. And since {]{m}, {N}};

Emp

< oo} C ~D{N}r~7)(S2-1 ), it follows that

P

7){N} r~7)(f2-1) is a core for ~2-1. Let 7-4 be the set of all rank one projections constructed from {[{m}, {N}}; E m v < oo}, and let A/" be the linear P

span of {RlJtdR2; R1, R2 E ~}. Since 7~ C fld, we have iV" c 3A_ And we have n(N') C D(A~), x t T ) { N } c 7)(f2), and the linear span of Aft7){N} is a core for f2. Let g be the linear span of {l{m}, {N});

Emp P

< oo). Then g is a subspace of 7){N} A 7)($2) such

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4. Physical Applications

that {4 | 7; ~, rl E $} c A4 and g is a core for f2. Thus, by Lemma 2.4.22, ,~s?((D(As?) N D ( , ~ ) t ) 2) is total in ~ | Furthermore, since M H B = H B M on I?{N}, we have J ~ i t ' D { N } C ~[){N}, Vt 6 and ~?uAdY2-~t = 3 4 ,

vt e 1~.

Therefore, by Theorem 2.4.23, AM is a standard generalized vector for

~{N}(M). There is, however, something more: we can find quasi-weights satisfying the KMS condition for the BCS model also in the thermodynamical limit. Let ~[ be the completion of the spin C*-algebra 9.1 with respect to the topology ~H defined in Lassner [3, 4]. Then 9./is still a *-algebra and each representation re(N} is continuous from 9.1 into s (SP{N}) endowed with the quasiuniform topology rq~ which makes of it a complete topological *-algebra. Hence ~r{N} can be extended by continuity to a .-representation #{N} defined on the whole ~ . Indeed, since s lows that ~r{N}(~ ) C s

is complete under rq~, it fol-

These facts above imply that #{N} is a

*-representation and, therefore, r } (!~1) is an O*-algebra on :D{N}. Then we can proceed as shown above and get KMS quasi-weights for the BCS model also in the thermodynamical limit.

4.5 Standard s y s t e m s in the W i g h t m a n q u a n t u m field theory The general theory of quantized fields has been developed along two main lines : One is based on the Wightman axioms and makes use of unbounded field operators (Streater-Wightman [1]). The other is the theory of local nets of bounded observables initiated by Haag-Kastler [1] and Araki [1]. Here we characterize the passage from a Wightman field to a local net of von Neumann algebras by the existence of standard systems obtained from the right wedgeregion in Minkowski s p a c e . The almost results stated here are based in the works of Bisognano-Wichmann [1,2] and Driessler-Summers-Wichmann [1]. Let @ be one scalar hermitian Wightman field, assumed to be an operatorvalued tempered distribution. It is regarded as a linear map of the Schwartz space S(]~ 4) into an O*-algebra s such that 4~(f)t = ~ ( f * ) for f E S ( ~ 4) adhering the assumptions : (A1) ]R4 is the Minkowski space R 4 with the Lorentz metric : xoy = x 4 y 4 -- x l y 1 -- x2y 2 -- z3y 3. The Hilbert space ~ obtained by the completion of i/) carries a strongly continuous unitary representation A ~ U ( A ) of the

4.5 Standard systems in the Wightman quantum field theory

213

Poincar6 group P. The subgroup of translations T(x) = U(I, x), x E ]~4 has the common spectral resolution T(x) = f ei(x~

x C ]~4

and the spectral measure # is contained in the closed forward light cone V+, where 17+ = {x E ]i~4 ," x o x > 0, x 4 > 0}. (A2) The existence of a vacuum state/2. For R c ]R4 we denote by 7)(R) the O*-algebra on :D generated by { ~ ( f ) ; f E S(]~ 4) and supp f C R}. It is assumed that ~ E :D, U(A)t9 = ~2 for each A E P, 7)(R)~2 is dense in for each open subset R of ~ 4 and :Do ~ 7)(]R4)~ is tu(R4)-dense in T). (A3) U(A)T~ = T) for each A E P and V ( A ) ~ ( f ) g ( d ) -1 = ~ ( d f ) for each A E P and f E S(]~4), where ( A f ) ( x ) = f ( A - l x ) for A E P and x E ~4. (A4) f --* O(f) is continuous of S(]~ 4) into ( p ( ~ 4 ) , T~). (A5) For each open subsets R and R1 which are space-like separated , i.e. R1 c R e =~ {y E ]~4 ," (X -- y)o(X -- y) < 0 for VX E R}, we have [X, X1] - X X 1 - X 1 X = 0 for each X E P ( R ) and X1 E 7)(R1). A local net is an assignment R -~ A(R) of regions R of the Minkowski space ]~4 with von Neumann algebras A(R) satisfying the conditions of isotony, i.e. -A(R1) c A(R2) if R1 c R2, locality, i.e. [A(R1), A(R2)] = 0 if R1 and R2 are space-like separated, and covariance, i.e. U ( A ) A ( R ) U ( A ) -1 = A ( A R ) for all A E P. D e f i n i t i o n 4.5.1. A Wightman field 9 is associated to a local net .4 of yon Neumann algebras if each field operator r has an extension to a closed operator, 4~(f)e C O(f*)*, that is affiliated with A(R) if the support of the test function f is contained in the interior of R. In the simplest case the field operators ~5(f) are essentially self-adjoint for all f* = f E S ( R 4) and @(f) and @(g) are strongly commuting if f* = f and g* = g in $(1~4) have space-like supports. In particular, this holds if certain growth conditions on the Wightman functions (Borchers-Zimmermann [1]) or the Schwinger functions (Driessler-Frhhlich [1] and Glimm-Jaffe [1]) are ! ! fulfilled. Then @ is associated to a local net R --~ (P(R)w) . However, the self-adjointness of the field operators cannot be expected, in general (counter examples are provided by Wick polynomial of free fields). Hence it is natural to look for more general ways of associating a Wightman field with a local net. Let WR and WL be the wedge-regions in Minkowski space ]R4 defined by W R = {x E ]i~ 4 ; x 3 > ] x 4 ]},

W L = { x C ]~4 ; x3

<~ _ ] x 4 ]}.

We denote by 142 the set { A W R ; A E P } of all wedge-regions Poincar6equivalent to WR, and denote by ~ the set of all closed double c o n e s / ( with a

214

4. Physical Applications

non-empty interior. For any K E K we have K = ~ { W ; W E l/P, W D K } and K ~ = [_J{W ; W c 142, W c K~}. It is sufficient for most purposes to restrict the choice of regions R to the following types : wedge-regions W, closed double cones K and the causal complements K ~. We investigate when the W i g h t m a n field r is associated to a local net W c VV ~ A(W) (K E lC ~ B(K)) of yon Neumann algebras by the existense of a standard system

T h e o r e m 4.5.2. The following statements hold: (1) ~2 is a cyclic and separating vector for "P(WR) and 7)(WL), and so X~2 ---+xt[2, X C P(WR) (resp. 7)(WL)) is closable and its closure is denoted by S~(wR)~ (resp. S~(w~)~). A1/2 be the polar decomposition. (2) Let Sp(wn)~ = ~,~(wn)~p(w~)a~

Jp(wn)ga equals the antiunitary involution J = U(re3, 0)O0, where 713 denotes the rotation by angle re about the 3-axis and O0 denotes the canonical T C P operator , and A1/2 --p(wR)ca equals a positive self-adjoint operator V(irr) obtained by analytic continuation of a one-parameter unitary group {V(t)}t~]R of velocity transformations in the 3-direction. (3) S~(wL)~ = S~(wR)n = JV(-ire). (4) JP(WR)J = P(WL), V(t)P(WR)V(t) -1 = P(WR) and V(t) ~D(WL) V(t) -1 ----P(WL) for all t C ]R. We first consider when there exists a v o n Neumann algebra A(WR) on such t h a t (P(Wn), g2, .A(Wn)) is a standard system. T h e o r e m 4.5.3. Let .A(WR) be a von Neumann algebra on 7-/. Then ('P(WR), ~, A(WR)) is a standard system if and only if A(WR)' c P(WR)w and .A(WR) C ~(WL)'w. P r o o f . Suppose t h a t (P(WR), ~,.A(WR)) is a standard system. Then it follows from Theorem 2.2.4 that S~(wR)n = S.a(wR)n, which implies by Theorem 4.5.2, (4) t h a t

A(WR) ---- JA(WR) J c JP(WR)wJ

---

5[)(WL)w 9

p

!

Conversely suppose that .4(WR)' c T'(WR)w and .A(WR) C P(WL)w. By Theorem 1.4.2 and Proposition 1.2.3 e.a(wn),(X) is affiliated with .A(Wn) for each X c P(Wn). Hence, since P(Wn)S2 is dense in 7-/, it follows t h a t

A ( W n ) ~ is dense in 7-/.

(4.5.1)

Similarly, we have that .A(Wn)'$2 is dense in T/. Furthermore, we have

S~(wn)~ c SA(wR)~

and

which implies by Theorem 4.5.2, (3) that

Sp(wL)~ C SA(WR)'~,

4.5 Standard systems in the Wightman quantum field theory

215

SA(wR)n = S~(wR),9 a S~,(wL) ~ = S~(wR)n a SA(wa)e. Hence, SA(wn)n = Sp(wn)~. Therefore it follows from Theorem 4.5.2, (4) that ( P ( W n ) , ~, At(Wn)) is a standard system. This completes the proof. T h e o r e m 4.5.4. Consider the following statements. (i) ~b is associated to some local net K 9 IC --+ B ( K ) of von Neumann algebras. (ii) There exists a standard system ( P ( W R ) , ~, At(WR)) such that (a) U(A)At(Wn)U(A) -1 = At(WR) for each A c P s.t. AWR = WR ; (b) U(A)At(Wn)U(A) -1 c A ( W n ) for each A E P s.t. A W R C W R ; (c) U(A)At(WR)U(A) -1 C At(WR)' for each A 9 P s.t. A W n c WR ~ = WL ; (d) {U(A)At(Wn)U(A) -1 ; A W R C g o } '' C P ( K ) ' w, K 9 IC. (iii) ~b is associated to some local net W 9 W --+ At(W) of yon Neumann algebras. (iv) There exists a standard system (79(WR), YI, At(WR)) satisfying the conditions (a), (b) and ( c ) i n (ii). Then the following implications hold : (i) ~

(ii)

(iii) +==~ (iv) . P r o o f . (iii) => (iv) Since 9 is associated to a local net W c 142 --+ At(W), it follows that At(Wn)' c P ( W R ) ~ and At(WR) C At(WL) t C ~(WL)Pw, which implies by Theorem 4.5.3 that (:P(Wn), $2, At(Wn)) is a standard system. It is clear that ( P ( W n ) , ~ , A t ( W R ) ) satisfies the conditions (a), (b) and (c). Furthermore, since At(Wn) c At(WL)', At(WL)Y2 is dense in 7-I and V ( t ) A t ( W L ) V ( t ) - I = At(WL) for all t C ]~ by the covariance of W --+ At(W), it follows from Theorem 2 in Bisognano-Wichmann [1] that At(WL) = At(Wn)', which implies

At(W c) = At(w)',

w c w.

(4.5.2)

(iv) =~ (iii) We put At(W) = U(A)At(Wn)U(A) -1,

W -- A W n c )/V.

Then we see by (a) that A(W) is well-defined for each W C ~/V, and W --+ At(W) has the covariance property. The condition (b) implies that W --+ At(W) has the isotony property. Furthermore, by (c) we have A ( W L ) C At(Wn)', which implies by the covariance of W --+ At(W) that At(W c) C At(W)' for each W c IV. Hence it follows that W --+ At(W) is local. Since

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4. Physical Applications

(P(WR), Y2, .A(WR)) is standard and W --~ A ( W ) has the eovariance property, it follows that A ( W ) ' c P(W)~w for each W E 142.. Thus we see that is associated to a local net W ~ A ( W ) . (i) =* (ii) We put A(w)

= {•(K)

; K c W}", W c W.

Then it follows from the isotony and A ( W ) has the same properties. Take Then it follows from the locality of K K , K1 E ~ s.t. K C W and K1 C ~ c , r

covariance of K ~ B(K) that W arbitrary W, W1 ~ /42 s.t. I4/1 C W ~. ~ I3(K) that B(K1) C B(K)' for each which implies

= {~(/~1); K1 C wC} H C A ( W ) ' = N { ~ ( / ~ ) ' ; K C W}.

Hence, W --~ A ( W ) is local. Since B(K)' C P ( K ) 'w for each K E tO, we have

•(w)'

= n { s ( K ) ' ; K c w } c ~{p(K)'w; K c w } =

~(w)'

w

for each W E 142, which implies that q5 is associated to the local net W A ( W ) . By the equivalence of (iii) and (iv) (P(Wn), ~, A(WR)) is a standard system satisfying the conditions (a), (b) and (c) in (ii). The condition (d) follows from {A(W); W c K~} '' = {B(K1); K1 C Kc} H C U ( K ) ' C V ( K ) 'w for each K E/C. (ii) ~ (i) We put

.A(W) = U(A).A(WR)U(A) -1, W = AWR E 142. As showed in the proof of (iv) =~ (iii), 9 is associated to the local net W -~ A ( W ) of yon Neumann algebras. We now put

~ ( K ) = I " ] { A ( w ) ; ~ c w } , K c ~c. Then it follows from the isotony and covariance of W --+ A ( W ) t h a t K --+ B(K) has the same properties. By (4.5.2) we have B(K)' = {A(W)';Kc

W}" = { A ( W ~) ; W ~ c K ~ } ''

c ~(K)'w,

K e tO,

and so it is proved as in (4.5.1) t h a t

t3(K)s

is dence in 7-/.

(4.5.3)

4.5 Standard systems in the Wightman quantum field theory Hence, .4 - {/3(K); K c more, it follows from the for each t C ]~, and so A [1], which implies that K

217

WR}" C A(WR) and .4/2 is dence in 7-/. Furthercovariance of K --* /3(K) that V ( t ) A V ( t ) -1 = A = A ( W R ) by Theorem 2 in Bisognano-Wichmann ~ / 3 ( K ) is local. In fact, we have by (4.5.2)

/3(K)' = {A(W)'; K c w } " = { A ( W C ) ; W ~ c KC} '' :

{ / 3 ( K 1 ) ; ~ ( 1 C K c } ''

/3(Ko) for each K, Ko E ~ s.t. K0 C K r Thus we see that @ is associated to the local net K --~/3(K). This completes the proof. R e m a r k 4.5.5. (1) Suppose that 9 is associated to a local net K C ]C --~/3(K). Then Y2 is a cyclic and separating vector f o r / 3 ( K ) , a n d / 3 ( K ) C p K( ~')w for each K E K:. In fact, by (4.5.3) S? is cyclic f o r / 3 ( K ) . It follows from the association of/3 with 4~ and the locality of/3 that

P(KC)~ = ~{7~(K1)~w; K1 C K c} D ~{/3(K1)'; K1 C K c}

/3(K), which implies as in (4.5.1) that B(K)'$2 is dense in 7-/. (2) Suppose that @ is associated to a local net W c 142 --~ A ( W ) . Then, A ( W c) = A ( W ) ' for each W E 142. This was already proved in the proof of Theorem 4.5.4 (in equation (4.5.2)). C o r o l l a r y 4.5.6. Suppose that "P(WR)~wis a v o n Neumann algebra. Then the following statements are equivalent. (i) (P(WR), f2, (P(WR)~w) ') is standard. (ii) (P(WR)rw) ' ---=~D(WL)Iw. (iii) r is associated to a local net W -~ (P(W)'w)'. P r o o f . (iii) =~ (ii) This follows from Remark 4.5.5, (2). (ii) ==~ (i) This follows from Theorem 4.5.2. (i) ~ (iii) It is clear that the conditions (a) and (b) in Theorem 4.5.4 hold. By Theorem 4.5.1 and the standardness of ('P(WR), f2, (T'(WR)~w)') we have P(WL)~w = JTP(WR)tw J = (7~(WR)~w)', which implies that

U(A)('P(WR)~)'U(A) -1 = (7)(AWR)~w)'C (P(WL)'w)' = P(wR)'w

for each A C P s.t. AWR c WL. Therefore, it follows from Theorem 4.5.4 that 9 is associated to a local net W --~ (T'(W)'w)'.

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4. Physical Applications

C o r o l l a r y 4.5.7. Suppose t h a t P(K)~w is a v o n N e u m a n n K E h:. T h e n the following s t a t e m e n t s are equivalent. (i) ( P ( W R ) , f2, (P(WR)'w)') is a s t a n d a r d system. (ii) 9 is associated to a local net W c W --* ( P ( W ) ~ ) ' of algebras. (iii) ~b is associated to some local net K E ~ -* B ( K ) of algebras. (iv) ~b is associated to a local net K E )~ ~ (P(K)~w) ' of algebras. (v) ~b is associated to a local net K E ~ ~ P(KC)'w of algebras.

algebra for a l l

von N e u m a n n von N e u m a n n yon N e u m a n n von N e u m a n n

P r o o f . Since "P(WR)~ = N { P ( K ) ~ w ; K c WR} and P ( K ) ~ is a v o n N e u m a n n algebra for all K c ~ , it follows t h a t "P(WR)'w is a v o n N e u m a n n algebra. (i) => (v) By Corollary 4.5.6 the s t a n d a r d s y s t e m ( P ( W R ) , ~2, (P(WR)~w) ') satisfies the conditions (a), (b) and (c) in T h e o r e m 4.5.4 and (7~(WR)~w) ' ---"P(WL)~. F u r t h e r m o r e , we have U ( A ) ( P ( W R ) t w ) ' U ( A ) -1 = "P(AWL)~w C "P(K)~

for each A C P s.t. A W R c K ~, and so it follows since "P(K)'w is a yon N e u m a n n algebra t h a t { U ( A ) ( ' P ( W R ) ~ ) ' U ( A ) - I ; A W R C KC} '' C "P(K)~w . Therefore, it follows from T h e o r e m 4.5.4 t h a t 9 is associated to a local net K --* U ( K ) , where B ( K ) = N{(P(W)~w)'; K C W } , K e / ~ . By R e m a r k 4.5.5 we have B(K) c

for each W 9 associated to a (v) ~ (iii) (iii) => (iv)

,

,w C P ( W

)w = ( P ( W ) ' w ) '

//V s.t. K C W, and so B ( K ) = P ( K ~ ) ~ . Therefore 9 is local net K ---* "P(K~)~. T h i s is trivial. B y the association of B with 9 w e have (P(K1)~w) ' C B(K1) C B ( K ) ' C : P ( K ) ~

for each K, K1 9 K: s.t. K1 c K ~, and so K ---* (P(K)~w) ' is local. It is clear t h a t K ---* ( P ( K ) ~ ) ' is isotony and covariant. F u r t h e r m o r e , since P ( K ) ' ~ is a voh N e u m a n n algebra, it follows t h a t # is associated to a local net K ~ (P(K)'w)'. (i) ~ (ii) This follows from Corollary 4.5.6. (iv) ~ (i) It follows from T h e o r e m 4.5.4 t h a t ('P(WR), S2, A ( W R ) ) is a s t a n d a r d s y s t e m for some von N e u m a n n algebra .A(WR), which implies by T h e o r e m 2 in B i s o g n a n o - W i c h m a n n [1] t h a t A ( W R ) = ~r p lt W RJwl ~' ~, 9 R e m a r k 4.5.8. (1) A W i g h t m a n field 9 is said to be satisfy a generalized H-bounded if there exists a nonnegative n u m b e r a < 1 such t h a t q s ( f ) e -H~

4.5 Standard systems in the Wightman quantum field theory

219

is a bounded operator for all f , where H denotes the Hamiltonian of 4~. Driessler-Summer-Wichmann [1] showed that if ~b satisfies a generalized Hbounded then T'(K)'w is a yon Neumann algebra for every K e ]C. (2) The Botchers tensor algebra S_ is defined by S = { f = (fo, f l , " " ); f0 E C, fn

9

S(]t~4n) for n _> 1}

equipped with the natural algebraic operations. The sequence of Wightman distributions Wn of the field ~b defines a positive linear functional 142 (Wightman functional) by )/V(f) = ~ - ~ W n ( f n ) , f C S (Borchers [2]). Borchersn

Yng'cason [2] defined the notion of centrally positivity of a positive linear functional ca on a *-algebra A with respect to a hermitian element a0 of A as follows : co is centrally positive with respect to ao if ca is positive on all elements of the form ~ a~a,,, an E A such that Z t'~an C 7)(A) for all t E ~ . rt

tt

This is an adaptation of a centrally positive operator defined by Powers [2]. Borchers-Yngvason [2] showed that ca is centrally positive with respect to a0 if and only if zr~(a0) has a self-adjoint extension ~-'~(a0), eventually in an extended Hilbert space 7-/~, such that the restrictions to 7)(zr~) of all bounded functions of ~ ( a 0 ) are contained in ~r~(fl,)~w, and using this, they obtained the following result: Let 9 be a Wightman field such that 7)(K)~w is a yon Neumann algebra for every K 9 K~. The following statements are equivalent: (i) ~5 is associated to a local net K E ]C ~ (P(K)'w)'. (ii) Let f be any element of S ( ~ 4) such that support of f is contained in the interior of a double cone K, and 8 ( f , K c) the subalgebra of _S generated by f and {g 9 ,S(]~4); support of g c KC}. Then W[SS_(I,KC) is centrally positive with respect to f . We investigate the connection between a standard system and the association of a local net of von Neumann algebras with ~b considering a commutant ~b(WR)/~, which is stronger than the weak commutant "P(WR)'w and is always a von Neumann algebra. For K 9 we denote by 9t-(K) the set of all real test functions f such that qb(f) 9 P ( K ) and the Fourier transform ] of f satisfies the condition ](p) 0 for all p. For f c 5~(K) we denote by T~(f) the yon Neumann algebra, generated by r i.e. 7r -- {C 9 ~ ( f ) C ~ C~(f)}', and denote by 9Vt(K) the set of all elements f of 9t-(K) such that U(A)TC(f)U(A) -1 c 7~(f)' for each A ~ P s.t. A K C K c. For R c M we define a commutant ~b(R)~ of P(R) which is always a yon Neumann algebra as follows : {c 9

; CV c

-

('/

X e'P(R) and C*:D C ~(~~

220

4. Physical Applications Theorem

4.5.9. Consider the following statements. ' ' is standard. ('P(WR)ss)) ~' ~, = P(WL)w. (ii) 73(WR)'~ = 7)(Wn)~w and r~p r~W R~wJ (iii) q5 is associated to a local net W E ~V -* ( p ( w G,) / . (iv) 9 is associated to a local net K E ~ -* B ( K ) such t h a t (i)

(7)(WR),.(2,

~

!

(~(wR)8~)

!

= {~(K); K c WR}".

(v) 5Vt(K) = ~ - ( K ) for each K 9 K. T h e n the s t a t e m e n t s (i) ~ (iv) are equivalent, and they imply the s t a t e m e n t

(v). P r o o f . It is easily shown by the definition of P(Wn)r~ and T h e o r e m 4.5.2 that

~ ( p ( w ) ) = U(A)~(P(WR)), 7)(W)~ = U(A)75(WR):.~U(A) -1, W = AWR ; J~(p(w,~)) = # ( p ( w L ) ) , JT)(WR)~J = P(WL)~ ; p(n2)l~ c P(RG~ ' if R, c R2. -

!

-

!

!

4.5.4) 4.5.5) 4.5.6)

(i) ==> (iv) By (4.5.4) t h e conditions (a) and (b) in T h e o r e m 4.5.4 hold. B y (4.5.5) and the s t a n d a r d n e s s of ( P ( W R ) , ~ , (13(Wt~)'~) ') we have I

(I)(WR)~8)

/

-

I

= J'P(Wn)ss J = T'(WL)~,1

(4.5.7)

which implies t h a t the condition (c) in T h e o r e m 4.5.4 holds. Furthermore, we have by (4.5.4), (4.5.6) and (4.5.7)

{(P(w)2)'; w c ~:~}" c P ( K ) 2 c P(K)'w for each K 9 ~ , and so the condition (d) holds. (iv) ~ (iii) This follows from the proof of (i) ~ (ii) in T h e o r e m 4.5.4. (ii) ::~ (i) This follows from T h e o r e m 4.5.4. Thus we see t h a t the statemerits (i), (iii) and (iv) are equivalent. (i) ~ (v) Let f 9 b~(K). We put -

!

B(K) = N { ( P ( W ) ~ )

!

; KcW}, K9

T h e n it is clear t h a t T~(f) c B(K), and so it follows from the locality of K 9 IC --+ B(K) t h a t

U(A)7~(f)U(A) -~ c B(AK) c B(K)' c 7~(f)' for each A 9 P s.t. A K c K ~. Hence, f ~ ~ t ( K ) . (ii) ~ (i) This follows from Corollary 4.5.6.

4.5 Standard systems in the Wightman quantum field theory

221

(i) ~ (ii) Let f E 5r(K). Then, f 9 9re(K) as seen above. We put

A ( W ) = {U(A)TZ(I)U(A) -1 ; A K c W } , W 9 142. Then it follows from Theorem 4.6, Lemma 4.7 and Theorem 4.8 in DriesslerSummers-Wichmannc[1] that (P(WR), f2, A(WR)) is a standard system such that A (W) = ~ ~ f)'w for all W E ~12, where ~~ f ) denotes the 0 % algebra on I9 generated by {U(A)qS(f) U(A) -1 ; A K C R}. Hence we have

p(wR)" c

(wR,

A(WR)' c p(wR) ,

= A(WL) =

and so P(WR)'w = A ( W R ) ' and it is a yon Neumann algebra. Hence, (7)(Wn), / ! f-2, (~o(WR)w)) is a standard system, and so we have by (4.5.10) ~

p(wR)

/

!

c P(WR)w

and -

I

!

-

J75(WR)~s~J C JT'(WR)~J

/

= P(WL)'w =

(~(WR)')'.

Hence, 75(WR)'s = T'(Wn)~. This completes the proof. R e m a r k 4.5.10. Suppose 5re(K) r r for some K c]C. Then, by the proof of (i) ~ (ii) in Theorem 4.5.9 9 is associated to a local net W --~ (P(W)'w)' of yon Neumann algebras. Driessler-Summers-Wichmann [1] have showed that ! is not necessarily associated to a local net K ~ P ( K C )w, but it is weakly associated to a local net K ~ p(KC)'w in the following sense : c

I

(P(K)w)

I

c

!

7)(K,f)w, K E/C.

C o r o l l a r y 4.5.11. Suppose that P ( W n ) is essentially self-adjoint. Then the following statements are equivalent. (i) (7)(WR), f-2, (P(WR)'w)') is standard. (ii) ~b is associated to a local net W E W --~ ( P ( W ) ~ ) ' of yon Neumann algebras. (iii) 4~ is associated to some local net K c ~ --* B ( K ) of von Neumann algebras. (iv) ~5 is associated to a local net K E K: --* :P(KC)'w of yon Neumann algebras. P r o o f . Since P ( W R ) is essentially self-adjoint, it follows that 7:'(Wn)'w = 73(WR)~8 and it is avon Neumann algebra, so that by Corollary 4.5.6 and Theorem 4.5.9 the statements (i) and (ii) are equivalent.

222

4. Physical Applications

(iv)

==~ (iii) This is trivial. (iii) ~ (i) This follows from Theorem 4.5.4 and Theorem 2 in BisognanoWichmann [1]. (i) ~ (iv) This is proved as in the proof of (i) ~ (iii) in Corollary 4.5.7. We finally give some examples of modular systems and quasi-standard systems for a Wightman field. E x a m p l e 4.5.12. Let ~b be a Wightman field. We could give some examples of standard systems (7)(W), ~2, A(W)) for wedge-regions W. But, it is difficult to give examples of standard systems for domains except wedgeregions. Suppose that ('P(WR), f2, .A(WR)) is a standard system. Then, for each open subset R of WR (79(R), ~, .A(WR)) is a quasi-standard. But, we don't know whether ('P(R), f2, A(R)) is a standard (or modular) system for some local net R ~ A(R) of von Neumann algebras. There is only one example known in case of a massless free field where the domain in the double cone. By Theorem 2 in Hislop-Longo [1] we see that for a massless free field (P(O), f2, (P(O)~)') is a modular system for each open double cone O in It{4 . Notes 4.1. The quantum moment problem was first studied by Sherman [1] who gave an affirmative answer for a countably generated O*-algebra fl,t which contains the restriction to 79 of the inverse of some compact operator. Woronowicz [1], [2] proved that the O*-algebra/:t(S(N[)) and the O*algebra generated by the position and momentum operators on S(]~) are QMP-solvable. Schmiidgen [2], [3], [21] generalized these results to more general O*-algebras. Theorem 4.1.6 extends the corresponding results of Sherman [1], Woronowicz [11, [2] and Lassner-Timmermann [1], and the proof is according to that of Theorem 2 in Schmiidgen [2]. We here use the algebraic conjugate dual of 79t of a dense subspace of 79 in a Hilbert space 7-/ and the ordered *-vector space ~;(79,79t). These have also appeared in Inoue [6, 16] and Tomita [2]. Let AA be a closed O*-algebra on 79 in 7-/ and 79~ a topological conjugate dual of the locally conver space 79M ~ 79[tM]. The rigged Hilbert space 79[tz4] c 7Y c 79~ and the space of all continuous linear operators C(79z4,79~) from 79z4 to the locally convex space 79~ equipped with the strong topology ~ whose subspace/2(79~, 79~) of all continuous sesquilinear forms on 79z4 x 79~ have been studied in Kiirsten [1, 3], Trapani [1] and Schmiidgen [21]. Theorem 4.1.7 and Theorem 4.1.10 are due to Schmiidgen [2], [21]. Theorem 4.1.13 is due to Lemma 5.2 in InoueUeda-Yamauchi [1]. Theorem 4.1.14 is due to Lemma 5.2 and Theorem 5.3 in Inoue-Ueda-Yamauchi [1]. Quantum moment problem for partial O*-algebras has been studied in Antoine-Inoue [1] and Kiirsten [3]. 4.2. This work is due to Inoue-Kiirsten [1].

4.5 Standard systems in the Wightman quantum field theory

223

4.3. Lemma 4.3.6 and Theorem 4.3.7 are due to Lemma 8.2 and Theorem 8.3 in Powers [i], respectively. Guder and Hudson [I] have considered the Tomita-Takesaki theory in the CCR-algebra 7r0(~4), and obtained the same results as those in Theorem 4.1.3. Theorem 4.3.14 is due to Example 5.2 in Antoine-lnoue-Ogi-Trapani [I]. 4.4 This work is due to Antoine-lnoue-Ogi-Trapani [i]. 4.5 This work is due to Inoue [14].

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[1] The Radon-Nikodym theorem for von Neumann algebras. Acta Math. 130(1973), 53-88. R.T. POWERS [1] Self-adjoint algebras of unbounded operators. Commun. Math. Phys. 21(1971), 85-124. [2] Self-adjoint algebras of unbounded operators II. Trans. Amer. Math. Soc. 187(1974), 261-293. [3] Algebras of unbounded operators. Proc. Sym. Pure Math. 38(1982), 389406. M. REED and B. SIMON

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[1] A necessary and sufficient condition for a yon Neumann algebra to be in standard form. J. London Math. Soc. 15(1977), 147-154. D. RUELLE

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Index

absolutely convergent series, 32 B C S - B o g o l u b o v model, 209 canonical algebra for s-degree of freedom, 196 canonical representation of a trace class operator, 31 canonical T C P - o p e r a t o r , 214 causal complements, 214 closed double cones, 214 closed forward light cone, 213 closure of O-algebra, 9 c o m m u t a n t s of generalized vectors, 43 c o m m u t a n t s of O*-algebras - strong, 12. - unbounded, 26 weak, 12

-

modular, 58 quasi-standard, 57 regular, 44 relative modular, 94 semifinite, 88 standard, 57 strongly cyclic, 42 strongly regular, 44 tracial, 61 generalized von N e u m a n n algebra, 27 GNS-construction - for (quasi-)wights, 115 - for positive linear functionals, 30 graph topology, 9, 27 -

Heisenberg c o m m u t a t i o n relation, 196 hereditary positive subcone, 115

-

essentially self-adjoint Ad-invariant subspace, 16 e x t e n d e d W*-algebra (EW*-algebra), 26 extension - of generalized vectors, 42 - of O-algebra, 8 - of (*)-representation, 27 of (quasi-)weights, 143 generalized Connes cocycle t h e o r e m - for generalized vector, 92 - for (quasi-)weights, 136 generalized H-bounded, 218 generalized Pedersen-Takesaki R a d o n Nikodym theorem - for generalized vector, 103 - for (quasi-)weights, 145 generalized vector, 42 - cyclic, 42 - essentially standard, 57 full, 48 - induced by a quasi-weight, 124, 125 -

induced extension of O*-algebras, 19 induced topology, 9 induction of O*-algebras, 15 interacting Boson model, 206 intertwing space, 28 KMS-condition, 54 Lebesgue decomposition t h e o r e m for weights, 144 Ad-invariant subspace, 15 Minkowski space, 212 m o m e n t u m operator, 18 net property, 119 O-algebra, 8 O*-algebra, 8 - algebraically self-adjoint, 10 - closed, 9 - defined by a generalized vector, 42 - essentially self-adjoint, 10

240

-

-

Index

h a v i n g t h e von N e u m a n n d e n s i t y t y p e p r o p e r t y , 153 i n t e g r a b l e , 10 Q M P - s o l v a b l e , 177 self-adjoint, 10 semifinite, 70 s p a t i a l l y semifinite, 104 s t a n d a r d , 10 s y m m e t r i c , 26

P o i n c a r 6 - e q u i v a l e n t , 213 P o i n c a r 6 group, 213 p o s i t i o n o p e r a t o r , 18 p o s i t i v e linear f u n c t i o n a l o n , - a l g e b r a s , 29 - c e n t r a l l y positive, 219 o n O*-algebras, 30 - a - w e a k l y c o n t i n u o u s , 130 q u a n t u m m o m e n t p r o b l e m , 171 Radon-Nikodym theorem for (quasi-)weights, 143 for p o s i t i v e linear functionals, 153 r e g u l a r p a r t of a vector, 165 r e p r e s e n t a t i o n , 27 * - r e p r e s e n t a t i o n , 27 - a l g e b r a i c a l l y self-adjoint, 28 - closed, 27 - essentially self-adjoint, 28 self-adjoint, 28 - s t r o n g l y positive, 199 -

S c h r 6 d i n g e r r e p r e s e n t a t i o n , 4, 9, 197 S c h w a r t z space, 9 s i n g u l a r p a r t of a vector, 165 space-like s e p a r a t e d , 213 s t r o n g l y c o m m u t i n g self-adjoint o p e r a t o r , 20 s t r o n g l y p o s i t i v e linear f u n c t i o n a l o n C C R - a l g e b r a s , 199 - o n O*-algebra, 30 system - cyclic a n d s e p a r a t i n g , 44 - essentially s t a n d a r d , 56 full, 48 - g e n e r a l i z e d m o d u l a r , 109 - generalized s t a n d a r d , 107, 109 - m o d u l a r , 58 q u a s i - s t a n d a r d , 57 regular, 43 s t a n d a r d , 52, 56 s t r o n g l y cyclic a n d s e p a r a t i n g , 45 - s t r o n g l y regular, 44 -

topolog:r o n O * - a l g e b r a s - A-topology, 25 p r e c o m p a c t , 36 - q u a s i - u n i f o r m , 24 p-topology, 25 - strong*, a - s t r o n g * , 23 strong, a - s t r o n g , 23 u n i f o r m , 24 - weak, a - w e a k , 23 t r i o - c o m m u t a n t s for a quasi-weight, 120 -

u n b o u n d e d b i c o m m u t a n t s of O*algebras, 26 u n b o u n d e d C C R - a l g e b r a , 4, 82, 196 u n b o u n d e d T o m i t a algebra, 204 u n b o u n d e d vector, 177 u n i t a r y equivalence, 29 v a c u u m state, 213 vector - cyclic, 17 - m o d u l a r , 61 q u a s i - s t a n d a r d , 61 s t a n d a r d , 61 - s t r o n g l y cyclic, 17 - ultra-cyclic, 17 wedge-regions, 213 (quasi-)weight 114 o n A4+, 115 ~,-absolutely c o n t i n u o u s , 143 - defined by a v e c t o r in 7-l, 116 - Q - d o m i n a t e d , 143 essentially s t a n d a r d , 126 faithful, 125 - {at~'}-invariant, 148 {c~t }-KMS, 128 - { a r } - K M S , 127 - on P(Tr~,(JM)~r i n d u c e d by a q u a s i - w e i g h t o n T'(JM), 127 q u a s i - r e g u l a r , 120 - q u a s i - s t a n d a r d , 126 regular, 120 semifinite, 125 - singular, 120 - Q-singular, 143 - s t a n d a r d , 126 - s t r o n g l y faithful, 141 - a - w e a k l y c o n t i n u o u s , 130 Weyl f o r m u l a t i o n of t h e c o m m u t a t i o n relations, 198 -

-

o n

7 ~ ( A d ) ,

Index Weyl representation, 198 Wightman field, 212

-

associated to a local net of von Neumann algebras, 213

241

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