Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris
1699
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
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
Cataloging-in-Publication Data applied for
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 - (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
.,
- 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 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-
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, 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~)) > =
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
(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 ~ = < 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 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 , 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:,~ 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 "-- 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)
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 ) 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 (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 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)
=
,
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 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| 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~anda.
Taking rn ~ 0o, we have ( T ~ n ]~n) < e
for all n > ArE and ~,
and sup ( T ~ I ~ , ~ ) 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
190
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) [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) 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
] x 4 ]},
W L = { x C ]~4 ; x3