SEMI-METRICS ON THE NORMAL STATES OF A W*-ALGEBRA by ' S. DAVID PR.OMISLOW B. Comm., U n i v e r s i t y of Manitoba, i 9 6 0 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department â€¢v of MATHEMATICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard The U n i v e r s i t y of B r i t i s h Columbia . â€¢ March 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r equ i r emen t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree tha p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l owed w i t hou t my w r i t t e n p e r m i s s i o n . Department o f M A T H EM A T J^S The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada Date W f r C H 1910 i i . Supervisor: D. C. Bures <â€¢ ABSTRACT i In t h i s thesis we investigate certain semi-metrics defined on the normal states of a W -algebra and t h e i r ap-p l i c a t i o n s to i n f i n i t e tensor products. This extends the work of Bures, who defined a metric d on the set of normal states'by taking d(n,v) = i n f [||x-y||) , where the infimum i s taken over a l l vectors x. and y which induce the states and v respectively r e l a t i v e to any representation of the 1 algebra .as a- von-Neumann algebra. He then made use of t h i s metric i n obtaining a c l a s s i f i c a t i o n of the various incomplete tensor products of a family of semi-f i n i t e W -algebras, up to a natural type of equivalence known as product isomorphism. By removing the semi-finiteness r e s t r i c t i o n form Bures' "product formula", which delates the distance under d between.two f i n i t e product states to the distances between t h e i r components, we obtain t h i s tensor product c l a s s i f i c a t i o n f o r f a milies of a r b i t r a r y W -algebras. Moreover we extend the product formula to apply to the case of i n f i n i t e product states. * For any subgroup G of the -automorphism group * G of a W -algebra, we define the semi-metric d on the set of normal states by: d (n,v) = i n f [d(u ,v ) : a,(3 â‚¬ Gl ; where V.a i s defined by u a(A) = .Â»j(a(A)) . We show the s i g n i f i c a n c e i i i . o f d i n c l a s s i f y i n g i n c o m p l e t e t e n s o r p r o d u c t s up t o weak p r o d u c t isomorphism, a n a t u r a l weakening of t h e concept o f p r o d u c t isomorphism. I n t h e case o f t e n s o r p r o d u c t s of s e m i - f i n i t e f a c t o r s , we o b t a i n e x p l i c i t c r i t e r i a f o r such a c l a s s i f i c a t i o n by c a l c u l a t i n g d ( g , v ) i n terms o f t h e Radon-Nikodym d e r i v a t i v e s o f the s t a t e s . I n the course o f t h i s c a l c u l a t i o n we i n t r o d u c e a concept of c o m p a t i b i l i t y w h i c h y i e l d s some o t h e r r e s u l t s P about d and d . Two s e l f - a d j o i n t o p e r a t o r s S and T are s a i d t o be c o m p a t i b l e , i f ' g i v e n any r e a l numbers a and Â£ , e i t h e r E ( a ) <. F((3) or F ( 0 ) _< E ( a ) ; where ( E ( x ) l , ( F ( \ ) } , a r e t h e s p e c t r a l r e s o l u t i o n s o f S,T , r e s p e c t i v e l y . We o b t a i n some m i s c e l l a n e o u s r e s u l t s c o n c e r n i n g t h i s concept. TABLE OP CONTENTS PAGE INTRODUCTION 1 HISTORICAL NOTE ON INFINITE TENSOR PRODUCTS j 4 CHAPTER 0. PRELIMINARIES.. ON W*-ALGEBRAS 11 CHAPTER I . THE. METRIC d 26 1. B a s i c p r o p e r t i e s o f d and p ' ...26 â€¢2. The p r o d u c t f o r m u l a f o r p 29 CHAPTER I I . THE SEMI-METRICS d G 34 p c 3. D e f i n i t i o n of d and . p 34 4. An i n e q u a l i t y 36 5. Monotone f u n c t i o n s 40 6. D i s t r i b u t i o n f u n c t i o n s and a 44 â€¢7. C o m p a t i b i l i t y .'. 56 â€¢ v 8. C a l c u l a t i o n o f p f o r s e m i - f i n i t e f a c t o r s .'.68 CHAPTER..III. APPLICATIONS TO INFINITE TENSOR PRODUCTS ..80 1 9. Isomorphisms o f p r o d u c t s : D e f i n i t i o n s 80 10. Main ; r e s u l t s on st e n s o r p r o d u c t s 83 BIBLIOGRAPHY â€¢ 95 ACKNOWLEDGEMENT I am gratef u l to Dr. Donald C. Bures who suggested the topic of t h i s t h e s i s , and who provided valuable assistance and. encouragement throughout i t s preparation. ^ â€¢:: s l \ t INTRODUCTION In [3] Bures defined a metric d on the set of a l l normal states of a W -algebra G by taking d(ii,v) = i n f (||x-yl|3 i where the infimum i s taken over a l l vectors x and y inducing the states \A and v respectively r e l a -t i v e to any representation of 0 as a von Neumann algebra. In t h i s thesis we .generalize t h i s d e f i n i t i o n by defining, for any subgroup G of the group of -automorphisms of G , a semi-metric d G , given by d G(|i,v) = i n f ( d ( u a , v P ) : a,fi e G} , where V*a(A) = ^i(a(A)) . In p a r t i c u l a r we are interested i n the case where G i s either the group of a l l -automorphisms, or the group of inner automorphisms, i n which case d G i s denoted by d or cf respectively. G We then investigate certain aspects of d and d with 'a view towards applications to i n f i n i t e tensor products. We f i r s t prove the "product formula", / P(H 1 Â® \x2 , \>1 Â® v 2 ) = pC|i1,v1). p(|a 2,v 2) for normal states and of G-^ and and v 2 of . Gg where G^ and G 2 are a r b i t r a r y W -algebras. Here 1 2 p(y,v) =-^[d(^,v)] . (See Theorem 2.4 below). This extends the r e s u l t of [3] where t h i s formula i s proved under the assumption that G-, and G 9 are semi-finite. As a major application of t h i s r e s u l t we can complete the c l a s s i f i c a t i o n of the incomplete tensor products of von' Neumann [12] up to product isomorphism [4]. In f a c t i n the notation of [4] we have that Â® (G ,u..) i s product i s o -i e l 1 2 morphic to Â® (C,,v.) i f and. only i f E [d(u.,v.)] < Â» . i e i 1 1 i e l In the notation of [12] the equivalent statement i s that Â® (G.,x ) i s product isomorphic to Â® (G.,,y.) i f . and only i e l 1 1 i e l 1 1 â€¢ i f the states and induced by the vectors' x^ and y^ respectively s a t i s f y the above condition^ In connection with t h i s r e s u l t we extend the pro-duct formula to apply to the case of i n f i n i t e product states. In Theorem 10.8 below we indicate the sign i f i c a n c e of d i n the c l a s s i f i c a t i o n of incomplete tensor products Â» up to weak product isomorphism. This type of isomorphism (see D e f i n i t i o n 9.1 below) i s a natural weakening of the con-cept of product isomorrlhism. For a countable family ( G ^ ) ^ e j of W -algebras, Â® (G^,u.) and Â® (G.,v.) are weakly pro-i e l 1 1 i â‚¬ l 1 1 2 , duct isomorphic i f and only i f Â£ [ c f(Li.,v. )] < Â» i e l 1 1 In the case where each G^ i s a semi-finite fac-tor we make thi s a more e x p l i c i t c r i t e r i a by c a l c u l a t i n g d i n terms of the Radon-Nikodym derivatives of the states (see Theorem 8.6 below). In the course of thi s c a l c u l a t i o n we introduce a concept of compatibility which y i e l d s some other r e s u l t s concerning v d . Suppose that S i s an a r b i t r a r y set and that f and g are any r e a l valued functions on S . We can c a l l f and g compatible i f f o r any pair x,y e S , f (x) j< f ( y ) implies g(x) _< g(y) . We extend t h i s idea to define compatibility between two s e l f - a d j o i n t operators on a H i l b e r t space. We then define two states on a semi-finite factor to be compatible i f t h e i r Radon-Nikodym derivatives are compatible. We show that f o r a fac t o r of type I or 1 1 ^ . -compatibility minimizes the distance between two states, i n the sense that d(|i,v) = d(u,v) whenever u and v are compatible. Moreover we show that for t h i s class of factors d * d . Following t h i s introduction i s a h i s t o r i c a l note on i n f i n i t e tensor products. We indicate there i n more d e t a i l the Connection between some of the problems considered i n th i s thesis and the previous work i n t h i s subject. In Chapter 0 we review some basic r e s u l t s on von < Neumann and W -algebras and introduce some notation and terminology. The main references f o r t h i s material are [5] and [ 9 ] . I t should be noted that reference [5] i s f o r the second e d i t i o n of Dixmier's book. HISTORICAL NOTE ON INFINITE TENSOR PRODUCTS The theory of tensor products of von Neumann alge-bras was originated by von Neumann, f i r s t f or the case of a f i n i t e family of algebras i n [ 9 ] , and then f o r the case of an i n f i n i t e family i n [12]. Consider f i r s t the case where we have von Neumann . algebras G-^ and G 2 acting on H i l b e r t spaces H-^ and H 2 respectively. The usual vector space tensor product of H^ â€¢and H 2 can be completed, by means of the inner products on the respective ,spaces, to another H i l b e r t space. This space i s known a s i t h e H i l b e r t space tensor product of H^ . and H 2 and usually denoted by H^ Â® H 2 . Then we can form a -algebra G Q , generated by elements A-^ Â® A 2 where A^ e G^ and A 2 e G 2 , with the usual b i l i n e a r re-l a t i o n s , and with m u l t i p l i c a t i o n and involution .defined by â€¢" (A-j^ Â® A 2) (B1 Â® B 2) = A ^ Â® A 2B 2 (A-^ Â® A 2 ) * = A^ Â® A 2 ; . G ', i s then a , -algebra tensor product of the -algebras G^ and G 2 . The representations of G-^ on H-^ and G 2 on H 2 ^Induce.in a natural way a representation of GQ 1 on H.^ Â® H 2 , and the von\ Neumann algebra generated by t h i s representation of G0; i s known as the tensor product of G-^ and G 2 .â€¢ Next, suppose we have an i n f i n i t e family of von Neumann algebras ( ^ i ) i e j acting on H i l b e r t spaces ( ^ i ) i â‚¬ i In [12], von,Neumann extends the construction i n the f i n i t e case to produce a H i l b e r t space known as the t o t a l tensor product of the family (H. ) and denoted by Â® (H. ) Again we can construct G Q , a -algebra tensor product of the family (G i) , and represent G Q on Â® (H.^ ) to obtain a von Neumann algebra, known as the t o t a l tensor pro-duct of the family (G^ .) . This t o t a l tensor product how-ever f a i l s to possess many of the properties which are en-joyed by the f i n i t e tensor product. For example: the t o t a l tensor product of'factors i s not i t s e l f a factor; the t o t a l tensor product of a family of f u l l operator rings i s not i t s e l f a f u l l operator ring ; the usual associativety laws for tensor products f a i l to hold; the algebra does not act on a separable space e^en i i f each H/ i s separable and I i s countable. A remedy f o r this' s i t u a t i o n i s provided by the decomposition of von ( Neumann. He showed that Â® (H^) s p l i t s up into a d i r e c t sum of orthogonal subspaces each of which i s invariant for the algebra G Q . These subspaces are known as incomplete tensor'products for the family (H i) and the von, Neumann algebras generated by G Q on each of these subspaces are known as incomplete tensor products of the family (G i) . I t i s the incomplete tensor pro-ducts which appear to be the proper generalization of the f i n i t e tensor product. 6. Each incomplete space i s completely determined by choosing a family of unit vectors (x i) with x i â‚¬ H i . If (x^) and (y i) are two such fa m i l i e s s a t i s f y i n g Â£ | l - (x.|y.)| < Â» they determine the same incomplete â€¢ i e l 1 1 space and therefore the same algebra (More d e t a i l s are given i n Chapter 0). Consequently, i n the f i n i t e case there i s only one incomplete tensor product algebra which coincides with the tensor product mentioned above. In the i n f i n i t e case however there are many such algebras, and a basic prob-lem i s to determine the relationships between them. Much of the theory can be s i m p l i f i e d by removing consideration of the underlying H i l b e r t spaces. Misonou showed i n [7] that for a f i n i t e family-of von Neumann alge-bras the r e s u l t i n g tensor product was independent, as an algebra, of the p a r t i c u l a r representations of the given family. One could then' speak of a W -tensor product of a f i n i t e family of W -algebras. Nakamuru i n [11] and Takesaki i n [19] gave space-free d e f i n i t i o n s of the f i n i t e tensor product i n cer t a i n s p e c i a l cases. Takeda i n [17] construc-ted a W -tensor product of an i n f i n i t e family of W -alge-bras by using inductive l i m i t s . Bures i n [4] gave a general d e f i n i t i o n f o r the W -tensor product of an a r b i t r a r y family of W -algebras and showed how to obtain the W -analogue of von Neumann's incomplete tensor products, c a l l e d l o c a l tensor products, by using f a m i l i e s of states i n place of f a m i l i e s of vectors. Let (G..)., T be a family of W -algebras. For every family ( ^ i ) ^ 6 j where i s a normal state on G i one obtains a l o c a l tensor product of the family (G^) , denoted by Â® (G. ) . A fundamental problem i s then to i e l 1 1 determine when two such f a m i l i e s of states produce tensor products which are e s s e n t i a l l y the same ( i . e . product isomor-phic). The solution, obtained p a r t i a l l y i n [3] and completed i n t h i s t h e sis, shows that t h i s happens i f and only i f the re spective components of the fa m i l i e s are s u f f i c i e n t l y "close to each other". The metric d i s used to make t h i s idea precise. When we apply t h i s r e s u l t to von Neumann algebras we see that an incomplete tensor product does not r e a l l y de-pend on the family of vectors determining i t , but rather on the family of states induced by these vectors. This explains von Neumann's consideration of weak equivalence, ([12] def. . 6.1.1), since the vectors e x. and x always induce the "same state. The c r i t e r i a which we- develop for weak product isomorphism can be i l l u s t r a t e d by looking at countable ten-sor products of type I factors. This i s a s i t u a t i o n which has been considered by many authors, [1], [2], [8], [13], [16] â€¢ and [18]. In p a r t i c u l a r we can extend the re s u l t s discussed i n ([12] Chapt. VII) concerning tensor products of l 2 factors. Let G be a fi x e d I 2 factor represented on a two dimensional H i l b e r t space H . F i x any orthonormal basis ,(x,y) i n H . L e t iu x and ur denote the states on G 8 induced by the vectors x and' y respectively, and f o r each - i < a < 1 - l e t uu = auu + (1-a) uu . Let P be the <- c t x y set of a l l sequences of. numbers belonging to the i n t e r v a l [-Â£ i ] , arid for f = (a-^cxg. .. ) e F l e t 3"F denote the tensor product Â® (G. ,to ) > ' where Gi = G f o r 1=1,2,... 1 a i 1 a l l i . Then every countable tensor product of 1^ fac-tors i s weakly product isomorphic to f o r some f e F If f = (a i) and g = 6 F , 3"f and 3" are weakly product isomorphic i f and only i f they are product isomor-phic, which occurs i f and only i f E [ (VaT - + Ul-a, i = l . .- yr^ )2] < Â« . â€¢ Next, consider the more general case. Suppose that G i s a fa c t o r of type I , n < Â» , with trace T â€¢, i n normalized so that, i t s value on a minimal projection = 1 It i s well known that any state u on G i s given by u.(A) = T(SAS) where St' is, a unique p o s i t i v e element of G . I f S and T are two p o s i t i v e elements of G we l e t n a(S,T) = T. a.b. where a, > a 0 . .. > a are the eigen-values of S , and b i 2 . ^2 ' *' â€” ^n a r e t h e e iSenvalues of T ( i n both cases counted' with respective m u t i p l i c i t i e s ) . Now suppose we have a family ( G s ) . _ n 0 where 1 1 â€” JL y Â« â€¢ â€¢ Â« G i i s a factor of type ' I , 1 < n i < Â« . Let X = the set of a l l sequences ('S-^Sg. . . ) where S I i s a p o s i t i v e operator i n G^ such that the sum of the squares of i t s eigenvalues (counted with respective m u t i p l i c i t i e s ) = 1 .. Then,'by the remark above there is_..a 1-1 correspondence ^between f a m i l i e s of states where i s a 9. . normal state on G^ , and the elements of X . We define an equivalence r e l a t i o n ~ on X by s e t t i n g S = (S-^Sg...) â€¢ 09 ~ T = (T-,To...) i f and only i f E [l-o(S.,T.] < Â» . Then . . i * l 1 x 'â€¢ i t turns out that the l o c a l tensor products determined by S and T are weakly product isomorphic i f and only i f S ~ T . Using t h i s approach we can rephrase the r e s u l t s of [2], [8], [16] . and [18], concerning the type of the f a c -tor which arises from the tensor product.of f i n i t e type I fac t o r s , i n the following form. The ~ equivalence classes pf X which produce a semi-finite factor are p r e c i s e l y those which contain a se-quence consisting of scalar m u l t i p l i e s of projections. The â€¢ v unique equivalence class f o r which these projections are a l l equal to the i d e n t i t y producesthe h y p e r f i n i t e 11^ factor ' P . The unique equivalence class f o r which these projec-tions are a l l minimal produces the I factor. A l l the r e s t produce P Â® I , a factor of type II It follows then that each of the ~ equivalent classes which does not contain a sequence of scalar multiples of projections produces a type I I I fa c t o r , and no two of these are weakly product isomorphic. However there may be some that are a l g e b r a i c a l l y isomorphic. ( D e f i n i t i o n 9.1 below). The problem of determining c r i t e r i a f o r algebraic isomorphism i s a much more d i f f i c u l t one. Powers i n [1J>] considered e s s e n t i a l l y those tensor products of I 0 factors given i n our notation above by 3 ,^ where f i s a constant sequence. He showed that f o r d i f f e r e n t choices of the constant the ten-sor products are a l l non-algebraically isomorphic and . thereby produced an uncountable family of non-isomorphic type III factors. His r e s u l t s have been extensively generalized by Araki and Woods i n [ l ] . I f we take i n f i n i t e tensor products of Abelian a l -gebras the problem reduces to the study of i n f i n i t e product measure spaces. Suppose we have a family (G^) where each i s an abelian von Neumann algebra and therefore can be considered as L * ( n ^ ) f o r a suitable measure space 0^ The t o t a l tensor product of t h i s family i s just Lw(fi) where fl i s the i n f i n i t e product measure space of the . Sup-pose 0 = (X,IB,H ) i . Then the set X s p l i t s up into a union of subsets (X ) which are e s s e n t i a l l y d i s j o i n t ( i . e . H(X a n Xp) = 0 f o r a ^ P) > and each of which corresponds to a product isomorphism class of incomplete tensor products. In t h i s case, the c r i t e r i a f o r product isomorphism i n terms of the. metric d was shown i n [33 to y i e l d the Kakutani theorem on i n f i n i t e product measures [63. 11. ' â€¢/".'''..'â€¢ '. '.CHAPTER 0 PRELIMINARIES ON W*-Algebras â€¢ â€¢ von-Neumann Algebras. Let H be a H i l b e r t space over the complex numbers with norm, || || , and inner product, (|) . In addition to the usual norm topology on H we have the weak topology, whereby a net of vectors ( x a ) converges to a vector x i f (x |y) converges to (x|y) f o r a l l y â‚¬ H . This induces Ct j on the set of a l l bounded l i n e a r operators on H the weak operator topology whereby a net of operators (A ) converges to the operator A i f (A x) converges weakly to Ax -:' f o r v a a l l x â‚¬ H ' ;â€¢ An algebra G of bounded l i n e a r operators on H i s c a l l e d a von >, Neumann algebra on H i f ';. (a) The i d e n t i t y operator 1 e G i (b) G i s s e l f adjoint ( i . e . A e G implies that i t s adjoint A e G .) (c) G i s closed i n the weak operator topology. P a r t i c u l a r examples of von Neumann algebras are Â£(H) , the set of a l l bounded l i n e a r operators on H and (j; (H) , the^set of a l l scalar multiples of the i d e n t i t y opera-tor. v For an a r b i t r a r y subset , Â§ of x(H) , the commu-1 12. tant of Â§ , denoted by Â§' , i s defined as (T e Â£(H) : TS = ST f o r a l l S e S} . The fundamental double commutant theorem of von'Neumann states that a s e l f - a d j o i n t algebra G with i d e n t i t y i s a von,'Neumann algebra i f and only i f G = G " . For any subset Â§ of Â£(H) and any x e H , ' [Sx] w i l l ' , denote the norm closed l i n e a r span of (Sx : S e S} . If G i s a von Neumann algebra on H , then [G'X] â‚¬ G" = G . (We w i l l follow throughout the convention employed i n the sentence above of using the same symbol f o r a subspace and for the orthogonal projection onto that subspace whenever the meaning i s clear from the context.) The following i s a well known and important r e s u l t . Â» Lemma 0.1 . Let G and H be H i l b e r t spaces and l e t G be a von Neumann algebra on H . Suppose <j> i s a 1-1 algebra homomorphism of (F into X(G) s a t i s f y i n g (|>(A ) = (<j>(A))* f o r a l l A e G , and d>( 1) = 1 . Then (}>(G) i s a von;Neumann algebra on G .. Proof. See e.g. ([53 Chapt. 1 , Â§ 4 , Cor. 2 ) . / W -algebras By a -algebra we mean a li n e a r associative alge-bra G over with i d e n t i t y (always denoted by 1 ) , which possesses an involution; that i s a,map from G to i t s e l f , sending ;'-A.7 into /A , which s a t i s f i e s f o r a l l A,B â‚¬ G and a. (XA + B) = XA + B , (X denotes complex â€¢5 conjugation). b. A <= A , c. (AB) = B A By a homomorphism from G to Â© where . G and B are -algebras we w i l l always mean an algebra homomorphism a â€¢X- # s a t i s f y i n g a ( l ) = 1 , and a(A ) = (a(A)) f o r a l l A e G . * * A -algebra G i s ca l l e d a W -algebra i f f o r some Hil b e r t space H there i s a 1-1 homomorphism <j> from G. into the -algebra Â£(H) so that 4>(G) i s a von Neumann algebra on H . By a representation of a W -algebra G we w i l l always mean a homomorphism d) s a t i s f y i n g the above conditions. Consider now a fi x e d W -algebra G * I f H and ^ are sub -algebras of G which are them-selves W -algebras, Lemma1 0.1 shows that B fl 3 i s a W -alge-bra. Consequently,for any subset Â§ of G we may speak of the W -algebra generated by S .. ; s An element A e G i s called â€¢ a. s e l f - a d j o i n t i f A =/A* , , b.' a projection i f A i s s e l f adjoint and â€¢ ' ' A 2 = A 3 â€¢ ,:c/ p o s i t i v e i f A = B B f o r some B â‚¬ G , . di -â€¢ a p a r t i a l isometry i f A,A i s a projec-tiori (neccessarily AA i s also a pro-j e c t i o n )i e. a unitary i f A A = AA = 1 ; .j Two projections E and F are said to â€¢'( - â€¢â€¢ â€¢ â€¢ . . . ' â€¢ ' ' ' ( be orthogonal i f EP ? 0 It i s e a s i l y v e r i f i e d that any representation of G takes elements of the various types mentioned into opera-â€¢ i tors of the same type i n the usual sense. We w i l l denote the set of a l l p o s i t i v e elements of G. by G + .We w i l l denote by _< the usual p a r t i a l order defined on the set of s e l f adjoint elements of G whereby A _< B i f B - A â‚¬ G + . Any-element A of G + has a xonique + h square root â‚¬ G which we denote by A E From the well known polar decomposition theorem any A e G can be;written as U|A| , where | A | e G + ' and t U i s a p a r t i a l isometry e G s a t i s f y i n g AU U = A . We w i l l denote by Aut(G) > the group of a l l auto-morphisms of G and by *' Int(G) the subgroup of a l l a e Aut(G) f o r which there exists some unitary U â‚¬ G s a t i s f y i n g a(A) = UAU* for a l l A e G Any representation of G induces a norm on G ; , -(the usual operator norm) which makes i t into a Banach alge-bra. It i s easy to v e r i f y that t h i s norm i s independent of the p a r t i c u l a r representation and we may speak uniquely of the norm of an element A 5 which we denote by ||A|| For any projection E e G we l e t Gg = (A e G : A = EAE] . ^ Then G-c i s - a *-algebra with i d e n t i t y E . * Moreover Gg i s a W -algebra since any representation <J> of G on a H i l b e r t space H induces a representation of 15. G E as a von-Neumann algebra on,the subspace <f)(E) ([.5] Chapt. 1, Â§2, Prop. 1). States and Traces . A l i n e a r f u n c t i o n a l y on a W -algebra G i s cal l e d a state i f | j ( A ) J> 0 f o r a l l A e G + and u ( l ) = 1 . It follows that n ( A ) i s r e a l f o r any s e l f adjoint A . Moreover |i i s automatically bounded (considering G as a Banach space) with ||u|l = 1 A state n i s said to be f a i t h f u l i f M(A)' > 0 for a l l non-zero A â‚¬ G + . It i s said to be normal i f for any upwardly directed set ( A Q ) of s e l f adjoint elements with l e a s t upper bound A , ( i ( A ) = sup{ | j(A a)3 We w i l l deal e x c l u s i v e l y with normal states and w i l l denote by Eg the set orf a l l normal states of G A trace on G i s a mapping T from G + to the closed i n t e r v a l [0,Â«] s a t i s f y i n g "(a) T(kA + B) = k T ( A ) + T ( B ) , f o r a l l k _> 0 and A,B e G + /(we define 0-<Â»> = 0 ). (b) T ( U A U * ) = T (A) , f o r a l l unitary U â‚¬ G and a l l A â‚¬ G + We define a trace to be f a i t h f u l or normal i n an analogous way to the d e f i n i t i o n f o r states. We^say that a trace T i s f i n i t e ; i f T ( A ) < Â» f o r a l l A e G + , semi-finite i f f o r every non-zero S e G + there exists T _< S such that ( o < T ( T ) < Â« . 16. We say that the algebra G Is f i n i t e (semi-flnlte) . i f f o r every non-zero T â‚¬ G + there i s a normal f i n i t e (semi-f i n i t e ) trace T on G such that 0 < T(T) . D e f i n i t i o n 0.2 Let H be a l i n e a r f u n c t i o n a l on G : . For any o e Aut(G) we define a f u n c t i o n a l \xa by LiÂ°(A) = u(a(A)) , f o r a l l A e G For any T e G we define a f u n c t i o n a l |a^ , by U T(A) = n(TAT*) , f o r a l l A e G Lemma 0.5 Let (i 6 E G , a e Aut(G) , and T e G with U (TT*) = 1 . Then\ \xa and | i T â‚¬ y J Q and 1 a (T) Proof. Obvious calculations. For any trace., T on G we define i n an analogous way the mappings T A and T t /from G + to [0,Â»] â€ž If T i s normal then T q i s also a normal trace for a l l a â‚¬ Aut(G) If T i s normal and i f S e G i s such that T(SS .) = 1 then T s i s f i n i t e valued and may be extended by l i n e a r i t y to a normal state of G , which we w i l l denote also by TC In p a r t i c u l a r for a f i n i t e - t r a c e T with T(1) = 1 we can take S = 1 /. The r e s u l t i n g state (denoted simply by T ) has the property.that 17. T ( A B ) = T ( B A ) , f o r a l l A,B e G . . ' . ' \ - V \ \ For any state y> on G we have the ine q u a l i t y * (0.1) |M(AB) I < M ( A A " ) ^ | i ( B * B ) * , for a l l A , B e G , i which we obtain by applying the Cauchy-Schwartz inequality to the p o s i t i v e conjugate b i l i n e a r form, ( A , B ) - u ( B A ) We use th i s to prove some fa c t s about a f i n i t e trace. i * Lemma 0.4. Let G be a W -algebra with a f i n i t e trace T . Let A e. G and l e t E be a projection i n G . Then (a) T ( | A | ) = max ( | T ( A V ) | : V a p a r t i a l isometry e GI such that A W * = A ] . ,._ (b) | T ( E A ) | < | lA | | T ( E ) . Proof. (a) Let A =TU|A| be the polar decomposition of . A Then f o r any V s a t i s f y i n g the stated conditions we obtain from (0.1) |T(AV)| = |T(U|A|* \A\*V[ < [T(U|A|U*)3* [T(V*|A|V)]* [T(|A|U*U)]* [T(|-A|W*)]* = T|A| . â€¢ . Then taking V = U we obtain N |T(AU )| -..T(U|A|U ) = T(|A|) ' (b). Using (0.1) and the f a c t that EA*AE _< HA*A|| E :. we have 'V: ... ' |T(EA)| = |T(E2A) I = |T(EAE) I . ' < [T(EA*AE)]* [T(E.)]* < ||A*A||* [T(E)]* [T(E)]* = ||A||T(E) . We next consider r e l a t i o n s h i p s between states and vectors. If \x e T,^ and d) i s a representation of G on a H i l b e r t space H, we say that a vector x e H induces U r e l a t i v e to cj> I f u ( A ) = (<J)(A)x|x) , for a l l A e G . D e f i n i t i o n 0.5. Let ( H j ^ i e i b e a n y nonvoid family of ele-ments of T,n . We define i ( x i ) i â‚¬ ] [ ] : d) i s a represen- . tat ion of G ,on a H i l b e r t space H and (x^) "' i s a family of vectors i n H such that x^ induces r e l a -tÂ£ve._tb (J)--for a l l i e 1} , This set i s never empty (see e.g. the proof i n [5] of Th. 1, Chapt. 1, Â§4). / . Lemma 0.6. Let \A c T,^ and l e t <j> be any representation of G . Suppose [d),x] e Q(u) .. Then [<f>,y] e Q(u) i f and only i f y = u'x f o r some p a r t i a l isometry u ' i n ( d ) ( G ) ) ' such that U'*U'x = x . Proof. Suppose y = U'x - f o r such a U' . Then if or any A e G . â€¢ : ' - \ " (<l>(A)y|y) = (<|>(A)U ;x|U'x) -.'(â€¢.(.A)U**u'x|x) -. (*(A)x|x) = u(A) Therefore [<j>,y] e Q(u) . Conversely, i f both x and y induce u i t i s a well known r e s u l t that we can define an isometry from the subspace [<}>(G)x] to the subspace [<}>(G)y] by taking U { ( <KA )X ) = <f>(\A)'y and extending to the closure (see [5]j Chapt. 1, Â§4, Lemma 3). Extending .U-Â£ to an operator U' on the whole space by s e t t i n g U' = 0 on the orthogonal complement of [<j)(G.)x] gives us the desired p a r t i a l isome-try. I f M â‚¬ E G the support of ^ i s defined to be the smallest projection E of G f o r which u ( l - E) = ;0- . By the normality of u the support always exists. The following i s a well! known r e s u l t (see [5] p. 58). Lemma 0.7. Let |U> e Â£ G . For any [<l>,x] e Q(^) , the support of \x = 4>-1 [(<j)(G))'x] . . , v Factors A W -algebra G i s ca l l e d a factor i f i t s center ( i . e . ; {T e G : TA = AT f o r a l l A e G} ) consists only of , scalar multiples of the id e n t i t y . A factor G can be e f f e c t i v e l y analyzed by looking at i t s set of projections. Two projections E and F are said to be equivalent, written E ~,F , i f f o r some p a r t i a l isometry U e^G i E = U U - and F = UU â€¢. We say that E ^ F i f E ~ G for some G < F . While these d e f i n i t i o n s can be made for any W -algebra, factors have the important property that for any two projections E and F either E ^ F or F ^ E . A projection E i s said to be f i n i t e i f E ~ F and F _< E imply that F = E . A projection which i s not f i n i t e i s said to be i n f i n i t e . A factor i s said to be of type I i f i t contains a minimal non-zero projection. Any such factor i s isomorphic to the algebra of 1 a l l bounded l i n e a r operators on some H i l -l bert space and we then say i t i s of type I where n i s the dimension of the space. A fa c t o r with no minimal projection i s said to be of type; a. H-^ â€¢Â» i f the i d e n t i t y i s f i n i t e , b. II , i f the i d e n t i t y i s i n f i n i t e but some CO non-zero projection i s f i n i t e , c. I l l !, i f a l l non-zero projections are i n f i n i t e . Factors of type I', II, and II are semi-finite. 1 CO In f a c t they each have a semi-finite trace, which i s unique up to a p o s i t i v e scalar multiple,' and which i s normal and f a i t h f u l . The only trace on a type III factor i s that which takes value Â» on a l l non-zero p o s i t i v e elements. If E i s a projection i n a semi-finite f a c t o r G , then G E i s a semi-finite factor. G E i s f i n i t e i f and only i f E i s a finite'.projection and t h i s occurs i f and only i f the trace of E i s f i n i t e . It follows that the f i n i t e factors are those of type I , n a f i n i t e c a r d i -n a l , and of type I I 1 . A factor of type I I , i s said to be hy p e r f i n i t e X 1 i f ; i t i s generated by an increasing sequence of subalgebras ( G i ^ i = l 2 â€¢ where G^ i s a factor of type I i < . Any .21. two hyp e r f i n i t e 11^ factors are isomorphic./ ([5], Chapt. 3, Â§7, Th. 2). â€¢ ' ; \ Tensor Products of von Neumann algebras Let I be an a r b i t r a r y indexing set and l e t ( G i ) i e I be a family of von1 Neumann algebras acting on the Hi l b e r t spaces ( H i ) i â‚¬ j A family of vectors ( x ^ with xÂ± e H i i s c a l l e d a C -sequence i f ' Â£ |l-||x. ||| < Â» . An equivalence r e l a t i o n Â° i e l 1 i s defined on the set of a l l C 0-sequences by taking ( x ^ equivalent to (y.) i f f Â£ | l - ( x . | y . ) | < Â» . : ' i e l 1 1 Let (x^) be a C 0-sequence and l e t r be i t s equivalence c l a s s . \ For each (y.) e r we l e t Â® (y.) 1 i e l 1 denote the multi-anti l i n e a r mapping'from T to $ /defined voy v Â® ( y 1 ) [ ( z i ) ] = T T ( y 1 | z 1 ) â€ž , f o r a l l (z Â± ) e r . i e l â€¢-. " i e l These mappings generate a l i n e a r subset H Q of the space of a l l multi-anti l i n e a r functionals on r , and H be-o comes a pre-Hilbert space by defi n i n g (Â®(y.Â» ) |&>(z. )) = | 1 (y.. |z.\,) 1 1 i e l 1 1 and extending t h i s inner product to a l l of H Q by l i n e a r i t y . The completion of H Q i s known as the tensor product of the family (H^)^ with respect to the C 0-sequence (x.^) and denoted by Â« (H., ,x. ) â€¢â€¢. -A" "^'"'-'-v 22. An orthonormal basis f o r Â® (H.,x.) can be i â‚¬ l chosen i n the following manner. F i r s t note that we can assume that ||x^ || = 1 f o r a l l i since replacing each . x^ by .'x^/llx^||' y i e l d s an equivalent C Q-sequence. For' 'each i e l , l e t ( x i ) j â‚¬ j ( i ) t e a n orthonormal basis for which includes x^ . Then the set of a l l elements of the form Â® (y.) where f o r each i â‚¬ I y. = x^ f o r â€¢ i e l 1 1 1 some j â‚¬ J ( i ) , and for a l l but a f i n i t e number of i â‚¬ I y^ i n p a r t i c u l a r = x^ , constitutes an orthonormal basis for Â® (H. , x. ) . This shows that Â® (H. ,'x. ) i s se-i e l 1 1 i â‚¬ l 1 1 parable i f each H i i s separable and I i s countable. , , Now, for, any k e I and . A k â‚¬ G k we define the â€¢ i . operator A f c on (U^,X^) by taking * \ ( Â® ( y Â± ) ) = Â® (y[) w n e r e " ^ 1 = ^1 l f 1 + k A^ i s then extended to the whole space by l i n e a r i t y and continuity. ' Let G be the von Neumann algebra on Â® (H^,x^) generated by {A^ : i e I and A^ e G i l . Then the tensor product of the family , (G^) with respect to the C Q-sequence (x.) , denoted by <59 ' (C,*x.) , Is defined to be the von-i e l 1 1 Neumann algebra G. . , together with the i n j e c t i o n s of each G 1 into G given by A. .-â€¢ X . . ' . :; .:\/ f . ' . . â€¢ . 1 â€¢'â€¢ 23. Note that i f (x^) and (y^) are equivalent : C 0-sequences i t follows immediately from the d e f i n i t i o n s that 1 (H i,x j L) = <S5 (H 1,y 1) and Â® (G 1,x i) = Â® (G^y.^ . Complete d e t a i l s and proofs of t h i s material can he found i n [12] or [2]. Products and Tensor Products of W -algebras Let I be an a r b i t r a r y indexing set and l e t , t. ( G i ) l â‚¬ j be a family of W -algebras. : A product f o r t h i s family i s an object (ct^)^ â‚¬j) * where G i s a W,-algebra, and f o r each i e l i s a \ 1-1 homomorphism1 from G.Â» into G s a t i s f y i n g . "' (a) a i ( G 1 ) and aj(Gj) commute pointwise f o r v;i" i + j â€¢ .' (b) {a^G^ t ' : i e l ] generates G as a W - a l -gebra. .:. We say that the products (G, (a^)) and (ft, (3^)) si are product isomorphic i f -there exists-an isomorphism <j> from ,G onto B such that (f>a^ =/P^ f o r a l l i e l ; We l e t A = A ( ( G i ) ) denote the set of a l l f a m i l i e s where e Â£ for a l l i e l . 7 jWe say that n i s a product state of the product (G, (a., )) *â€¢> i f u i s a normal, state of G such th&i f o r some â€¢ii ( F f . . o 1 ( A 1 ) . ) ^ T T M A . ) , ieF 1 1 ieF . â€¢ 24. â€¢for a l l f i n i t e subsets F of I and a l l A^ e G^ â€¢ We w i l l denote such a |i by Â® (u. ) . I t follows from the norma- \ i e l l i t y of u that i f Â® ((i, ) exists i t i s unique. i e l D e f i n i t i o n 0.8. For any e A we define a product, Â® (G . ,u. ) , as follows. For each i e l choose [c|>.,x. ] i e l 1 1 i 1 1 â‚¬ Q(u.) . Then Â® (G . ,u 1 ) i s defined to be the algebra .; i e l Â® (<j). (G. ),x.,)-' with i n j e c t i o n s a. given by i e l 1 1 1 x = T]TA7T f o r a l l AÂ± e GÂ± . I t i s shown i n [3] or [4] that Â® ( G . , u 1 ) i s unique up to i e l product isomorphism. In [4] a purely algebraic d e f i n i t i o n of Â® (G.,u.) I ' i e l 1 1 i s given. * I t i s easy to see that the product state Â® (y.) i e l 1 exists on Â® (G.,,u.,) In f a c t i n the construction above i e l 1 1 i t i s induced by the vector Â® ' (x.) â€¢ â€¢ i e l 1 Following the usage i n [4] we w i l l r e f e r to these products, constructed from' an element of A , as l o c a l ten-sor products. / Suppose that the indexing set I i s f i n i t e . For ease i n notation l e t I = {1,2} . Then a l l l o c a l tensor pro-cucts of a given family are product isomorphic and we may speak of a unique tensor product (G^a-^ag)) Â°f the family (G^Gg) . ^It i s usual to denote G by G^ & G 2 and the element a-j^A-^) a 2 ( A 2 ) by'' -A^ Â® A 2 . Every element (y^,^ 2) of A determines a product state, denoted by ^1 Â® w 2 ' â€¢â€¢ 25. ' Lemma 0.9. Let G and a be W -algebras, and l e t u e Â£ ( and v e rfi . Then f o r a l l S e G + and T â‚¬ i B + with u(SS*) - v(TT*) - 1 * V)(S*T) = H v T Proof. Choose any A e G and B e B . . Then (U Â® v) (SÂ®T) [A Â» B] = (u Â® v)[(S Â® T) (A Â® B) (S Â® T)*] = (w Â« v)[(SAS*) Â£ (TAT*)] = u(SAS*) v(TAT*) = U s(A)v T(B) ' ^ U g Â® v T [A Â® B] The lemma.now follows from the uniqueness property of pro-â€¢t> duct states. 26. CHAPTER I THE METRIC d 1. Basic properties of d and p The quantities d and p were Introduced i n [ 3 ] . In t h i s section we review some basic properties. * D e f i n i t i o n 1.1. - Let G be a W -algebra and l e t u and v â‚¬ t~ . We define. G V 'd(u,v) = i n f {||x-y|i : [4>,(x,y)] â‚¬ Q(n,v)} , ' P(u,v) = sup {|(x|y)| : [(j>,(x,y)] e Q(u,v)} . â€¢ i I t i s proved i n [-3]' that d . i s a metric on and that d and p are connected j by the formula ( 1 . 1 ) â€¢ [ d ( u , v ) ] 2 = ' 2 [ 1 - P ( L I , V ) ] - : . The number d(u,v) can vary from 0 to J2 . To' gain some insight into the d e f i n i t i o n we can consider the extreme cases. â€¢.'â€¢â€¢! Lemma 1.2. ' ; (a) d(|i,v) = 0 ' i f and only i f u = v . \ (b) d(n,v) Â« i f and only i f the support of u. and the support of v are orthogonal. Proof. (a) This follows from the f a c t that d i s a metric. (b) Let E = the support of u , F = the support of v and supposethat FE = 0 . Choose any [<t>,(x,yy] e QXu,v) Since ||o>(E)x||2 = n ( l ) = ||x||2 , we have that <|>(E)x = x and s i m i l a r l y <|>(F)y = y . Then 1 (x|y) =j= (<j>(E)x|o>(F)y) = (o)(FE)x|y) = 0.' Therefore p(u,v) = 0 . and from (1.1) we have that d(u,v) = 72* . v Conversely suppose that d(ui,v) = . Then p(u,v) = 0 . Choose any ' [<|>, (x,y) ] e Q(u>v) . For any unitary U' i n ( < | > ( G ) ) ' ' Lemma 0 .6 shows that [$Â£\j'x3y)] e Q(u,v) and therefore 0\</| (U'x|y) | _< p(u,v) = 0 . We have then that (A'x|y) = 0 f o r a l l A' e (<J>(G))"' by wr i t i n g each such element as a li n e a r combination of unitaries. By Lemma 0 . 7 , y i s orthogonal to a dense subset of the sub-space 4>(E) and therefore <|>(E)y = 0 j . Then, <j>(E)A'y A'd)(E)y = N0 , fo r a l l " A ' e (d)(G))' , and by Lemma 0.7 again d)(E) <j>(F) =.0 ; . Since <j) i s a . ; i - l mapping we have that EF = 0 .'â€¢ . ,' â€¢ v 28. Lemma 1.3 Let G be a W -algebra and l e t u*v and Then (a) For any 0 <_ a <_ 1 , d(u.', ( l-a)u + av j , _<-'VSa (b) |p(u),u) - p(uu,v)| < J2 d(u,v) Proof. (a) Let u' denote the state (l-a)(i + av Choose any t<j>*(x,y)] e Q(M>,V) . A d i r e c t c a l c u l a t i o n shows 'that [$ Â© (j), (-/1-a x e ,/a x , ./I-a x e y)] e Q(u,u') . Here <J) Â© (j) denotes the d i r e c t sum representation of <J> with i t s e l f on the space Hâ€¢ Â© H where H i s the underlying : space of (J> . We have then d(w,M ; ) WJT^a x Â© Ja x - "yi=a x Â© Ja y|| \ = Va ||0 Â© (x-y)|| Â± J2a (b) Using (1.1) and the backwards t r i a n g l e Inequa-l i t y , f o r d we obtain, -2 2 |p(Â«J,u) - p(iu,v)| = ^|[d(u),u)] - [d(uu,v)] | / = i(d(Â«J*u) + d(uj,v)) |d(Â«j,u) - d(uj,v) | < VS d(u,v) :2. The product formula for p In t h i s section we w i l l prove i n general the pro-duct formula f o r p which was obtained i n ([3], Theorem 2. 5),: for semi-finite algebras. We do t h i s by means of Lemma 2.1 which i s simi l a r i n statement and proof to ([3], Lemma 1.6). .However by dealing with only one element of G we are able to avoid the use of a trace. * 4 Lemma 2.1. Let G be a W -algebra, LA e and l e t T e G 2 be such that LA(T ) '= 1 . Then, P ( U , U T ) = H(T) I i Proof. Choose any [<|>,(x,y)] e Q(u,u T ) . A d i r e c t c a l -c u lation shows that [(f), (x,d)(T)x) ] e Q(u,(j^,) ., Therefore,, v (2.1) P ( H , H T ) > |(x|4.(T)x| = u(T) . On the other hand, since y and <})(T)x induce the same state r e l a t i v e to d) , Lemma 0.6 shows that y = U/<J>(T)x fo r some p a r t i a l isometry U' i n (4>(G))' . Therefore |(.x|y)| = |(x|u'o>;(T)x| - .|(U' + ( T)*xK ( T)*x)| â€¢I' < |i<J>(T)*x||2 - li(T) Taking the supremum over a l l - [<j>, (x,y) ] Q(LI,V) we obtain 30. p ( | i i M T ) < H(T) . This formula and (2.1) complete the proof. In the remainder of t h i s section we consider two W -algebras G^ and G 2 and t h e i r tensor product G^ Â® G 2 We also consider states _ and â‚¬ and u 2 and v 2 â‚¬ E G . We want to prove the following product formula for p . (2.2) P d ^ Â® u 2 !, v x Â® v 2 ) = p f w - ^ v ^ - p f n ^ V g ) Lemma 2.2. Let j = 1 or 2 . Suppose that Uj = ( v j ) T f o r some T\j â‚¬ Gj . Then (2.2) holds. Proof. From Lemma 6.9, U x Â® W2 = ( ^ 1 ^ Â® ^ V2^T 2 = ( v l Â®-^T ^ - S'Tg. and by sucessive applications of. Lemma 2.1 0 P(U 1 Â® (J 2 , v 1 -.Â® v 2 ) = v a Â® v 2 Â® T 2) =,v 1(T 1) v 2 ( T 2 ) = p(u 1,v 1) p(u 2,v 2) . Lemma 2.5. For any 6 such that 0 <_ 6 Â£ 1 l e t vj = (1-6)vj + , 6Wj :'V where j = 1 or 2 . Then d(v^ Â® v 2 ,.-..vj Â® V 2) <, 2756 .. . . . . Proof. By the tria n g l e inequality, d ( v 1 Â® v 2 , â€¢ vÂ£ Â® > V g ) _< dCvjL Â® V g , v 1 Â® v 2 ) - + d'(v-j_ Â® vÂ£ * v-[ V g ) From Lemma 1.3 (a), ' â€¢ ' .. . . d ( v 1 Â® V g , s>1 Â® V g ) = d ( v x Â® vg , [ (1-6 )v1 Â® V g + 6( vÂ± e* Ug) ]) S i m i l a r l y , ^(V^ Â® Ivg Â» "v-{ Â® V g ) < , which completes the proof. - We now remove the r e s t r i c t i o n s of Lemma 2 .3 Theorem 2 .4 . The product formula ( 2 . 2 ) holds i n general. ^ 2 Proof. Given any Â«' such that 0 < e < 1 , l e t 6 = c T"4T Let V j = ( l - 6 ) v j + 6Mj .3 where j = 1 or 2 . Then f o r n > 7j Â» Uj(A) < 'n V j (A) ; f o r a l l A e G j . From Sakai's Radon-Nikodym Theorem . ( [ 5 ] , Chapt. 1, Â§4, Th; 5 ) , , (2.3) N U j = ( v j ) T ', fo r some N T^ e G"J . -..-j;:-, y , ^ â€¢ . â€¢ .â€¢ , . : ... ./â€¢ ,'.. 32. , For ease i n notation l e t â€¢ i. . i-V" a = P ( w i 9 u 2 ' v l Â® v 2 ^ V r; - b = P ( u 1 , v 1 ) , â€¢ â€¢ â€¢ y ; c = p ( u 2 , v 2 ) , i-,'.. and l e t a 7 , b' and c' be defined s i m i l a r l y ! w i t h re-p l a c i n g V j . We want to show that (a-bc) = 0" . From Lemmas 1.3, 2.1, 2.3 and formula (2.3) we have â€¢ l ' v 1 ' â€¢ â€¢ â€¢ â€¢ â€¢ . ' â€¢ 1 ' â€¢ â€¢;r' â€¢ '/ (a'-b'c') = o) / ' -' |a-a'| < 46* |'b-b'| < 26* ; y and |c-c/| < 26* . " i â€¢ â€¢ â€¢ 1 Then, since 6 , b i and c are a l l _< 1 we have . .|a-bc| = | (a-bc) - (a'-b'c'-) | ' â€¢; " . t' ' â€¢ . . - r l y j < I a-a' I + I b c - b ' c ' l , :; x < . '''"â€¢'â€¢_<â€¢ |a-a'| + b|c-c'| + c|b-b,'j + |b-b'||c-c'| _< 2[6*(2+b+c) +2b] < 26*(4+b+c) 1 126* < c y. Since c was chosen a r b i t r a r i l y we conclude that (a-bc) = 0 . ^ Remark 2.5 We can of course define d and p f o r any p o s i -t i v e normal l i n e a r functionals as was done i n [3], instead of r e s t r i c t i n g the d e f i n i t i o n to states. Since a vector x i n -duces a functional-/ji i f and only i f the vector k x induces ku. , we have that p(ku,v) = k* p(u,v) f o r a l l k Oi . 33. â€¢Moreover \x Â® v i s defined f o r functionals and the mapping , i â€¢ ( M J V ) to u Â® v i s b i l i n e a r . I t follows from these remarks \ that the product formula w i l l s t i l l hold f o r p o s i t i v e normal l i n e a r functionals. â€¢v-/ 34, â€¢ â€¢ â€¢ ' v â€¢ CHAPTER II .' THE SEMI-METRICS d G 3. Def i n i t i o n s of d u and p" D e f i n i t i o n 3.1. Let G be a W -algebra and l e t G be any subgroup of Aut(G) . For any u and v â‚¬ EQ we de-f i n e V ! .. dG(M,v) = i n f {d(na,vP) : a,p e G} , , P G ( M , V ) = sup {p(u a,v P) : a , 0 e G} . . .., From (1.1) we obtain (3.1) [ d G ( u , v ) ] 2 = Â£ [ l - p ^ ( u , v ) ] â€¢ < ; Lemma 3.2. ^ (a) d G(^i,v) = i n f {d(u,v a) : a e G} (b) p G(u,v) =sup{p( u, v a) : a e G} Proof. Let\ a e G . For any [<j> ,(x,y)] e Q(u,v) <j>a i s a l -so a representation of G and a d i r e c t c a l c u l a t i o n shows [4>a,(x,y)] e Q(|i a,v a). Therefore d(u a,v a) < ||x-y|| and tak-ing the infimum over a l l elements of Q(uiv) we obtain 35. N d ( M a , v a ) < d(u,v) ; Then replacing \i and v by | i a and v a r e s p e c t i v e l y and _1 replacing a by a we have A â€¢ d ( n a,v a) = d((a,v) , f o r a l l a â‚¬ G . Therefore d(y a,v^) = d(u,v^ a ) which proves the lemma. c It follows e a s i l y that f o r a l l G , d i s a semi-metric on E~ . * ' " ^Notation. I f G = a l l o f Aut(G)_ we w i l l l e t p and d denote p.. and d respectively. , ; 'â€¢_ ^ !>. . I f G =jIht(iG) we w i l l l e t .*p and' cf denote p G 'and d respectively. â€¢v V â€¢ / .4. An Inequality. In t h i s section we prove a matrix in e q u a l i t y which c i s a fundamental r e s u l t f o r the c a l c u l a t i o n of d . Notation. For any integer n l e t R denote the r e a l vec-n | tor space consisting of. a l l sequences a = (a^,a 2,...a n) of r e a l numbers. We l e t D* = [a e R n : &1 > a . for i < J â€¢ and a j L 2 0 f o r a l l i} Note that D* i s closed under addition, and m u l t i p l i c a t i o n by a non-negative scalar. For a p o s i t i v e integer r _< n , l e t 6 denote ' i the element a of\ with a^ = l , l X i _ < r ; a ^ = 6 r r < i _< n . Then any element of D* can be written as a 'linear combination of 16 ' s with non-negative c o e f f i c i e n t s . In f a c t f o r a e D + '; ~ n Z ( a = Jx { a i " a i + l } 6 i + V n â€¢ . . . ' â€¢ / f r . F i n a l l y we l e t M C<|) denote the set of a l l n by n matrices k = (k^j) over the complex numbers. Definition;4. 1. :" For a,b e D + and k' â‚¬ M (<Â£:) we define . n n\ 1 , n T k ( a , b ) ' = - i , j = i a i b J k i J ' Lemma 4.2. Let c be an element of R such that c. > o . â€¢> . . n i â€” f o r 1 < i < n , and l e t . : k â‚¬ MN{$) be such that' f o r a l l r with 1 < r < n r Then f o r a l l a.b, e D + n n (4.1) |T R(a,b)| < i 2 i a i b i c 1 _ 4 . Proof. Let a,a,b ,D e D n and l e t X be a p o s i t i v e number. Suppose that (4.1) holds f o r the pai r s (a,b) and (a,b) Then |T.(Xa+a,b) | = | Z (Xa+a), b,k | ':y^- y fa:; { i , j=i 1 J 1 J - ' f a < X|T R(*,b)| + |T k(a,b)| / .;- < x J x Â¥ 1 = 1 + J x a i V i , n _ = T, (Xa+a).b,c. i = l 1 1 1 Therefore (4.1) holds f o r ((xa+a),b) . S i m i l a r l y , i f (4.1) holds for (a,b) and (a,"b") i t w i l l hold f o r (a,(b+xl))) . I t follows then, from the remarks above concerning that i t i s s u f f i c i e n t to prove (4.1) f o r a = 6 r and b = 6 g . Moreoyer by the symmetry of the conditions on k we may as-sume that, r < s We have i n t h i s case; 38." â€ž â€¢" r s r s \ |T k(a,b)| = | Â£ T k | < Â£ | Â£ k. . | . : K i = l j=l 1 J i = l j=l 1 J â€¢ n < Â£ c. = Â£ a.b.c. , ~ }=1 1 i = l 1 1 1 and the lemma i s proved. \ Theorem 4.3. Let G be a W -algebra with a f i n i t e trace T . Let S and T be elements of G + such that : n S = Â£ a i E i Â» where a = (a 1,a 2...a n) e and the E ^ s i = l n are mutually orthogonal projections i n G ; T = Â£ b . F . , i = l 1 1 . with s i m i l a r conditions; and T(E^ ) = T ( F 1 ) for i = 1 , 2 , ...n We w i l l denote t h i s common value by c^ . Then, for any V e G with ||v|| < 1 | T ( S T V ) I < Â£ a.b.c. . - 1 = 1 1 1 x Proof. n Therefore STV = Â£ a,b,E,F,V i , j = l 1 0 1 J n T ( S T V ) = Â£ a ^ j k ^ ^ j : /where k ^ = r ( E Â± F j V ) . The r e s u l t w i l l now follow from Lemma 4.'2 a f t e r we v e r i f y the appropriate conditions on ^(k. ..j . Choose any. 1 j< r < n r â€¢ â€¢ .â€¢ J r :... Since Â£ F, i s a projection, || Â£ F-.V|| < 1 . . Then, using ' j = l J ^ - ;V j=l J ~ ' â€¢ ' â€¢Lemma 0 . 4(b) ; I.r. k VI = |T[E ( Â£ F.V)] I j=i 1 J 1 j=i 3 < T(E )!| Â£ F v|| '"'<: c. : i . 1 j=i 4 . - 1 â€¢ 39.; ..Similarly, since T ^ E J F J V ) = T ( P JV E j . ) we have r I; E k. . 1=1 1 J 5. Monotone Functions , , Notation Let R + denote the non-negative r e a l numbers and -let ( R + ) ' = (R + U +Â») â€¢ Let M = the set of a l l functions f from ( R + ) ' : to (R +) ' such that x _< y implies f (x) > f (y) . D e f i n i t i o n 5.1. For any â€¢ f â‚¬ M we define a function â€¢%)â€¢''. on ( R + ) ' by " â€¢ â€¢ 7(a) = i n f [x â‚¬'(R +)' : f ( x ) < a) (Note that inf{^} = +Â« , sup{/zJ] = 0 ) The following f a c t s .about 7 - are either well known or e a s i l y v e r i f i e d ; " â€¢ Lemma 5.2 (a) 7 e M a"nd moreover i t i s r i g h t continuous ( i . e . f ( i n f (S)) = sup(f(S)) , f o r a l l S c (R +) '. ) X (b) 7(a) = x , i f and only i f ( i ) f o r every y < x , f ( y ) > a ( i i ) f o r every/ y > x , f ( y ) _< a (c) Let k e R + . Then g(x) = kf(x) f o r a l l x implies g(x) -= f ( ^ ) for a l l k' x f ( x ) = f ( f ) f o r a 1 1 x implies g(x) = k7(x) for a l l x (d) Let ( f - L n o be a sequence of functions i n M such that f, < f . f o r i < j } . : Define f on ( R + ) ' by f ( x ) = s u p ( f n ( x ) ) . Then, " . â€¢! ' \';;->;;^f:â‚¬.:M and (Tn)f 7 . : , ' â€¢ 41. (e) Let cf> be a continuous function defined on , the i n t e r v a l [a,b] and l e t f â‚¬ M be f i n i t e valued on t h i s V i n t e r v a l . Then the S t i e l t j e s i n t e g r a l <KX) df(x) = ,f(b). J a J <â€¢ +(7(o))da. . â€¢ f ( a ) We need ah extension of (e) to improper integralsi' Lemma 5.3 Let be a continuous function on R + such that (j>(xj >Z0' f o r a l l x and c}>(.0) = 0 . Let f e M be such that f(x) < Â» f o r x > '0; and f(x) -â€¢ 0 as x â€”â€¢Â» <Â» Then ' â€¢ - J; cjÂ»(x)df(x) = f f ( 7(a))da . J o J 0.'- - ... Â» -! ([ denotes lim f| } and i s allowed to take the value +Â» )* Q m-~ J 1 , n-Â»oo n Proof. Let f + ( 0) = sup (f(x) : x e (R +) '} and n o t e ' f i r s t that, y e f + ( 0) implies by Lemma 5.2(b) that f ( y ) = .0. > and therefore that <J>("f(y).) = cj)( 0) = 0 . Then using Lemma . 5.2(d) â€¢ '', , , m - J c|>(x)df(x) = - . l i m (f <f)(x)df(x)) /â€¢ V0 ! m~Â« J 1 ' .:-n-*Â« n â€¢'' f ( - ) â€¢ ' ' â€¢ â€¢ f + ( 0 ) ;:- : = lim (J n i(T( a))da = f '" 4>(T(a))da â€¢ v m-Â» f(m) . i ,^ ' J,0 -; ;;â€¢ " > â€¢ . . â€¢ â€¢ " , . ' ' ' 0'* . . . : \ - ; ^>fv' .:â€¢>':,â€¢â€¢â€¢â€¢' i. , ' 'L :V;- ' "v; The main r e s u l t of t h i s section i s the following. \[ V Theorem 5.4. Let <{> be any continuous function from R + x R + ; :â€¢ > ;to R + . Let f , g and h e M be such that f o r a l l . x,y e R + . . i . h(cj>(x,y)) e [f(x),g(y)] . (We l e t [,a,b] denote the closed i n t e r v a l with end-points a and b even i f a > b ). Then f o r a l l . a â‚¬ ( R + ) ' such that 7(a) and g(a) e R + I h(a).= <J)(7(a)/g(a)) . v â€” 'â€¢" i 'â€¢â€¢' : Proof. Let 7(a)* = x and g(a) = y . Given any e > 0. choose, by the continuity of <|> , 6 > 0 so that ;(5..l) d>(x-6â€žy-6) 2 <t>(x,y)-c â€¢ (5 .2) cj)(x+6,y+6) _< 4>(x,y)+s : v '. ,' By our hypothesis . h(<j>(x-6,y-6)) e [f (x-6 ), g(y-6) ] and from Lemma 5.2(b) both' f(x-6) and g(y-6) are > a . Using (5. l)^we' obtain . â€¢ ; '"â€¢;.[. . (5 .3 ) h(<|>(x,y)-e). _> h(<J)(x-6,y-6 )) > a \; 43. Prom Lemma 5 .2(b) again both f(x+ 6 ) and g(y+6) are _< a and using ( 5 . 2 ) we obtain s i m i l a r l y that . . ( 5 . 4 ) h(<j>(x,y)+e ) _< a From ( 5 . 3 ) , ( 5 . 4 ) and the other d i r e c t i o n of Lemma 5 .2(b) we have TT(a) = <|>(x,y) = <f>(f(a),g(a)) . â€¢ X 6 . D i s t r i b u t i o n Functions and q ,â€¢['" j . â€¢ ' â€¢â€¢ â€¢ v .Spectral Resolutions. Let A be a s e l f - a d j o i n t operator (not necessarily bounded) oh a H i l b e r t space H . Let J}^ denote the domain of A . (For terminology concerning unbounded operators refer to [20], ) By a spectral resolution of A we w i l l always mean the unique; family of projections i n H , ( E ( x ) } , X a r e a l number, which s a t i s f i e s the following properties: ' " ' I i - - 1. X <. u; implies E ( x ) < E ( u ) ;â€¢â€¢ 2. U,E(X!) = i â€¢ , 0,E(X).= 0 ; â€¢â€¢ - '. t , , ' ' A. \^ K . . â€¢ . 3- E ( x ) commutes with any operator i n Â£(H) which commutes with A : r , |'/ 4. 0 X > X E(X) = E ( X Q ) , forÂ« a l l X Q j ' ' . , 5. â€¢ For -oo < x _< u'' < +Â» , x e E ( y ) - E(Xi); . implies that x. e J&A and if-,||x||= 1 , x < (Ax|x) < u ".. â€¢ ' - â€¢ " " ' \ '.'.'J Integrable Operators and Radon-* Nikodym Derivatives. <â€¢ Suppose that G i s a von Neumann algebra on a H i l b e r t space H . An operator S on H:' (not necessarily bounded) i s said to be; af f i l a t e d with G, , written/ S n G , i f ' -U S U = S ' f o r every/unitary U e G' â€¢ If S i s i n f a c t bounded and everywhere defined, the double' commutaht theorem shows that S n G..; i f and only i f S e G > V For any1 s e l f - a d j o i n t S with spectral resolution (E(\)} , i t is, easy to see that,'. S'n C. i f and only i f 'E ( x ) e G f o r a l l r e a l X . .. \ Now suppose that T i s a semi-finite, f a i t h f u l , normal trace on G Let \7\ = [A e G;: T (A A ) < Â«} . It i s shown i n ([5] Chapt. 1, Â§6, Prop. 1) that 7\ i s a two-sided. 2 i d e a l of G and that the i d e a l 7h = 7\ consists precisely, of those elements of . G which are f i n i t e l i n e a r combinations of p o s i t i v e elements of f i n i t e trace. Then T can be extended by l i n e a r i t y to a l i n e a r functional T' on 7!\ . It .satisfies (6.1) T '(ST) = T '(.TS) , for a l l S,T e 71, . We can define an inner product, on 7} by taking . (s|T) = T '(T S) and t h i s produces a norm on 71 given by . '[ \\S\\2 = . [ T ' ( S * S ) ] * - .;. , I t i s shown i n [[5]* Chapt. 1, Â§6, Th. 8) that we can also define ' a norm on' TJi. by v â€¢ ;'.' 1 ! % = T'(|SD . : In [15.], Segal defines two subclasses of the operators; a f f i l i a t e d with G which are 'known as square integrable and integrable, with respect to T , and which contain 71 and 7f[ respectively. Moreover he extends, T' to a'functional on the class of integrable operators. These classes can be considered as completions of .71 â€¢ and 7h with respect to the above mentioned .norms. We summarize a few basic facts, that, we need i n the following lemma., Proofs can be found i n [15]. .' Lemma 6.1 â€¢ \ '' (a) Let S be a po s i t i v e s e l f - a d j o i n t integrable -\ . .operator with spectral resolution ( E ( X ) } "... Then . v , -; - ( i ) > ( i - E ( x ) ) < Â« , for \ > o , ' â€¢ ( i i ) T'(S) X d T ( 1 - E ( \ ) ) < Â» . 1 (b) Let S be a p o s i t i v e s e l f - a d j o i n t square-integrable operator with spectral r e s o l u t i o n ( E ( x ) } . Then ( i ) T ( 1 - E ( \ ) ) < Â« for X > 0 , i ( i i ) S 2 i s integrable and T ' ( S 2 ) =-f X 2 d T ( 1 - E ( X ) ) ' < 0 The key r e s u l t which we need i s the following theorem of Segal ( [ 1 5 ] , Thebrem 1 4 . 1 ) . It i s a generalization of both the Radon-Nikodym theorem i n measure theory and of a we l l known r e s u l t concerning X(H) . Lemma 6.2. Let G be a von Neumann algebra with a semi-f i n i t e , f a i t h f u l , normal trace x , and l e t u e. Â£_ . Then . â€¢<â€¢;â€¢ there exists a unique square integrable, p o s i t i v e s e l f - a d j o i n t operator S such that u ( A ) = T '(S. 2A) , for a l l A e G . 2 The operator S i s commonly known as the Radon-Nikodym derivative of y with'respect to T . We w i l l denote thi s by wr i t i n g ;; x ' w = T s .. ".' â€¢ . Note that i f S e G > formula ( 6 . 1 ) shows that t h i s agrees with our notation of Chapter 0.''â€¢;,.;; ' V x X s : : " : ' X . ' ' i.,, . . . ' . â€¢ y i . Remark. In the above d e f i n i t i o n s and r e s u l t s we considered a von Neumann algebra G on a H i l b e r t space. However a square integrable, p o s i t i v e s e l f - a d j o i n t operator can be â€¢ V i viewed as simply an abstract spectral resolution; that i s a family of projections i n G s a t i s f y i n g the appropriate conditions, a l l of which r e l a t e only to the algebra G i t -s e l f and the trace. Accordingly we can apply these r e s u l t s to a W" -algebra G without regard to any p a r t i c u l a r repre-s e n t a t i o n of G . D i s t r i b u t i o n Functions â€¢ ' D e f i n i t i o n 6.3. Let G be a von Neumann algebra and l e t T be a f a i t h f u l , normal semi-finite trace on G . For any po s i t i v e s e l f - a d j o i n t AnG we define the function f ^ on (R + V by . â€¢â€¢; ' f A ( X ) = T(1-E (X)/) , for 0 _< X < Â» f A(Â») = 0 . where (E ( x ) } i s the spectral resolution of A . f A of course depends on the 'trace T; . We^call f A the d i s t r i b u t i o n function of A with respect to T , by analogy to the p r o b a b i l i t y d e f i n i t i o n . Note that the d e f i n i t i o n above i s preferable to taking the increasing function, . X - T ( E ( X ) ) , since i f A i s square 48> integrable f" A i s f i n i t e valued, except perhaps at 0 . Obviously, i n the notation of Section 5> f A i s a ri g h t continuous function e M and we can form the function 7 A â€¢ ,â€¢â€¢ Lemma 6.4. Let A be a p o s i t i v e s e l f - a d j o i n t operator i n G, and l e t ( A ^ n - i 2 be a sequence of mutually com-muting elements of G + such that (1) AÂ± < Aj for 1 < j ( i i ) A = sup ( A j n Then, Proof. Let {E ( x ) } be the spe c t r a l r e s o l u t i o n of A and n v n l e t (E (x)} be the spe c t r a l r e s o l u t i o n of A . ( I t i s not hard to v e r i f y that E n ( iX ) jrE ( x ) f o r a l l X . Therefore (1-E n ( x ) ) t (l-E ( x ) ) . By the normality of T , we have that f o r " a l l X > 0 f - ( X ) = T( l-E n (x)) '1> T(1-E(X)) = f A ( x ) . Lemma 6.5 Let T e G + and l e t P be a projection i n G such that ,TP = T . Let T' and T' denote the r e s t r i c t i o n s of T and T r e s p e c t i v e l y to the algebra G p . Then f^, , computed with respect to the algebra G p and trace T' i s equal to f ^ computed with respect to T . P r o o f L e t (E(x)} be the sp e c t r a l r e s o l u t i o n of T.- '. Then the spectral r e s o l u t i o n of T' as an operator on the Hil b e r t space P i s { E ' ( x ) } where E ' ( X ) = E ( x)P f o r a l l X . Now f o r any unit vector x i n l - E ( x ) , (Tx|x) > X .: For any.unit vector x i n 1-P , (Tx|x) = (TPx|x) = 0 . . This shows that ( l - E ( X ) ) flV(l-P) =0 t or l - E ( x ) < P for a l l X > 0 . Then for any X > 0 , . f T > ( X ) = T ' ( P - E ( X ) P ) = t ' ( P ( l - E ( x ) ) - = T ( P ( 1 - E ( X ) ) = T ( 1 - E ( X ) ) = f T ( X ) , and by r i g h t continuity frp/(0) = f T ( 0 ) . i ' ' â€¢ â€¢ . D e f i n i t i o n s of q ' De f i n i t i o n o. 6. Let G be a W -algebra and T a normal v semi-finite trace on G . For any p o s i t i v e s e l f adjoint operators S and T a f f i l i a t e d with G and square i n t e -grable with respect to T we define o(S,T) = J o 7 s(x) 7 T(x) dx . By Lemma 5.'3> and Lemma' 6.1(b) we have . that "f g and Trp' are square integrable functions on [0,Â»)" . Therefore_-.q(S,T) i s always < Â» . ' ' . Note that ; a depends on the trace T . When G i s a factor we can eliminate t h i s dependence by defi n i n g o .for states.'/ .. ;'â€¢/;.â€¢; 50." ' ... IJDefinition 6.7. Let G be a factor and l e t u. and v e E^ . Suppose that for some semi-finite trace T on G , u = TI'V" V and v = Tip . We then define / cj(ia,v) = a(S,T.) computed with respect to T To j u s t i f y t h i s d e f i n i t i o n we need to show that i t i s independent of the p a r t i c u l a r trace chosen. Let T' be another semi-finite trace on G . Since G i s a fa c t o r , T' = kT , f o r some k > 0 , and i t follows that u = Ts//jyj^ p If; the spectral r e s o l u t i o n of S i s ( E ( X ) } the : s p e c t r a l r e s o l u t i o n of /S/Vk" i s e a s i l y seen to be { E ' ( \ ) } where E " ( \ ) = E(^/k\) f o r a l l \ . Letti n g primed symbols .denote calculation^ with respect to r' we have Prom Lemma 5 .2(c) / , . . â€¢ ./; ' . â€¢â€¢' ; "' ' T s / J z M ~ p s & After a s i m i l a r c a l c u l a t i o n for v we obtain /â€¢ a ' ( u , v ) = o'(S/JZ , TA/k") V ; i 3 ' : 0 = 7s'(!y); t T ( y ) dy - a(S,T) = ,a( U,v) . â€¢-. .1,-.â€¢C,â€” St. â€¢ â€¢ 5i. â€¢ D e f i n i t i o n 6 .8 . Let G be a factor with a semi-finite . â€” â€” â€” â€” â€” â€” â€” â€” â€” â€” : I ' â€¢ 'â€¢. -trace T . We l e t ".-\ : â€¢ v ; \ â€¢ . ' " â€¢-. â€¢ â€¢ \ â€¢ ; r ; ' ; C P(G) = {a â‚¬ Aut(G) : TÂ° = T} ; (Note that P(G) i s obviously independent of any p a r t i c u l a r trace). Now l e t G be any subgroup of Aut(G) . Let % be . the homomorphism from G onto the m u l t i p l i c a t i v e group of .. p o s i t i v e r e a l numbers defined by f(a) = k i f T a = k T . The kernel of 4 â€¢ i s obviously G 0 P(G) . ' Then G/G fl P(G) can be i d e n t i f i e d as a subgroup of the m u l t i p l i c a t i v e group . of reals which we denote by r(G) . D e f i n i t i o n 6 .9 . Let G be a factor with a semi-finite . 4 trace T , l e t G be a subgroup of Aut(G) , and l e t y. and v e . Suppose that u. = Trp . - 1 . 'k . . We define, v >v r Â« â€¢ , ' ; ' - a k ( u , v ) = J% \ 7 g(x) 7 T(kx) dx , f o r a l l k > 0 , . .'. ;. a G ( u,v) = sup (a k ( u,v) 1 :/k e r(G)} As above we can show that a i s independent of the :â€¢ '\ p a r t i c u l a r trace. :â€¢ y'.'.-) The motivation for these d e f i n i t i o n s i s shown by ' â€¢ the following. ., ' Lemma 6 .10 . Let v â‚¬ Â£~ = T m for some T e G + and some 1 G â€¢ JL semi-finite trace T .. Let a e Aut(G) with TÂ° = kj . Then V o{\X,\>a) = a k ( u , v ) . . . Proof. By lemma 0.3, v a = ( T ^ ) 0 = T a = kT n , (ot~ (T)), ( a ^ T ) ) . = T . - i . L e t T = J^(a (T)) . I f T has s p e c t r a l ^ ( a " 1 ( T ) ) r e s o l u t i o n (E(X)} , T' has s p e c t r a l r e s o l u t i o n (E ' ( x ) } where E ' ( x )V ; =..a_1(\A/K)) f o r a l l X â€¢ / Then f T / ( \ ) = T(1 - a^iEiX/JZ)) = -Â£<ra(l - cf^EfxA/E) ) . = -| f T ( \ A / k ) ! By lemma 5.2(c), ^ / ( X ) = 7k f " ( k x ) , f o r a l l X and the' lemma f o l l o w s from the d e f i n i t i o n s of a and a k Remark 6.11. I f G 1$ a f a c t o r of type I or II-j^ , P(G) = Aut(G) . So f o r any G c A u t ( c ) , r ( G ) = {1}' and : G G o = a . The purpose of d e f i n i n g o i s t h e r e f o r e t o d e a l w i t h f a c t o r s of type I I where t h i s i s not the case. An example of Suzuki p. 188) shows t h a t i f G = P Â® I v where P i s the h y p e r f i n i t e 1 ^ factor, then r ( A u t ( G ) ) = the e n t i r e group of m u l t i p l i c a t i v e r e a l numbers. The same r e a s o n i n g employed i n t h i s example shows t h a t i f G = fil Â® I 00 f o r a 11^ f a c t o r ft then r ( A u t ( G ) ) = the fundamental group of 0 . (See [10] Theorem V I I I f o r the d e f i n i t i o n of funda-mental group"). In the r e s t of t h i s s e c t i o n we w i l l c o n s i d e r a f i -n i t e f a c t o r G w i t h normalized t r a c e T [ ( i . e . T(1) =1 ). 53.' We want to develop some continuity properties of a \ â€¢ . i The following i s a well known r e s u l t . (See [9] . \ lemma 14.22). Xemma 6.12. Let E and P be projections i n G with T(E) < T( F) . Then (1-E) n F + 0 . The next lemma i s e s s e n t i a l l y proved i n ([91 Lemma 15.21). Lemma 6.13. Consider any representation of G . For any ; A e G + and 0 _< a _< Â« , .(6.2) f.(a) = inf(sup{(Ax|x) : ||x|| = 1 , Ex = x)) H ,E -as E runs over the projections of G with T(1-E ) _< G . ! - â€¢ Proof. I f a = Â»' both sides of (6 .2 ) are e a s i l y seen to be 0 , so assume a < Â» . Let "f^(a) = b and l e t the ri g h t side of (6.2) = b* â€¢ . Let [E(\)} be the spectral ' o / . resol u t i o n of A . Choose any b' and b" with b' < b <,b" Then <r(l-E(b')) = f A ( b ' ) > a by Lemma 5.2(b) . So; f o r : any projection E with T(1-E) _< a, (l-E(b')) fl E ^ 0 by â€¢ ; â€¢â€¢Lemma 6.12. Therefore E contains a unit vector x such that (Ax|x) > b' . I t follows that b Q > b' . On the other hand, T(l-E(b")) = f A ( b " ) _< a by â€¢-. Lemma 5.2(b). So for any unit vector i n E(b") , (Ax|x) _< b". . I t follows that b Q _< b" . Since b' and; b'' were chosen a r i b i t a r i l y we have, that b Q = b . : ; Lemma 6.14. Let G be a f i n i t e factor.. Suppose that S,s',T and T' e G s a t i s f y ||S-S' I! < 6 , UT-T* II < 6 , where 0 < 6 < 1 . Then . |o(S,T) - a(S',T')| < 6(|lS|| + ||T|| + 1) , >[ where a i s calculated with respect to the normalized trace T Proof. Note f i r s t that since T(1) = 1 , f o r any A e G + f A _< 1 , and therefore 7 A(a) = 0 f o r a l l a > 1 . There fore the integral! i n the d e f i n i t i o n of_ a can be taken over the i n t e r v a l [ 0 , 1 ] . . Since fg(!|S||) = fT(||T||) = 0 we have â€¢ (6.3.) f s ( a ) < ||S|| , 7 T(a) < ||T|| , for 0 < a < ' Now consider any representation of G. and any unit vector x Then-. |(Sx|x) - (s'x| x) | = . |^ (S-S' )x 1 x) | _< |1S-S' || /j 6 . From Lemma 6.13 we can conclude (6.4:)' | 7 s(a ) -7 s/(a )J < 6 and JT T(a)-T T* (a) | < 6 , f o r 0 < Using (6.3) and (6.4) we. have f o r a l l a > 0 55. | f g ( a ) f T ( a ) - f s , ( a ) T T , ( a ) | . < 7 s(a) |7T(.a)-7T, (a) | + 7 T(a) |7 s(a)-7 g, (a) | + | f s ( a ) - f s / ( a ) | | f T ( a ) - f T ( a < 6 ( | | S | | + ||T|| +. 6.) < 6 ( | | S H + ||T|| + 1) . Therefore !a(S,T) - a ( s',T)| < J- |7g(a)7T(a) ~"7g, (a)7 T, (a) |da < 6 ( | | S | | + ||T|| + 1) . v <r 1 56. 7. C o m p a t i b i l i t y , .'â€¢ . D e f i n i t i o n 7.1. L e t S and T be s e l f - a d j o i n t o p e r a t o r s (not n e c e s s a r i l y bounded) on a H i l b e r t space H . Suppose t h a t (E(X)} i s the s p e c t r a l r e s o l u t i o n o f S â€¢, and {P(\')} i s t h e s p e c t r a l r e s o l u t i o n o f T We say that' S and T ar e c o m p a t i b l e , i f g i v e n any o r d e r e d p a i r (a,|3) o f r e a l num-b e r s e i t h e r ( l ) or (2) h o l d s : â€¢ â€¢ (1) E(cx) < P(f3) , ( 2 ) ' F(f3) < E ( a ) . . Remarks. I t i s not ; h a r d t o v e r i f y t h a t the above d e f i n i t i o n .-remains unchanged i j f we use l e f t c o n t i n u o u s s p e c t r a l r e s o l u -. t i o n s i n p l a c e o f r i g h t c o n t i n u o u s ones. Note t h a t i f S and T a r e c o m p a t i b l e t h e y ob-v i o u s l y commute w i t h each 5 o t h e r . -The main r e s u l t o f c o m p a t i b i l i t y w h i c h we need i s " t h e f o l l o w i n g . .-â€¢ â€¢ / â€¢. Theorem 7.2. L e t G be a von-../Neumann'" a l g e b r a w i t h a n o r m a l , s e m i - f i n i t e t r a c e T . L e t S ' and T, e G + be c o m p a t i b l e , 2 p and s a t i s f y T ( S ) < <* , T ( T ) <. *> . Then f o r 0 < a <_ Â« , f S T ( a ) ' .= f g ( a ) 7 T ( a ) P r o o f . We a p p l y Theorem 5 . 4 , t a k i n g <j)(x,y) = xy ., I t r e -mains to v e r i f y the conditions of that theorem. Let ( E ( x ) } ( F ( X ) } , and ( G ( x ) } be the spectral resolutions of S , T , and ST respectively. Since the operators a l l commute with each other any two of these spectral projections w i l l commute. Let ( a , B ) be any ordered pair of non-negative r e a l numbers. /â€¢ â€¢ â€¢ Case 1. Assume that E(a) _< F ( B ) For any unit vector x i n E(a) S^x â‚¬ E ( a ) _< F ( B ) , which shows that (STx|x) =' (TS^x|S^x) < P(Sx|x) < a B . For any unit vector y i n 1 - G ( q B ) . * (STy|y) > aB . *' ' ' This shows that E ( a ) ( l - G(aB)) = 0 , so (7.1) .(1 - G(aB)) jC (1 - E ( a ) ) . S i m i l a r l y , we see that f o r any unit vector x e (1 - F ( B ) ) _< (1 - E ( a ) ) , we have (STx|x) > a B , and we obtain (7.2). . V . / (1 - G(aB)) 2 . (il - F ( B ) ) . From (7.1) and (7,;2) we obtain by the p o s i t i v i t y of T that f T ( S ) < f g T ( a P ) < f s ( a ) Case 2. Assume that E(a) jÂ£ F ( B ) By compatibility we must have F ( 8 ) â€¢<_ E(a) and arguing as above we obtain ' I â€¢ â€¢ ' ' â€¢' . f s ( a ) < fg T(aP) < f T 0 ) . We have then that i n any event,-for a l l non-negative a and B f S T ( a i 3 ) e [ f s ( a ) ,. f T ( B ) J . . j , ' \ 2 '2 Now we note from Lemma 6.1 that since T ( S ) and T ( T ) are f i n i t e , "fg and . 7 T are f i n i t e valued except perhaps at 0 . We apply Theorem 5l.'4 to complete the proof. / Corollary 7 . 3 . If S and T â‚¬ C + are compatible, a n d / 2 p s a t i s f y T ( S ) < Â» , T ( T ) < Â» , then \:'. . ' T ( S T ) = o (S,T) . Proof. Prom Lemma 6.1(a), Lemma 5.3, and Theorem 7.2, â€¢CO X T ( S T ) = 7 S T ( a ) d a = o ( S,T) . . Remark. As a further a p p l i c a t i o n of Theorem 5.4 we may 59. o b t a i n t h a t f ( S + T ) = 7 g + 7 T f o r c o m p a t i b l e S and T , â€¢ by t a k i n g <}>(x,y) = x + y and p r o c e e d i n g as i n Theorem 7.2. S i m i l a r l y , t a k i n g (f>(x,y) = kx y i e l d s f k g = k7g f o r k > 0 , ( a r e s u l t w h i c h we e s t a b l i s h e d p r e v i o u s l y i n the remark f o l -l o w i n g D e f i n i t i o n 6.7). D e f i n i t i o n 7 . 4 . L e t G be a W - a l g e b r a . An element A e G w i l l be c a l l e d s i m p l e , i f i t can be w r i t t e n as a f i n i t e l i n e a r c o m b i n a t i o n o f m u t u a l l y o r t h o g o n a l p r o j e c t i o n s â‚¬ G . Remark. L e t A be s i m p l e and s e l f - a d j o i n t . Then n A = E c.E, , f o r some r e a l numbers ( c . ) and m u t u a l l y 1=1 1 1 1 , , n o r t h o g o n a l , non-zero p r o j e c t i o n s (E. ) w i t h T, E. = 1 . 1 â€¢ i = l 1 'â€¢(We can always o b t a i n t h e l a t t e r c o n d i t i o n by i n s e r t i n g i f n e c e s s a r y an a p p r o p r i a t e p r o j e c t i o n w i t h 0 c o e f f i c i e n t . ) v I t i s easy t o see t h a t t h e s p e c t r a l - r e s o l u t i o n ( E ( \ ) } /of A i s g i v e n by (7-3) E(X) = E E . , the sum,taken over a l l j such t h a t u " ' a n d ' i t ' f o l l o w s .that,*'; ' (7.4) 1 - E ( \ ) = E E. , the sum t a k e n over a l l j such t h a t 6o. Theorem 7 . 5 . Let S and T be simple, s e l f - a d j o i n t elements # V of a W -algebra G . Then S and T are compatible i f and. \ only i f there exist. ( i ) a set of non-zero, mutually orthogonal projec-m tions i n G (G. ), n 0 m . with Â£ G, = 1 ; x x â€” x ^ Â£ - j . * Â« m * ^ x t i . ' ( i i ) non-increasing sequences of r e a l numbers, (c , , c 0 . . . c ) and (d,,d 0...d ) : 1 2 m m m m such that = Â£ c.G. and T = Â£ d.G. i = l 1 1 i = l 1 1 Proof. Suppose.that S and T are of the above form. From ( 7 . f 4 ) , any non-zero, spectral projection of either S . ;v. â€¢ . â€¢ m . or T i s = E G . f o r some k such that 1 '< k < m . Ob-. i=k 1 . ~ ~ â€¢ viously S and . T< are compatible. Conversely, suppose that S and T are compatible, n k Let S = E a.E. , T'= E b .F . , where (E.) and (F .) are 1=1 1 1 j t i J J 1 J sets of non-zero, mutually orthogonal projections with sum >'= 1 For any i , j , ^ = E ^ j i s a projection and, ' S = t a ^ and T = E b j G ^ Deleting those G.. = 0 and suitably renumbering the remain-ing ones, we obtain . m m S =. E c.G. and T = E d.G. , X i=l' i = l where; ( i ) ' 2 m i s a s e t of non-zero mutually orthogonal projections with sum = 1 , ( i i ) c 1 > c 0. . . > c m .,' ( .^ i i + l 3 hen This of course is. a f a m i l i a r procedure which can be applied whenever S and T commute. We want to show now that the compatibility implies that the d^'s are non-decreasing. Suppose t h i s i s not the case. Then for some i , d^ < d i + ^ and necessarily, c^ > . Choose 'numbers c and d such that d i < d < and c^ > c > c.j_+l' ' ' L e t tE(^)3 and (F(X)} be the spectral resolutions of S and T re-spectively. From ( 7 . 3 ) and ( 7 . 4 ) , we obtain !G. < F(d) n (1 - E(c)) , I Gi+1 i E ( C ) 0 ( 1 " P ( 4 ) ) 'â€¢ â€¢tÂ» Since G i and are both .non-zero, we necessarily /have E(c) F(d) and F(d) j^'.E(c) a contradiction to .the compatibility. Therefore, we must have l i â€” d i + l ? ^ o r 1 = l ^ 2 ^ â€¢ â€¢ â€¢ m " " l and the theorem i s prove d. .We now need some lemmas concerning the comparison of projections i n a factor. In the proofs we use several standard*results which can be found i n ([93 Chapters VI and â€¢ > . . . > . 62. Lemma 7.6. Let E and H be projections i n a factor such â€¢'j that H i s i n f i n i t e and E _< H . Then either E ~ H or H-E ~ H Proof. Suppose E ^ H - E . Then H-E must be i n f i n i t e , f o r i f not, E would be f i n i t e and then H would be f i n i t e . Therefore H-E = P + Q where H-E ~ P ~ Q . Then H = E . + H-E-^P + Q = H-E , and i t follows that H ~ H-E . If E-^H-E ., .then H - E ^ E and we obtain simi-l a r l y that H ~ E '. D e f i n i t i o n 7.7. Let H be a projection i n a W -algebra G . i If E and â€¢ P are1 projections of G contained i n H we say . . . \ that E-: ~;F- , i f 'â€¢ E ~ F and H-E ~ H-F . We say that â€¢ â€¢ ( H ) E . F ,. i f E < P ~ Q < F , for some P and . Q . (H)-' ~ (H)t, Lemma 7.8. Let E i F , and H be projections i n a factor G such that E < H and F'< H '. Then either E ^ F , or. P ,<~~ E . ; (H) ' (H) Proof. If H i s f i n i t e , i t i s a well known r e s u l t that ~ i s the same as ~ , and we are done. So assume H i s (H) i n f i n i t e . By symmetry we may assume that f o r some P^ , E ~ F 1 < F . N I f H-E ~ H-F 1 we are done. â€¢ If not, H-E and H-F-L cannot both be . ~ H . From Lemma 7.6, either E or F^ i s ~ H and necessarily E ~ F ~ H . B y symmetry again we may assume that for some G. , H-E ~ G _< H-F . T Then. H-G contains F and must be equivalent to H . There^ fore, F < H-G ~ E , which shows that F E (H) (H) Corollary 7.9* Assuming the conditions of Lemma 7.8, there e x i s t s a unitary U e G such that: (a) U GU = G for a l l projections G <_ 1-H ; (b) Either ( i ) U*FU _< E or ( i i ) U*PU > E holds. (cf. [5], Chapt. I l l , Â§2, Prop. 6.) Proof. We can suppose by Lemma 7.8 that E F . Then, . (H) E < P ~ Q < F for some P and Q. . . . Let V and W be pa r t i a l , isometries: i n G. s a t i s f y i n g V V = P , W = Q ', â€¢ W*W = H-P , W*W =|H-Q . Then, U = V + W + 1-H i s a . unitary i n G which s a t i s f i e s (a) and (b) ( i i ) . Theorem 7.10. Let S and T be s e l f - a d j o i n t , simple elements of a factor G . Then, there exists a unitary U e G such that \S and U TU are compatible. m. / Proof. Let S = Â£ a,E, and T = Â£ b.F. , where (E, ) and' -~ :. k=l K K 1=1 1 1 K (F^) are sets of non-zero,, mutually orthogonal projections ".with ; sum -= 1 and a ^ C a 2^ . .. < a mâ€ž. and b^ < b 2 .. . < b p .. . .. ., We define the'pro jec.tions , 64. Then any spectral projection of S = E ^ ^ for some i = 0,1, , ...m . We define i n a similar way the projections ,, j = 0,1,...p , corresponding to T . Now, l e t A = {0,1,.. .m} B = {0,1,...p} , and order the set A x B lexico g r a p h i c a l l y . If S and T are not already compatible there must exis t i some ( i , j ) ,In A x B such that (*) E ^ ^ P^ J ) and. p ( j ) .| E ( i ) . Let ( i Q * J 0 ) b e t n e least such element. Obviously i > 0 and j Q > :0 .-Then E - { F Â° , so obviously ; ( i Q ) .. ( J 0 - l ) â€¢ , E jf.P . j This implies by the minimality of ( i Q , J 0 ) '.that â€¢ â€¢ â€¢ â€¢ â€¢ (7.5) P U ! Â° - J > < K ( i Â° ; S i m i l a r l y we obtain (7.6) E Â° . < F Â° Case 1. Suppose E <_ F Let . E' = E - F Â° , and l e t H = 1 - F Â° o From (7.5), E'. i s a projection contained i n H . /'Gb-o viously F. x< H â€¢. Choose a unitary U e G which s a t i s f i e s the - conditions of Corollary 7-9 for E'.. , F . and H . â€¢:. .'. . ,.,,' " 1 o Jo â€¢ -' 1 65. v. Consider the element T = U TU , and the corresponding projections, P' = U*F, U , and F'(J) = i F' w . Let . . K * k=l K '"(*'â€¢â€¢)â€¢ denote the statement (*) above with F ^ ^ replaced by p'(J) . ' From the conclusions of Corollary 7.9 ,we have that F ( J ) _ F ' ( j ) for j < j Q ; and that either . (a) . F' . < E.' , i n which case Jo . o or (b) E^ â€¢ \< P'. , i n which case o j Jo (Jo*-1) , ,<<U E = F + Ef .<_ F' Â° . o So (*') does not hold for ( i Q , J O 0 â€¢ Now consider any ( i , j ) < ( i 0 ^ J 0 ) â€¢ â€¢ ^ (a) If j < j Q , .F' (J) = p(J) , and by the mini-mality of ( 1 0 * J 0 ) >(*') does not hold f o r ( i , j ) . ~ (b) If j = j Q , -we must have i _< i Q - l > and. m ( i 0-l) ( j Q - i ) ( V 1 ) (1) So again, (*/ )^ does not hold for ( i , j ) â€¢'. ( i D - l ) ( J - l ) Case 2. Suppose E | F , â€¢ ( V 1 ) ( V 1 ) By the minimality of ( i 0 * J 0 ) > we have F _< E We can then argue i n a s i m i l a r manner to Case 1, using (7 .6) i n place of (7.5). In any event we can f i n d a unitary U â‚¬ G such that when we replace T by U TU , the number of pairs ( i , j ) for which (*) holds i s s t r i c t l y decreased. The theo-rem then follows e a s i l y by induction. Example 7.11. Theorem 7.10 w i l l not hold i n general i f S and T are hot simple. Let M be the von Neumann alge-bra consisting of L w[0,1] acting on L [0,1] by multiplica-tion. Let G be any free, ergodic, measure preserving group of transformations of [0,1] and l e t G be the factor constructed from M and G i n the standard manner (see [9], Chapter XII). * For any bounded measurable function h on [0,1] we l e t T^ denote .the corresponding element of G . There i s a f a i t h f u l trace T on G , which s a t i s f i e s (7.7) " ,, T ( T h ) = I * 1 h(x)dx . , o Iri p a r t i c u l a r consider the;functions f and g defined by . ' ; â€¢ L e t t i n g x denote t h e c h a r a c t e r i s t i c f u n c t i o n o f a s e t we have f o r 0 _< X Â£ 1 , the s p e c t r a l p r o j e c t i o n E ( x ) of T, t h e s p e c t r a l p r o j e c t i o n F ( x ) , o f T and E ( X ) = F ( x ) = 1 f o r X > 1 E ( x ) = F ( x ) = Q f o r X < 0 . A p p l i n g [ l . l ) s we see t h a t f o r any u n i t a r y U e G (7.8) T ( U * F ( X ) U ) = ' T ( E ( X ) ) 3 < X < + Â» . C o n s e q u e n t l y , i f t h e r e e x i s t e d a u n i t a r y U such * â€¢ ' I t h a t U T U and T y were c o m p a t i b l e , we would have f r o m . y (7.8) and the f a c t t h a t T i s f a i t h f u l , U * F ( x ) U = E ( x ) 3 f o r 0 < X < + â€¢ . By the u n i q u e n e s s o f t h e s p e c t r a l r e s o l u t i o n we would have t h a t * U T g U = T ^ and t h e r e f o r e t h a t / t h e von Neumann a l g e b r a s genera-t e d by T f and T are u n i t a r i l y e q u i v a l e n t . But t h i s i s a c o n t r a d i c t i o n , s i n c e T f g e n e r a t e s { T ^ : h â‚¬ L w[0,1]} w h i c h i s i" ; maximal a b e l i a n , w h i l e T 'â€¢ g e n e r a t e s { T n : h e L W[0,1] , and h ( x ) = h ( l - x ) } w h i c h i s n o t maximal a b e l i a n . T * [ o , x ] ' = T x([0 , X/2] U [ l - X/2,1]^ 68. G 8... C a l c u l a t i o n o f p f o r s e m i - f i n i t e f a c t o r s . The c a l c u l a t i o n s f o r p made i n t h i s s e c t i o n w i l l be based on the f o l l o w i n g r e s u l t f o r p p r o v e d I n [ 3 ] . Lemma 8.1 1 ( [ 3 L Prop. 2.3),. L e t G be a W - a l g e b r a w i t h a n o r m a l f i n i t e t r a c e T . L e t y and v e be such t h a t y = Tg and v = T t f o r some S and T â‚¬ G + . Then P(u,v) = T ( | S T | ) . f We want t o t a l k about the t y p e o f s t a t e s used i n the 'i above lemma, so we (make t h e f o l l o w i n g d e f i n i t i o n . â€¢ \ â€¢ D e f i n i t i o n 8 . 2 . L e t G be a W - a l g e b r a w i t h a n o r m a l , semi-f i n i t e t r a c e . T . An element y.. o f Z Q w i l l be c a l l e d e l e m e n t a r y w i t h r e s p e c t t o T , i f y, = T G f o r some S e G + such that.. T(E ' ) < Â» , where E i s t h e range p r o j e c t i o n o f S I n t h e r e m a i n d e r o f t h i s s e c t i o n G w i l l always be a f a c t o r and we can speak o f e l e m e n t a r y s t a t e s w i t h o u t r e g a r d t o T . " , â€¢ . â€¢ ' â€¢ / ' . â€¢ Lemma' 8 . 3 . L e t G be a f i n i t e ; f a c t o r w i t h n o r m a l i z e d t r a c e T and l e t SX and T be s i m p l e elements o f ; G . Then, f o r any V e G w i t h ||v|!'_< . l'; 'y ' / 'â€¢-â€¢u. '': 'â€¢' R ' |T(STV) I,â€¢< d ( S , T ) . Proof. By Theorem 7.10, choose a unitary U â‚¬ G so that S * m ' * and U TU are compatible. Let S = Â£ c.G. and U TU = ' i = l m E d.G. be the representations given by Theorem 7.5. Let i = l 1 1 ' -.gj_. J= TCGJ^) . By the unitary invariance of T and Corollary 7.3, . 1 â€¢ a ( S , T ) = o ( S , U T U ) = T ( S U T U . ) < m = E c i d . g 1 . 1=1 1 The lemma now follows e a s i l y from Theorem 4.3. Lemma 8.4. Let G; be a factor with a f i n i t e trace T . . Let u. and v be Elements of EQ. such that ti = T g and v = T t for some S and T â‚¬ G. . Then: (a) p(y,v) <_ d"(u,v) , and equality holds i f ^ S and T are compatiblej (b) Given any e > 0 , there exists a e,Int(G) . such that p(U,v a) > a(u,v) - e . . Proof. We can assume that T(1) = 1 Choose any e with . 0 < e.< 1 ' â€¢ â€¢ L e t r =-(||s|| + ||T|| + 1) i A It follows immedi-ately frpm the^spectral theorem that the simple elements of G+1 are norm dense i n G + . T h e r e f o r e , we can choose simple elements, â€¢ S ' and T' of G + such that Ils-s'll < h > llT-T/" <'e 70. Prom Lemma 6.14, (8.1) | a ( S , T ) - o ( S ' , T ' ) | < e/2 , and a similar type of ca l c u l a t i o n shows that | S T - S ' T ' || < e/2 . For any V e G with' ', ||v|! <\ we have I S T V â€¢- . S ' T ' V I ! _< H S T - S ' T ' ||||vl| < e/2 , and since the state T has norm 1 , v ( 8- 2| |- 1 T | ( \ S T V ) J ^ J T ( S ' T J V ) | " 1.'^ " I T ( S T V - ) . . - T ( S ' T ' V ) | . â€¢ " : s s - â€¢ v- â€¢ ':= | T ( S T V - S ' T ' V ) I; : . ""' / < e/2 .- ' "â€¢"'" From (8.1), (8.2), and Lemma 8.3, TISTV).|< |T(S'T'V) I + e/2 < a{S',T') + e/2 < a(S,T) + e 7 1 Then.from Lemma 0."4(a) and Lemma 8 . ' l , â€¢â€¢ â€¢ p(u,v) = T | ST | < o(S,T) +. e = o(M,v) +. e S i n c e e was chosen a r b i t r a r i l y , â€¢ : p(u,v) < o(H,v) . I f S and T a r e c o m p a t i b l e t h e e q u a l i t y f o l l o w s d i r e c t l y f r o m C o r o l l a r y 7 . 3 . T h i s p r o v e s ( a ) . By Theorem 7 . 1 0 , choose a u n i t a r y U e G such t h a t S and U T U a r e c o m p a t i b l e . C a l c u l a t i n g as i n ( 8 . 2 ) we.can show ( 8 . 3 ) JT(SU*T.U)| > |T(S'U*T'U)| - e/2 L e t . a be the i n n e r automorphism i n d u c e d by U . Then. vÂ° = T * . From ( 8 . 1 ) , ( 8 . 2 ) , Lemma 0 . 4 ( a ) , C o r o l l a r y u T U ; 7 . 3 , and Lemma 8 . 1 , p ( n , v a ) = T( | S U * T U | ) 2 | T ( S U * T U ) | ' :' \ -2 | T ( S ' U * T ' U ) I - e/2 - = .a(S',T' ) - e/2 > a ( S , T ) - e % â€¢/;.->'../\ = a(u,v) - e ; ; ... w h i c h p r o v e s ( b ) . ; / ; 72. Lemma 8.5. L e t G be a s e m i - f i n i t e f a c t o r and l e t u. and v be e l e m e n t a r y s t a t e s of J.^ . Then, (a) and (b) o f Lemma 8.4 h o l d . P r o o f . Suppose \A = T g , v = T t where S and T â‚¬ G + L e t E and .F be t h e range p r o j e c t i o n s of S and T r e - â€¢ s p e c t i v e l y . S i n c e E and F a r e f i n i t e i t i s a w e l l known r e s u l t t h a t the p r o j e c t i o n G = E U F i s f i n i t e (see [9] Lemma 7-35). L e t y,' and v' denote t h e r e s t r i c t i o n s o f U and v r e s p e c t i v e l y t o the f i n i t e f a c t o r G Q . L e t . a ' be any i n n e r automorphism of G^, . We extend a' t o an i n -ner automorphism a-, of G as f o l l o w s . I f the u n i t a r y U i n Gr, i n d u c e s a'' , we t a k e a as the i n n e r automorphism I i n d u c e d by t h e u n i t a r y U + C :(1 - G ) . Then ( v ' ) a i s the r e s t r i c t i o n of v a t o G Q . Moreover, â€¢, / .â€¢ v A ( G ) = v ( a ( G ) ) = v ( G ) = T ( S G S ) = T ( S 2 ) = u ( l ) = 1 , and s i m i l a r l y u ( G ) = 1 . From ([3] Prop. 1.10), â€¢'" / â€¢ P ( y , v a ' ) = P ( u ' , ( v ' ) a ' ) . and t a k i n g a ' = the, i d e n t i t y , automorphism^ ..â€¢ ' â€¢ 'â€¢; V â€¢ â€¢ : :.;;v â€¢<;â€¢â€¢:"â€¢) â€¢ /. ; P(W,V) =. ,'p(u',v') F i n a l l y , we see from Lemma' 6.5 t h a t a(u,v) = a(u'Â»v') The lemma now follows d i r e c t l y from Lemma 8.4. Theorem 8.6. â€¢ Let G be a semi-finite factor and l e t G be a subgroup of Aut(G) which contains Int(G) .' Then, f o r any u, v e Â£ Q p G(u,v) = a G(ti,v) â€¢ We f i r s t prove the r e s u l t i n a spec i a l case. Lemma 8.7- Theorem 8.6 holds i f u and v are elementary.: Proof. Choose any a e G . Then obviously i s also elementary. From Lemmas*58. 5 and 6.10 we see that P(u,v a) < a(u,v a) < a G(u,v) . Then from Lemma 3 . 2 . , (8.4) ' p G(u,v) < a G(u,v) . Now choose any e > 0 Let k â‚¬ r(G) be such that a'k(y, v) > o (u,v) - e/2 ',- and choose f3 e G so that T = kT for any trace T on G/.. 'Applying Lemma 8.5, there exists ye. Int(c) such that ...â€¢, :>X.:'.'.'" '; â€¢ ';â€¢:./;â€¢. â€¢â€¢â€¢ 'â€¢' â€¢ 74 (8.5) ' P ( U , V P y ) > o ( u , v 3 ) - e/2 = o k ( y , v ) . - e/2 r > a (n,v) - Â« . S i n c e Int(G.) c G , we have t h a t ^ e G , and t h e lemma f o l l o w s f r o m (8.4) and (8.5). Remark 8.8. The d e f i n i t i o n f o r a ( y , v ) can o b v i o u s l y be made f o r any p o s i t i v e , n o r m a l , l i n e a r f u n c t i o n a l s as w e l l as f o r s t a t e s . I f a s t a t e u = T c , the f u n c t i o n a l ky = T . S JXS f o r any p o s i t i v e k , and we have seen t h a t "F = Â»/K T C o n s e q u e n t l y , f o r any p o s i t i v e k and r , a ( k y > r v ) = Jkr a ( y , v ) â€¢ I n Remark 2.5 we n o t e d t h a t p ( k y , r v ) = Jkr p ( y , v ) â€¢ I t f o l l o w s t h a t Lemma 8.7 h o l d s f o r p o s i t i v e , n o r m a l , l i n e a r f u n c t i o n a l s , and we w i l l a p p l y i t i n t h i s - f o r m i n the f o l l o w i n g p r o o f . P r o o f of-Theorem 8.6. i , . ' , â€¢â€¢ / â€¢ L e t T be a s e m i - f i n i t e t r a c e on G. w h i c h we keep f i x e d t h r o u g h o u t the p r o o f . "Suppose t h a t u = and v = T t . L e t {E(x)} be the s p e c t r a l r e s o l u t i o n o f S . F o r any i n t e g e r n > 1 l e t P n,- ( E ( n ) - E(i)) , and l e t S = SP . S i n c e S i s s q u a r e , i n t e g r a b l e , . P i s a f i n i t e n n : -a ... , " n p r o j e c t i o n and S n i s a bounded p o s i t i v e o p e r a t o r w i t h f i n i t e range. We d e f i n e the f u n c t i o n a l u n = u p â€¢ . Then un = T S n n so i t i s el e m e n t a r y . S i n c e SnT S , we o b t a i n by Lemmas 5.2(d) and 6.4 t h a t '"' ' 'â€¢â€¢ (8.6) TSJ 7 S . A n a l o g o u s l y , we d e f i n e p r o j e c t i o n s , o p e r a t o r s T r f , and e l e m e n t a r y f u n c t i o n a l s v n c o r r e s p o n d i n g t o v . We. o b t a i n (8.7) fT t V ' n .From Lemmas 6.1 and 5.3 we have (8.8) T ( S 2 ) = f ' (T (a ) ) 2 d a , and T ( S 2 ) = J " ( 7 q ( a ) ) 2 d a Now u s i n g (8.6) - (8.8) and the monotone convergence theorem ; . ; u ( P N ) = T ( S 2 ) t T ( S 2 ) . = u ( i ) = I . / : v ( Q n ) \: 1 . r - : . a ( | i n , v n ) = a ( S n , T n ) / | a(S,T) = o(u,v) . S i m i l a r l y a k ( u n , v r i ) ^.â€¢a k(u,v) f o r a l l k > 0 and t h e r e f o r e o G ( u n , v n ) T a G ( u , v ) . Then g i v e n any e > 0 we can Choose an i n t e g e r - m so t h a t |i(P m) > 1 - (YQ)2 /and v C Q j > 1 - ( ~ ) 2 , and (8 .9 ) a G (M m,v m) > o(u,v). - | . Let a be any element of G . Then â€¢ â€¢ " V I ( v n , ) a = ( v f o ^ ) a = ^ - 1 Â»J by Lemma 0. ; , ( V .'â€¢(-a'1(01n)) . :. â€¢ and v a ( a - 1 ( Q m ) ) = v ( Q j > 1 - ( ^ ) 2 . We can then apply ([3] Prop. 1 . 9(c)) to obtain \ - . ; -,. . . . . |p(u,v a) - P ( ^ v m ) | <_ c/2 , fo r a l l a e G : , " Â» and therefore, ' â€¢ ' (8 .10) |p G(u,v) - P G ( % > ^ m ) l < Â« / 2 .. ' â€¢ " ' '' / Lemma 8 . 7 , applied to the elementary functionals u vand v gives â€¢' ( 8 - 1 1 ) = P G ( l V v m ) > ^ G ( u l n , v m ) . From formulas^ (8 .9) - (8 .11) we have |p G(u,v) - a G ( u , v ) | < e and since e was chosen a r b i t r a r i l y the proof i s completed.' D e f i n i t i o n 8.Q. Let G be a semi-finite factor and l e t u and. v e T^r ~. We say that > u .and v are compatible,^ â€¢ ' v. â€¢â€¢â€¢ â€¢ 7 7 . i f f or some semi-finite trace T on G Â» u = T G and v = T T where S and T are compatible operators. Lemma 8 . 1 0 . Suppose that p. and v â‚¬ are compatible. Then p(u,v) = a(p-,v) . Proof. This follows d i r e c t l y from Lemma 8 . 4 ( a ) i f the Radon-Nikodym derivatives of u and v â‚¬ G . For the general case we can employ the same type of approximation procedure used i n proving Theorem 8 . 6 . Theorem 8 . 1 1 . Let G be a factor of type I or of type 1 1 ^ . Then, f o r any u arid v 6 E Q : â€¢ ' * Proof. Immediate from Theorem 8 . 6 , Lemma 8 . 1 0 'and formula Example 8 . 1 2 . Consider the algebra G = Â£(H) where H i s separable. Let T be the trace on G , normalized so that i t s value = 1 on minimal projections. Let S be a p o s i t i v e , Hilbert-Schmidt operator ( i . e . T(S ) < Â» ). It i s we l l known (a) (b) d(ji,v\) = V2[ l - o ( n , v j I f p. and v)} are compatible, d(u,v) = d(p.,v) ( 3 . 1 ) . that S = i n the sense of strong convergence, where (a.) i s a non-increasing sequence of p o s i t i v e r e a l 78 numbers such that E (a.) < Â« , and ( E . ) i s a sequence' of non-zero, mutually orthogonal, minimal projections such that I E 1 = 1 . Morever the sequence (a.) i s uniquely 1=1 determined by S . Let ( E ( x ) } be the spectral resolution of S - . We can apply (7-4) to see that . j k 1 - E ( X ) = E E ^ , where k i s such that a k > X _> a k + 1 . It follows that f ( x ) = the number of a^'s > X '. Therefore, i f X > a n + 1 , f ( x ) < n ; and i f X _< a n + 1 , f(X) _> n + 1 â€¢ 1* follows from Lemma "5,2(b) that 7(n+r) = a n + ^ > where n i s any non-negative integer and' 0 <, r < 1 es oa Then, i f S = E a.E., and T = E b.F., are two such opera- ' I â€¢â€¢ i = l . . i = l 1 1 . tors we have '\i at a(S,T) = E a.b, .' i = l 1 1 -v-'M A similar r e s u l t of course holds when H Is f i n i t e dimensional. Since any state y On G = T g f o r some H i l b e r t - . V v + â€¢''-"'". Schmidt S e G N Â» t h i s gives us an immediate c a l c u l a t i o n ' . formula for d(u,v) i n the case of a type I factor. â€¢ : â€¢' ; '. . 79.. Example 8 .13 . L e t G. be a s e m i - f i n i t e f a c t o r w i t h t r a c e T . F o r any p r o j e c t i o n E â‚¬ G such t h a t T ( E ) = a < Â« , we can e a s i l y v e r i f y t h a t f E = / X [ o a) ^ ^ d e n o t i n g the c h a r a c t e r i s t i c f u n c t i o n o f a s e t ) . T h e r e f o r e , i f E and F a r e any two p r o j e c t i o n s e G such t h a t a = T ( E ) < T ( F ) = 3 < Â» , we have a(E,F) = a . Now i f we c o n s i d e r the s t a t e s u = l / a ( T E ) and v = 1 / B ( T F ) , we have from Theorem 8 .6 and Remark 8.8 t h a t s â€¢ " 1 â€¢ V p(u,v) = ;o (u,v) = 1/Ja& (o'(E , P ) ) -. = T O T P = 7 T ( E ) / T ( F ) . . ' Note t h a t i f a/B e r ( A u t ( G ) ) 1 , the same.type o f c a l c u l a t i o n shows t h a t p(u,v) = 1 . Of cour s e t h i s i s e v i d e n t ,' imme-d i a t e l y s i n c e we can i n t h i s case f i n d an automorphism s e n d i n g F onto E and t h e r e f o r e v onto u . _ â€¢?: ' CHAPTER I I I ; ; . APPLICATIONS TO INFINITE TENSOR PRODUCTS . 9. Isomorphisms of Products; Defi n i t i o n s. D e f i n i t i o n 9.1. Let (G,(c^)) and ( f l , ^ ) ) be two pro-ducts for the family of W -algebras ( G i ) i â‚¬ i â€¢ W e s a v that they are: (a) product isomorphic, If there .exists an isomorphism (}> from G onto 6 s a t i s f y i n g <j)ai = 8^ f o r a l l i e l In such a case we w i l l write (G, (a^).) ~ (Â©-, (6^)) ; â€¢> (b) weakly product isomorphic, i f there exists an isomorphism d) from G onto 0 s a t i s f y i n g <|)(ai(Gi)) = S i(G i) f o r a l l i e l ; (c) permutably product isomorphic, i f there exists an i s o - ; morphism dp from G onto fj.' and a permutation ir of I , â€¢ / s a t i s f y i n g <|Â»(ai(G1).) = P T( i')(G i r( i)) f o r a 1 1 1 6 1 > ( l n other words they are weakly product isomorphic after a rear- . rangement of the G^'s ); (d) a l g e b r a i c a l l y isomorphic i f there exists an isomorphism from G onto B I t i s obvious that:- (a) - (b) - (c) - (d) , and i t i s not hard to show.';that, a l l implications are s t r i c t . Remark 9.2. Note t h a t i f (G, (oj.')). and â€¢ (fiÂ»(^))' a r e wea k l y p r o d u c t i s o m o r p h i c , we can d e f i n e automorphisms^ <p^ of f o r each' i e I by '. . ; $Â±(AÂ±) = P5 L 1<|)a 1(A 1) , f o r a l l A Â± 6 G^ We. t h e n have . <J>aÂ£ = , o r i n o t h e r words t h e diagram ' x â€¢ â€¢ 1 V-v ^ i i s commutative f o r ( a l l i e l . A s i m i l a r ^ r e s u l t o f cours e h o l d s f o r p e r m u t a b l e â€¢product isomorphisms. ! ... â€¢ . / Remark ' 9 - 3 . I t f o l l o w s i m m e d i a t e l y from the d e f i n i t i o n s t h a t r a p r o d u c t isomorphism p r e s e r v e s p r o d u c t s t a t e s . That ' i s , i f v e = .^(v.^) f o r the p r o d u c t (iB, (P^)') Â». and i f <f> i s a p r o d u c t isomorphism f/om ( G , ( a ^ ) ) o n t o t h e n = ^ ( v ^ f o r .the. p r o d u c t .'; (G, ( c ^ ) ) . I f <j> i s i n * s t e a d a.-we,ak- p r o d u c t ^isomorphism,'^ v-'' = :â€¢ Â®j( v^-J'-.' f o r the .'.<j>^ ', . d e f i n e d i n Remark 9.2. â€¢/'['â€¢'i-:':: . 'â€¢â€¢.'â€¢â€¢â€¢.â€¢ â€¢â€¢â€¢ â€¢Â«'â€¢' / (G,(a,d),)) ~ Â® ( G ^ u / ) . i . 1 1 i e l 1 1 : * i 1 Proof. For each i e l choose [^^x^] e Q(u^ ) . Let â€¢.^â€¢%i^v ft be the von-Neumann algebra Â®(R i,x^) where = ^(G^) . ..; !' : i > â€¢" .Then obviously [ \ [ r i d)7 1 , x i ] e QCu^) , and since â€¢ d)^ i s an auto-morphism, = ^ < t ) ^ 1 ( G i ) . So by the d e f i n i t i o n of Â® ( G I > U I ) there exists an isomorphism is from G onto ft s a t i s f y i n g . â€¢I â€¢*. :.1 . M i \i<a1(A1) = i(fid)~1(Ai) , for a l l i e l - and A^ e G^ . Then, tya^^) = " ^ j ^ A Â± ) = â€¢ i ( A i ) . f o r a 1 1-. 1 â‚¬ 1 and'â€¢ Aj^ e G^ , which by; the d e f i n i t i o n of Â® ( G I , U I ) proves the â€¢ ! i lemma. ; ' -SS;; . . â€¢ â€¢' " A\ V ' - . â€¢ ' ' -;}>'. Lemma 9- 5. ^ (G - ^ U J ) and Â® (GJ,V^) are weakly product S$' ;- â€¢' i e l v ,' i e l 1 1 isomorphic, i f and only i f there exists for each i e 1' , an automorphism d> of G., such that Â® ( G 1 } u J ' and Â® (G^v., ) â€¢' i e l i e l 1 . 1 .â€¢ "/';â€¢â€¢};. are product isomorphic. Vs.â€¢â€¢..â€¢'. â€¢ â€¢ â€¢ ' ' / â€¢ ' s "'â€¢ â€¢//â€¢ Proof. ..This follows immediately from Lemma 9-4 and Remark 9.2. N v. â€¢â€¢â€¢ 10. Main Results on Tensor Products. . . 1 â€¢ â€¢ \ In t h i s section we apply the product formula for'.5 â€¢p which we obtained In Section 2 to develop necessary and s u f f i c i e n t conditions f o r two l o c a l tensor products of a . given family of W -algebras to be product isomorphic. The same r e s u l t was proved i n ([3], Theorem 4.1) under the assump-tions that the algebras are semi-finite, and i t was shown to be a non-commutative extension of Kakutani's Theorem on i n f i -n i t e product measures [6]. The only need of the semi-finiteness r e s t r i c t i o n was i n invoking the product formula for p , and ^ as we have i n Theorem 2.4 removed t h i s r e s t r i c t i o n , we can appeal d i r e c t l y t o i [ 3 ] for a proof of t h i s present r e s u l t . : We w i l l however present a complete proof here, s i m p l i f y i n g part of i t , and at the same extending the product formula f o r -p to the case of an i n f i n i t e number of factors,. F i n a l l y we obtain necessary and s u f f i c i e n t conditions of a similar nature for two. l o c a l tensor products to be weakly product isomorphic . and to be permutably product isomorphic. â€¢ / ' ' . D e f i n i t i o n 10.1. Let (G,(a i)) be a product for the family â€¢ ( G ^ ) i â‚¬ l . Let (|> be any representation of G on a H i l b e r t space H . Let u e . For any non-void subset J of , i!' we define: â€¢â€¢' ' ; â€¢ ' ' ' v .<' ;: G x.= the . W -algebra generated by {aj,(A1) ; , i e J , J--' â€¢ . . A i â‚¬ G i } ' A - / ' ^ (j) = the r e s t r i c t i o n cj> to '. G:-: :!:: 1 ' U J = the r e s t r i c t i o n of \ to, GJâ€¢ Lemma 0.1 shows t h a t d) J i s i n f a c t a r e p r e s e n t a t i o n o f G J on H Lemma 10.2. Suppose (G,(a.,)) ~ Â® (G.*,|i.) â€¢ L e t u. he " ~ 1 i e l 1 1 the element of -2 G = Â® (y. ) and l e t v be any element o f i e l '^EG. . Then t h e r e e x i s t s some s u b s e t J o f I w i t h a f i n i t e compliment such t h a t p ( y J j V ^ ) > 0 . P r o o f . Suppose t o the c o n t r a r y t h a t p ( y J , v J) = 0 f o r a l l Ji c I w i t h f i n i t e compliments. F o r any J c I ' l e t E^ denote t h e s u p p o r t o f '. J 1 ;v ' , and l e t E denote t h e s u p p o r t o f v .. â€¢ We can c o n s i d e r each G^ as a von Neumann a l g e b r a on a H i l b e r t space H^ w i t h a v e c t o r x^ i n H^ i n d u c i n g y. ', and G as a von Neumann a l g e b r a on H = Â® (H. ,x. ) . ; . 1 v . i e l 1 1 -Choose an o r t h o n o r m a l b a s i s , . f o r H -as d e s c r i b e d i n Chapter 0. L e t x' be any element o f t h i s b a s i s . R e c a l l t h a t t h e r e e x i s t s a ^ s u b s e t J = J ( x ' ) r o f I , w i t h f i n i t e compliment, such t h a t x' = â€¢ Â® , ( x i ) , where x' = x. f o r a l l i e J . i e l 1 7 1 1 S i n c e Â® (x . ) i n d u c e s y on G i t I s o b v i o u s t h a t x' i e l . â€¢ i i n d u c e s y ^ on G^ . Then, by our assum p t i o n and Lemma 1.2 the s u p p o r t o f y i s o r t h o g o n a l i n G t o E . T h e r e f o r e , ||EJx'||2 = y J ( E J ) = 0 . S i n c e v ( E J ) = v J ( E J ) ; = 1 , E J 2. E N and we must.have Ex' = 0 ... . S i n c e t h i s i s t r u e f o r a l l x' i n , some o r t h o n o r m a l b a s i s o f H we have E. = 0 , and t h e r e f o r e y = 0 , w h i c h i s a c o n t r a d i c t i o n t o t h e f a c t t h a t u i s a s t a t e . â€¢ '.;>-'' Remark 10.3. We next r e c a l l some elementary f a c t s about I n f i n i t e products of numbers. - I f ( r j . ) i e-~I " i S a f a m i l y of p o s i t i v e numbers, we say that TTr., converges i f C' >. j lim (TTr.) as F runs over the f i n i t e subsets of I . exi s t s F i e l as a p o s i t i v e number r . We then define TTr;., = r . : â€¢ i e l 1 â€¢ ; . .Suppose . (rj_)j_e;r/' i s family of non-negative numbers with at l e a s t one r. = 0 We then say that TTr; converges \ . .... i e l 1 ' _ '; i f there are-only a f i n i t e number of zero factors and a f t e r deleting.these the'resulting family of p o s i t i v e numbers con- . verges ' a s above. In t h i s case we define TTr, = 0 . I t ; i e l follows from these (definitions that TTr, converges i f and ' j i e I - ' . : '.-M-only i f E | l - r . | l < Â» . i t i s easy to v e r i f y that i f i e l 1 â€¢ TTr. - r and TT s. = s then I I ( r . s . ) = rs i e l 1 i e l 1 i e l - 1 1 . â€¢ ':; We now consider the following s i t u a t i o n ; the no-ta t i o n w i l l remain f i x e d f o r the remainder of t h i s section. Let I be an a r b i t r a r y indexing set and.let ( G i ) i e i b e a -family of W -algebras . Let^ (Mj.)i ej_ 8 1 1 ( 1 (vi)ieiÂ» b e t w o elements of A[(G i)] . As a notational convenience we w i l l l e t Â®'.; denote Â® ( s i m i l a r l y f o r E and "\~\ j.: -i e l ' 1 â€¢ , Lemma 10.4. . The following conditions on (u.) and (v..) are equivalent: G i , n 2 . ; 09 (a) E[d x ( u i , v i ) ] . < X (b) E [ l - p i ( u i , V 1 ) l - < Â« ', (c) T T P i ( u i 5 - v i ) . converges] for any subgroups G^ . of A u t ( G 1 ) . Proof: Immediate'from (3.1) and Remark 10.3. ! Theorem 10.5. Suppose that v = Â® ( ) exists as a product state on fl^G^y.^) . Let y = ^(UJ^) . Then p ( l l , v ) ^["^(u^^v^^) , a convergent product. Proof. Suppose ( G , ( a ^ ) ) ~ Â®(G i,y j L) . We consider y i and v as elements of E . â€¢ Choose any [<Mx,y)] e Q(y,v) and l e t F be any f i n i t e subset of I â€¢ . A d i r e c t calcula-t i o n shows that [<|> â€ž', (x,y) j , e Q(y ,v ) and therefore F F i ' p(u,^v ) _> l ( x l y ) l â€¢ Taking the supremum over a l l elements of Q(y,v) we obtain ' â€¢ . (10.1) p ( u P , v F ) > p(u,v) . " F F It i s obvious that y = Â® (y. ) and v = Â® (v., ) on ieF 1 ieF 1 â€¢pi Â® ( G J ) = G i . Therefore, by Theorem 2.4, which extends ieF 1 < . yS-y from two to any f i n i t e number of factors', by an obvious i n -duction and standard associ'atlvety arguments> (10.'2) P ( U F , V F ) ^ T T p ^ i V i ) .;^f'V By (10.1) and (10.2), (10.3) p(w,v) < i n f {TTp^., ) : P a f i n i t e subset of 1} ~, ieF 1 1 Now by Lemma 10.2, choose a subset J of I with f i n i t e com-pliment such that p(u J, v J) .>; 0, . By applying the above ar-gument to the algebra G1^ , we obtain from (10.3) that (10.4) i n f { I 1 p(u,,v,) : F a f i n i t e subset of J} > 0 ieF Let J'.= {i e I : ' p ( u i , v j L ) > 0} . By (10.4), J c j ' , so j ' has f i n i t e compliment. Moreover j ' - J i s f i n i t e i 1 so (10.4) holds with J replaced by J'- ; Then, since the value o f p i s always 1 ' , lim ( T T p ( u v )) = i n f ( T T . p C u ^ V i ) ) , F ieF 1 1 p' ieF' where F , F' runs over the' f i n i t e subsets of i ' , j ' re-spectively. Therefore | |p(u^,v^) converges and we may write (10.3) as / ' ' '.V (10.5) P(u,v) < [ \ P ( U Â± , V Â± ) . We now prove the other d i r e c t i o n . Let k be any po s i t i v e number < 1 . Choose any sequence (k f i) of posi -t i v e numbers < 1 so that 1 X k = k , Let I = f i e I : n=l n Â° d(u i,v i) > 0} . T h e convergence of | |p(u 1,v i) and Lemma 88. 10.4 show that I i s at most countable. Let y be an i n -f e c t i o n from I Q into the p o s i t i v e integers. Define a f unc-. â€¢ ti o n g on I by {1 , for i e I - I Q \ ( i ) > f o r 1 6 Xo ' Then choose, f o r each i e l , an element [ $ , ( x , y ) ] of Q ( u I 3 V I ) s a t i s f y i n g | ( x i | y i ) | > g(i)[p(uÂ±,yÂ±)} . This i s : c e r t a i n l y possible by the d e f i n i t i o n of p and the f a c t that i e I - I Q implies that u i = v j _ â€¢ By multiplying the vec-tors by suitable scalars i f necessary we can assume th a t . 1 (10.6) ( x i l y i ) 2 g ( i ) [ p ( u 1 3 v i ) ] , f o r a l l i e l " v. It i s easy to see thatTTg(i) converges and that i t s value i s 2 TT k n = k â€¢ Therefore, (10.6) shows that "^(x^^ j y 1 ) n=l â€¢ converges and that ': â€¢ / ( 1 0 . 7 ) TT , ( x i l y i ) I'kCTTpd^iVi)]; y for a l l i e I . ,We have then that E [ l - (x^|y^)j] converges, which shows that fo r the family of H i l b e r t spaces (H^) , where Is the underlying space of dp^ , (x^) and (y^) are equivalent C Q-sequences. Therefore, the vectors x = Â®(x^) and y = Â®(y i) are elements of H = .^H^x.^) which induce the states y and v respectively. Then-using ( 1 0 . 7 ) , 8 9 . P ( u * v ) > | ( Â® ( x 1 ) | Â« ( y i ) j = | | ( x i | y 1 ) > k [ ^ p ( u l V i ) J . Since k was chosen a r b i t r a r i l y we have that p(u,v) 2]Tp(Hi*vi) > f o r a 1 1 i e l The above formula and 10.5 complete the proof. Corollary 10.6 ( [3] Lemma 3.6) Suppose that E [ d ( u i * v i ) ] <â€¢Â» . Then Â®(G 1,v i) and Â®(G^,u^) are product isomorphic. . Proof. In the second part of the proof of Theorem 10 .5 2 the convergence of E[d(y^,v^)] was a j l that was necessary to produce the H i l b e r t sp:ace H = Â®(H^,x^) = Â®(H^,y^) , and the tensor product Â®(<|>^(G^),x^) = Â®(<^(6i)* vi) which i s pro-duct isomorphic to both S^G^u^) and ^ G ^ v ^ ) . . The corol-l a r y follows immediately. ' â€¢ ' / â€¢ ' ' We now w i l l state the three main theorems of t h i s section and then give the proofs. Theorem 10 .7 . The following â€¢conditions "bh>;.(u.) and r(vV)> are equivalent: . " ' ' / p (a) E [ d ( u i , v 1 ) ] 2 .< â€¢ ; ,V xi â€¢ ' (b) Â®(G^,U^) and (gi(G^,v^) are product isomorphic; 90. ( c ) Â®(v^) e x i s t s as a p r o d u c t s t a t e o f Â®(G^,y^) . Moreover, i f any o f t h e s e c o n d i t i o n s h o l d we have p(^(\iÂ±), Â®(vÂ±)) =JXp(ui,v1) , a conve r g e n t p r o d u c t . â€¢ I Theorem 10.8. The f o l l o w i n g c o n d i t i o n s on (u.^) and ( v i ) are e q u i v a l e n t : (a) E [ d ( u i , v i ) ] < Â» , and f o r a l l but a c o u n t a b l e number o f i e l the infimum i n the d e f i n i t i o n o f d i s a t t a i n e d ; (b) &(G^,u.j_) and ^ ( G ^ j v ^ ) a r e w e a k l y p r o d u c t . isomorphic;... ( c ) There e x i s t automorphisms (j)^ o f G 1 such t h a t ta <%(v^ ) e x i s t s as a| p r o d u c t s t a t e o f (g^G^,^) . ! I Theorem 10.9- The f o l l o w i n g c o n d i t i o n s on ( u i ) and (v^,) are e q u i v a l e n t ; (a) F o r some p e r m u t a t i o n ir o f I , E(d ( u i.jv' 7 r^ ij) ] < Â» , and f o r a l l but a c o u n t a b l e number o f i e l t h e i n f i - ' mum i n the. d e f i n i t i o n o f d i s a t t a i n e d ; (b) Â®(G i , u i ) and ,Â®(G i,v i) a r e ; p e r m u t a b l y p r o d u c t i s o m o r p h i c ; , â€¢â€ž-' ; >' ( c ) F o r some p e r m u t a t i o n TT o f I , t h e r e e x i s t â€¢ ta isomorphisms ^ of G i onto ^ ( i ) such t h a t , ^ ( v ^ ^ ^ ) e x i s t s as a p r o d u c t s t a t e o f (S^G^u^) . v P r o o f o f Thedrem 10.7. By Theorem 10.5 and Lemma 10.4, ( c ) i m p l i e s ( a ) . By C o r o l l a r y 10.6, (a) i m p l i e s ( b ) . I t i s im-mediate t h a t (b) i m p l i e s ( c ) (see Remark 9-3). The l a s t s t a t e -m e n t . i s immediate from Theorem 10. 5. . 'â€¢ â€¢ 91. P r o o f o f Theorem 10.8. Suppose t h a t (a) h o l d s . L e t I Q be the s u b s e t o f I f o r w h i c h the infimum i s n o t a t t a i n e d i n d(u^,y^) , and t h e r e f o r e by (3.1) the supremum i s n o t a t t a i n e d i n p(u^jV^) . By h y p o t h e s i s , t h e r e i s an i n j e c t i o n "y from I Q i n t o t he p o s i t i v e i n t e g e r s . D e f i n e a f u n c t i o n g on I by 0 , f o r i e T - I 0 g ( l ) l / 2 Y ( i ) , f o r i e l ' . o By Lemma 3.2,, we can choose ' automorphisms <j)^ o f G^ such t h a t â€¢ " ! . â€¢ i ta ~ (10.8) p ( w 1 , v i i ) > p ( u i , v i ) - g ( i ) . , Now s i n c e V. g ( i ) < 1 , and s i n c e Â£[ 1 - p(u 1,v j,)] < Â» by the h y p o t h e s i s and Lemma 10.4, we o b t a i n from (10.8) t h a t 4), â€¢ (10.9) S[-l - p ( u i J v i 1 ) ] < Z [ l / - p ( u i , v i ) ] + Â£ g ( i ) < â€¢ . Then from Lemma.10.4 a g a i n , . â€¢â€¢ Vi â€¢ â€¢ 'â€¢ \ ' ' '' "â€¢ ' , â€¢ . . ta o â€¢ (10.10) : -; . E [ d ( U i , V i 1 ) ] S < Â« . .... ." â€¢â€¢ ,n ' '.' 'xy-â€¢';;.;|; â€¢ â€¢ r V f | ; : ' . â€¢ ' C o n v e r s e l y , suppose,:there e x i s t automorphisms <j>. of â€¢ v ? ' â€¢ ~ 'â€¢' â€¢ <}>. â€¢ L G. such t h a t . (10.10) holds..' â€¢ S i n c e d(u.,v.,) < d ( y . , v . 1 ) we have . ' : : ' 2 E [ d ( u i , v i ) 3 < Â» > and moreover, f o r a l l but a . c o u n t a b l e number o f i e I we .â€¢ , must have d(u^,v^) = d ( ^ i , v i ) = 0. . So (a) i s e q u i v a l e n t . t o the statement t h a t (10.10) h o l d s f o r some automorphisms 4^ o f G^ . R e p l a c i n g (b) w i t h t h e e q u i v a l e n t ^ statement g i v e n by Lemma 9.5, we can a p p l y Theorem 10.7 t o ( y . ) and (Vj_ ) t o prove Theorem 10.8. P r o o f of Theorem 10.9â€¢ T h i s f o l l o w s i n an o b v i o u s manner from Theorem 10.8. Lemma 10.10. Suppose t h a t f o r each, i e I , i s a f a c t o r of type I. o r type 11^, and t h a t y^ and . are c o m p a t i b l e s t a t e s . Then Â®(G.Â£,Mi) and ^ ( G ^ v ^ ) are w e a k l y p r o d u c t i s o m o r p h i c i f and o n l y i f t h e y a re p r o d u c t i s o m o r p h i c . P r o o f . Immediate f r o m Theorems 10.7, Theorem 10.8, and Theo-r rem 8.11. Example 10.11. L e t G be a I g f a c t o r w i t h n o r m a l i z e d t r a c e T , and l e t E and F be two o r t h o g o n a l p r o j e c t i o n s i n G. .. Suppose t h a t G^ = G . f o r a l l i e l . U s i n g Example 8;12 and the' theorems of t h i s s e c t i o n we can e a s i l y o b t a i n t h e f o l l o w i n g . - â€¢â€¢ ,(a) L e _ t \lÂ± = 2 T E a n d v i = 2 TF f o r a l l i ' . e ^ I Then <5?(Gi>yi) and $>(G.^,v^) â€¢ a r e n o t p r o d u c t i s o m o r p h i c b u t t h e y a r e w e a k l y p r o d u c t i s o m o r p h i c . â€¢; ' ::': , : (b) Let I be the p o s i t i v e integers. Suppose that: 2T e ., f o r i odd T 3 for i even ; v i = w ( i + l ) ' f o r . a 1 1 1 â‚¬ 1 5 Then ^(G^Uj.) and <^(Gi,vi) are not weakly product isomor-phic but they are permutably product isomorphic. Example 10.12. Let G be.a factor of type Iy with nor-malized trace T ,| and l e t uj. be a factor of type I 2 with normalized trace T^ ' . Let E be a minimal projection i n a Then G = A3 Â® (ft Â® fl . . Define, the states u and v on G by, - ;: u - ( 2 ^ ) Â« ( 2 ^ ) Â® T' = ^ ( E ^ 1 } . v = (2T e) Â® T' = 2 T ( E ^ 1 } . Let G i = G 3 Uj, = u and v.^ = v for a l l I e l . By Example b.13, p(u,v) = V Â£ , s o ^ ( G ^ , ^ ) and Â®(G 1,v i) are not weakly (and therefore i n t h i s case not permutably) product â€¢isomorphic' However there ex i s t "refinements" of each of these tensor^products which are permutably product isomorphic. Therefore they are a l g e b r a i c a l l y isomorphic. In f a c t , sup-pose I i s countable. Using standard associativety arguments, '.'â€¢we can f i n d i n each case c o u n t a b l e d i s j o i n t s u b s e t s J and . V '.<â€¢ : K w i t h . J u K = I so t h a t t h e r e s u l t i n g a l g e b r a ; , . ;\ . = [ Â® ( I B , T ' ) ] [Â®((S,2T4) ] ' ,â€¢ whi c h by s t a n d a r d r e s u l t s on type \ ' J K h ;â€¢ :;f V:1:'^ -' (see [2] e.g. .),-.=\P Â® I where P i s the hyper-finite' I I r V / . V.-; f a c t o r . , ' J . ' . ' ' 'â€¢ Example 10.13. The p r o d u c t f o r m u l a does n o t h o l d f o r p . 'â€¢As a t r i v i a l example, l e t y and v be any two s t a t e s of a W - a l g e b r a C such t h a t p ( y , v ) < 1 . Then on .G Â® G , p(u Â® v , v Â® u) = 1 . But p(y, v) p(-v,u) < 1 . . . . I t i s easy t o show however t h a t we always have -p ( u 1 u 2 > v x Â® v 2 ) 2 P ( W 1 > V 1 ) P ( U 2 , v 2 ) . . . T h i s i s a u s e f u l f a c t , s i n c e i t shows t h a t i n c e r t a i n c a ses ; we may be a b l e t o deduce t h e a l g e b r a i c isomorphism o f two non-.;,.;:?'. w e a k l y p r o d u c t i s o m o r p h i c t e n s o r p r o d u c t s by p a r t i t i o n i n g I ;,;.'. . i n t o d i s j o i n t s u b s e t s and computing t h e d i s t a n c e d .between the r e s u l t i n g p r o d u c t s t a t e s on the. r e s p e c t i v e p i e c e s . â€¢ ' â€¢ .; , . 95 BIBLIOGRAPHY ' . [1].' H. A r a k i and E.J. Woods, A c l a s s i f i c a t i o n of f a c t o r s , ; Publ. RIMS., Kyoto Univ. S e r i e s A, 3(1968), 51-130. [2] . D. Bures, C e r t a i n f a c t o r s constructed as i n f i n i t e tensor, products, Composito Math., 15(1963), 196-191. f2] D. Bures, An extension of Kakutani's theorem on i n f i n i t e product measures t o the tensor product of s e m i - f i n i t e W -algebras, Trans. Amer. Math. Soc. 135(1969), 199-212. D. Bures, Tensor.products of W -algebras, P a c i f i c Jour-n a l of Mathematics 27(1968), 13-37. [5] J. Dixmier, Les algebres d'operateurs dans l'espace h i l b e r t i e n , second e d i t i o n , G a u t h i e r - V i l l a r s , P a r i s , 1969; [6] S. Kakutani, On equivalence of i n f i n i t e product measures, Ann. of Math. 49(1948), 214-226. I T J Y. Misonou, On the d i r e c t prodtact of W -algebras,. To-hoku Math. J. 6(1954), 189-204. [8] C.C.'Moore, I n v a r i a n t measures on product spaces, P r o c ' . F i f t h B e r k e l y Sympos. Math. S t a t , and Prob. 1967. / [9] F. Murray and J. von-Neumann, On r i n g s of operators I , â€¢ A n n . of Math. 37(1936), 116-229. . [10] F. Murray and J. von-Neumann, On r i n g s of operators IV, . Ann. of Math. 44(1943),. 716-808. [11] M. Nakamura, On the d i r e c t product of f i n i t e f a c t o r s , Tohoku Math. J. 6(1954');, .205-207. N . â€¢ - â€¢ '. 9 6 f a : )[12] J . v o n : N e u m a n n , O n i n f i n i t e d i r e c t p r o d u c t s , C o m p o s i t o . \ / . ' " ' M a t h . 6(1938), 1-77. â€¢ .'. [13] R . T . P o w e r s , R e p r e s e n t a t i o n s o f u n i f o r m l y h y p e r f i n i t e . a l g e b r a s a n d t h e i r a s s o c i a t e d r i n g s , A n n . o f M a t h . 86(1967), 138-171. .'â€¢ [l4] N . S u z k i , A u t o m o r p h i s m s o f W - a l g e b r a s l e a v i n g t h e c e n t e r e l e m e n t w i s e i n v a r i a n t , T o h o k u M a t h . J . 7(1955), 186-191. [15] I . E . S e g a l , A n o n - c o m m u t a t i v e e x t e n s i o n o f a b s t r a c t i n t e g r a t i o n , A n n . o f M a t h . 57(1953), 4 0 1 - 4 5 7 . â€¢ '. [16] E . S t o r m e r , S y m m e t r i c s t a t e s o f i n f i n i t e t e n s o r p r o d u c t s V o f C - a l g e b r a s , J o u r n a l o f F u n c t i o n a l A n a l y s i s 3(1969), 4 8 - 6 8 . â€¢ â€¢ / [ 1 7 ] Z . T a k e d a , I n d u c t i v e l i m i t a n d i n f i n i t e d i r e c t p r o d u c t o f o p e r a t o r a l g e b r a s , T o h o k u M a t h . J . 7(1955) 67-86..''! [18] 0 . T a k e n o u c h i , . O n t y p e c l a s s i f i c a t i o n o f f a c t o r s c o n -s t r u c t e d a s i n f i n i t e t e n s o r p r o d u c t s , P u b l . R I M S , , ' . J K y o t o U n i v . S e r i e s A , 4(1968), 467-482. r l * [ 1 9 J M . T a k e s a k i , O n t h e d i r e c t p r o d u c t o f . W - f a c t o r s , T o h o k u M a t h . J . 10(1958), 116-119. [20] F . R i e s z a n d B . S z - N a g y / F u n c t i o n a l - A n a l y s i s , U n g a r , . : r : ' 'â€¢ y N e w Y o r k , 1955. ' . * V ' ^
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Semi-metrics on the normal states of a W*-algebra Promislow, S. David 1970
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Title | Semi-metrics on the normal states of a W*-algebra |
Creator |
Promislow, S. David |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | In this thesis we investigate certain semi-metrics defined on the normal states of a W* -algebra and their applications to infinite tensor products.
This extends the work of Bures, who defined a metric d on the set of normal states by taking d(Î¼,v) = inf { x-y } , where the infimum is taken over all vectors x and y which induce the states Î¼ and v respectively relative to any representation of the algebra as a von-Neumann algebra. He then made use of this metric in obtaining a classification of the various incomplete tensor products of a family of semi-finite W* -algebras, up to a natural type of equivalence known as product isomorphism. By removing the semi-finiteness restriction form Bures' "product formula", which relates the distance under d between two finite product states to the distances between their components, we obtain this tensor product classification for families of arbitrary W* -algebras. Moreover we extend the product formula to apply to the case of infinite product states. For any subgroup G of the *-automorphism group of a W*-algebra, we define the semi-metric d(G) on the set of normal states by: d(G) (Î¼,v) = inf {d(Î¼(Î±) ,v (Î²) : Î±,Î² Îµ G} ; where Î¼(Î±).a is defined by Î¼(Î±)(A) = Î¼(Î±(A)). We show the significance of d(G) in classifying incomplete tensor products up to weak product isomorphism, a natural weakening of the concept of product isomorphism. In the case of tensor products of semi-finite factors, we obtain explicit criteria for such a classification by calculating d(G)(Î¼, v) in terms of the Radon-Nikodym derivatives of the states. In the course of this calculation we introduce a concept of compatibility which yields some other results about d and d(G) . Two self-adjoint operators S and T are said to be compatible, if given any real numbers Î± and Î² , either E(Î±) â‰¤ F((Î²) or F(Î²) â‰¤ E(Î±) ; where {E(Î»)} , (F(Î»)} , are the spectral resolutions of S,T , respectively. We obtain some miscellaneous results concerning this concept. |
Subject |
Von Neumann algebras |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080513 |
URI | http://hdl.handle.net/2429/35276 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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