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Semi-metrics on the normal states of a W*-algebra Promislow, S. David 1970

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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 I N PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  the Department •v  of MATHEMATICS  We  accept  to  the r e q u i r e d  The  this  thesis  as c o n f o r m i n g  standard  U n i v e r s i t y of B r i t i s h . • M a r c h 1970  Columbia  In  presenting  this  an a d v a n c e d  degree  the  shall  I  Library  further  for  scholarly  by h i s of  agree  this  written  thesis  in p a r t i a l  fulfilment  of  at  University  of  Columbia,  the  make  tha  it  permission  available  representatives. thesis  for  It  is  financial  by  ATH  gain  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  WfrCH  EM  A T  Columbia  1910  the  shall  J^S  not  requirements  reference copying o f  I  agree  and  copying or  for  that  study.  this  thesis  Head o f my D e p a r t m e n t  understood that  permission.  M  for  for extensive  p u r p o s e s may be g r a n t e d  Department o f  Date  freely  British  the  or  publication  be a l l o w e d w i t h o u t  my  ii.  Supervisor:  D. C. Bures  <•  ABSTRACT  i  In t h i s t h e s i s we  investigate certain  d e f i n e d on the normal s t a t e s of a p l i c a t i o n s to i n f i n i t e t e n s o r  W - a l g e b r a and  on the set of normal s t a t e s ' b y t a k i n g  the s t a t e s  and  v  ap-  defined a metric  d(n,v) = i n f [||x-y||)  where the infimum i s taken over a l l v e c t o r s induce  their  products.  T h i s extends the work of Bures, who d  semi-metrics  x.  and  y  which  r e s p e c t i v e l y r e l a t i v e to  any  r e p r e s e n t a t i o n of the a l g e b r a .as a- von-Neumann a l g e b r a .  He  1  then made use  of t h i s m e t r i c  the v a r i o u s incomplete finite  i n obtaining a c l a s s i f i c a t i o n  tensor products  of a f a m i l y of semi-  W - a l g e b r a s , up to a n a t u r a l type  as product  of  of equivalence  known  isomorphism. By removing the s e m i - f i n i t e n e s s r e s t r i c t i o n form  Bures' "product d  formula",  which d e l a t e s the d i s t a n c e under  between.two f i n i t e product  t h e i r components, we  o b t a i n t h i s tensor product  f o r f a m i l i e s of a r b i t r a r y product  formula  a  subgroup  Moreover we  extend  W - a l g e b r a , we  i s d e f i n e d by  d (n,v) u (A) a  G  of the  the  states.  *  -automorphism group G d e f i n e the semi-metric d on the set of  *  normal s t a t e s by: V.  W -algebras.  classification  t o apply to the case of i n f i n i t e product  For any of a  s t a t e s t o the d i s t a n c e s between  = i n f [d(u ,v = .»j(a(A))  .  We  ) : a,(3 € Gl  ; where  show the s i g n i f i c a n c e  ,  iii. of  d  i n c l a s s i f y i n g incomplete  product  isomorphism,  product  tensor products  a n a t u r a l weakening of t h e concept o f  isomorphism. In t h e case o f t e n s o r p r o d u c t s  we o b t a i n e x p l i c i t  of s e m i - f i n i t e  factors,  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  d (g, v)  calculating of  up t o weak  i n terms o f t h e Radon-Nikodym  derivatives  the states. In the course  concept  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  of c o m p a t i b i l i t y which  y i e l d s some o t h e r  results  P  about  d  are s a i d £  and  .  Two s e l f - a d j o i n t o p e r a t o r s  t o be c o m p a t i b l e ,  , either  (F(\)}  d  E ( a ) <. F((3)  S  i f ' g i v e n a n y r e a l numbers or  F ( 0 ) _< E ( a )  , are the spectral resolutions  of  ,  T  a  and  (E(x)l  ; where S,T  and  ,  respectively.  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  H I S T O R I C A L NOTE ON I N F I N I T E TENSOR PRODUCTS CHAPTER 0.  PRELIMINARIES.. ON  CHAPTER I .  THE. METRIC  1. •2.  Basic properties  W*-ALGEBRAS  of  26 d  and  '  d  3.  D e f i n i t i o n of  4.  An  29 34  G  and . p  5. 6.  Monotone f u n c t i o n s Distribution functions  34  inequality  Compatibility  ...26  c  p  •7.  p  p  THE SEMI-METRICS d  4 11  d  The p r o d u c t f o r m u l a f o r  CHAPTER I I .  j  36 and  40 44  a  .'.  56 • v  8.  Calculation  CHAPTER..III. 9. 10.  of  p  f o rsemi-finite  factors  .'.68  APPLICATIONS TO I N F I N I T E TENSOR PRODUCTS ..80  Isomorphisms o f p r o d u c t s : D e f i n i t i o n s  80  M a i n ; r e s u l t s on s t e n s o r p r o d u c t s  83  BIBLIOGRAPHY  •  95  1  ACKNOWLEDGEMENT  I am g r a t e f u l t o Dr. Donald  C. Bures who suggested  the t o p i c o f t h i s t h e s i s , and who p r o v i d e d  valuable  and. encouragement throughout i t s p r e p a r a t i o n .  •::  s  l \  ^  assistance  INTRODUCTION  t  In [3] Bures d e f i n e d a m e t r i c a l l normal s t a t e s of a  W -algebra  G  d  on the s e t o f  by t a k i n g  d(ii,v)  = i n f (||x-yl|3 i where the infimum i s taken over a l l x  and  y  \A and  i n d u c i n g the s t a t e s  t i v e t o any r e p r e s e n t a t i o n of  0  v  vectors  respectively  rela-  as a von Neumann a l g e b r a .  In t h i s t h e s i s we .generalize t h i s d e f i n i t i o n by d e f i n i n g , f o r any subgroup a semi-metric a,fi e G}  d  G  of the group of -automorphisms , given by  G  , where  i n which case  d  G  a  .  a  a l l -automorphisms,  G  ,  P  In p a r t i c u l a r we are  i s e i t h e r the group of  o r the group o f inner i s denoted by  G  G  d (|i,v) = i n f ( d ( u , v ) :  V* (A) = ^i(a(A))  i n t e r e s t e d i n the case where  of  d  automorphisms, cf  or  respectively. G  We then i n v e s t i g a t e c e r t a i n aspects of  d  and  d  w i t h 'a  view towards a p p l i c a t i o n s t o i n f i n i t e tensor products. We f i r s t prove the "product formula", /  P(H  1  ® \x  2  , \> ® v ) = pC|i ,v ). p(|a ,v ) 1  f o r normal s t a t e s Gg  where  G^ 1  and  and  G  2  1  2  of  G-^  are a r b i t r a r y  1  2  and  2  and  W -algebras.  v  2  Here  2  p(y,v) =-^[d(^,v)]  .  (See Theorem 2.4 below). T h i s extends  the r e s u l t of [3] where t h i s f o r m u l a i s proved under the assumption t h a t  of .  G-,  and  G  9  are s e m i - f i n i t e .  As a major a p p l i c a t i o n o f 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 o f the incomplete t e n s o r p r o d u c t s of v o n ' Neumann [12] up t o product isomorphism  [4].  In f a c t i n the  n o t a t i o n of [4] we have t h a t  ® (G ,u..) i s product i s o iel morphic t o ® (C,,v.) i f and. o n l y i f E [ d ( u . , v . ) ] < » iei iel In the n o t a t i o n of [12] the e q u i v a l e n t statement i s t h a t 1  2  1  .  1  ® (G.,x ) i s product isomorphic t o ® (G.,,y.) i f . and o n l y iel iel • i f the s t a t e s and induced by the v e c t o r s ' x^ and y^ 1  1  1  r e s p e c t i v e l y s a t i s f y the above In c o n n e c t i o n  1  condition^  with t h i s r e s u l t we extend the pro-  duct formula t o apply t o the case o f i n f i n i t e product  states.  In Theorem 10.8 below we i n d i c a t e the s i g n 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 t e n s o r p r o d u c t s  up t o weak product isomorphism.  »  T h i s type o f isomorphism  (see D e f i n i t i o n 9.1 below) i s a n a t u r a l weakening of the concept of product isomorrlhism. of  For a countable f a m i l y  (G^)^ j e  W -algebras,  ® (G^,u.) and ® (G.,v.) are weakly p r o iel i€l , duct isomorphic i f and o n l y i f £ [ c f ( L i . , v . )] < » iel 1  1  1  1  2  1  In the case  where each  t o r we make t h i s a more e x p l i c i t  G^  1  i s a semi-finite fac-  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 d e r i v a t i v e s of the s t a t e s (see Theorem 8.6 below). In the course of 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 which y i e l d s some other r e s u l t s concerning d  .  v  that  f  can c a l l  and  g  f  and  Suppose t h a t  S  i s an a r b i t r a r y s e t and  are any r e a l v a l u e d f u n c t i o n s on g  compatible  i f f o r any p a i r  S  .  x,y e S  We ,  f (x) j< f ( y ) define  implies  g(x) _< g(y)  .  We extend t h i s i d e a t o  c o m p a t i b i l i t y between two s e l f - a d j o i n t o p e r a t o r s  a H i l b e r t space.  We then d e f i n e two s t a t e s on a s e m i - f i n i t e  f a c t o r t o be compatible i f t h e i r Radon-Nikodym are  compatible.  sense t h a t  compatible. d * d  d(|i,v) = d(u,v)  between two states, i n  whenever  u  and  v are  Moreover we show t h a t f o r t h i s c l a s s o f f a c t o r s  . Following  t h i s i n t r o d u c t i o n 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. the  derivatives  We show that f o r a f a c t o r of type I or 1 1 ^ . -  c o m p a t i b i l i t y minimizes the d i s t a n c e the  on  We i n d i c a t e there  i n more d e t a i l  Connection between some o f the problems considered i n  t h i s t h e s i s and the p r e v i o u s  work i n t h i s  subject.  In Chapter 0 we review some b a s i c r e s u l t s on von < Neumann and W - a l g e b r a s terminology. and  [9].  and i n t r o d u c e  The main r e f e r e n c e s  I t should  some n o t a t i o n and  f o r t h i s m a t e r i a l are [ 5 ]  be noted t h a t r e f e r e n c e  second e d i t i o n o f Dixmier's book.  [ 5 ] i s f o r the  HISTORICAL NOTE ON INFINITE TENSOR PRODUCTS  The  t h e o r y of t e n s o r p r o d u c t s of von Neumann a l g e -  bras was o r i g i n a t e d by von Neumann, f i r s t  f a m i l y o f a l g e b r a s i n [ 9 ] , and then f o r the case o f  finite  an i n f i n i t e  f a m i l y i n [12].  Consider f i r s t algebras  G-^  and  respectively. •and  H  f o r the case o f a  G  the case where we have von Neumann .  a c t i n g on H i l b e r t spaces  2  space i s known a s i t h e H i l b e r t H  and u s u a l l y denoted  2  form a where  -algebra A^ e G^  G  and  space t e n s o r product of by  H^ ® H  2  .  A  2  2  H^  e G  2  This H^ .  Then we can  , generated by elements  Q  H  by means of the i n n e r p r o d u c t s  on the r e s p e c t i v e ,spaces, t o another H i l b e r t space.  and  and  The u s u a l v e c t o r space t e n s o r product o f  can be completed,  2  H-^  A-^ ® A  2  , w i t h the u s u a l b i l i n e a r r e -  l a t i o n s , and w i t h m u l t i p l i c a t i o n and i n v o l u t i o n .defined by  •"  (A-j^ ® A ) (B 2  1  ® B ) = A ^ 2  (A-^ ® A ) * = A^ ® A 2  G ', i s then a G^  and  G  on  2  on  G H  2  .  and  G  2  .  , - a l g e b r a t e n s o r p r o d u c t of the The r e p r e s e n t a t i o n s o f  2  G-^  on  -algebras H-^ and  2  G  1 Q  , and the von\ Neumann a l g e b r a generated by t h i s  representation of G-^  2  ^Induce.in a n a t u r a l way a r e p r e s e n t a t i o n of  2  H.^ ® H  ; 2  ® A B  .•  G; 0  i s known as the t e n s o r p r o d u c t o f  Next, suppose we have an i n f i n i t e f a m i l y o f von Neumann a l g e b r a s  (^i)i j  a c t i n g on H i l b e r t spaces  e  (^i)i i €  In [12], von,Neumann extends the c o n s t r u c t i o n i n the f i n i t e case t o produce a H i l b e r t space known as the t o t a l product  o f the f a m i l y  (H. )  Again we can c o n s t r u c t of  the f a m i l y  (G )  G  and denoted by , a  Q  (H. )  -algebra tensor  , and r e p r e s e n t  i  ®  G  Q  on  tensor  product  ® (H.^) t o  o b t a i n a von Neumann a l g e b r a , known as the t o t a l t e n s o r product of the f a m i l y  (G^.)  .  T h i s t o t a l tensor product how-  ever f a i l s t o possess many of the p r o p e r t i e s which are enjoyed by the f i n i t e t e n s o r product.  For example: the t o t a l  t e n s o r product  o f ' f a c t o r s i s not i t s e l f a f a c t o r ; the t o t a l  tensor product  of a f a m i l y of f u l l  itself a full for  operator r i n g ; the u s u a l a s s o c i a t i v e t y laws  t e n s o r products f a i l t o h o l d ; the a l g e b r a does not a c t  on a separable I  operator r i n g s i s not  space e^en i i f each  H/  i s separable and  i s countable. A remedy f o r this' s i t u a t i o n i s p r o v i d e d by the  decomposition  of von Neumann.  He showed t h a t  (  ® (H^)  splits  up i n t o a d i r e c t sum of o r t h o g o n a l subspaces each of which is  i n v a r i a n t f o r the a l g e b r a  known as incomplete  G  Q  .  These subspaces a r e  t e n s o r ' p r o d u c t s f o r the f a m i l y  and the von, Neumann a l g e b r a s generated  by  of  these subspaces are known as incomplete  of  the f a m i l y  (G ) i  .  tensor  product.  Q  i  on each  tensor  I t i s the incomplete  ducts which appear t o be the proper finite  G  (H )  products  tensor pro-  g e n e r a l i z a t i o n o f the  6.  Each incomplete  space i s completely  choosing a f a m i l y of u n i t v e c t o r s If  (x^)  £ •iel  and  (y )  are two  i  | l - (x.|y.)| < » 1  space and  with  i  such f a m i l i e s  x  Consequently, i n the f i n i t e  incomplete  tensor product  incomplete given  case t h e r e i s  a l g e b r a which c o i n c i d e s  mentioned above.  In the  infinite  a b a s i c prob-  i s t o determine the r e l a t i o n s h i p s between them. Much of the t h e o r y can be  s i m p l i f i e d by removing  c o n s i d e r a t i o n of the u n d e r l y i n g H i l b e r t spaces. showed i n [7] bras  the r e s u l t i n g tensor product was  family.  One  could then' speak of a  f i n i t e f a m i l y of  independent, as  W -algebras.  Nakamuru i n [11]  and  gave s p a c e - f r e e d e f i n i t i o n s of the f i n i t e  product  i n c e r t a i n s p e c i a l cases. W - t e n s o r product  Takeda i n [17]  of an i n f i n i t e  bras by u s i n g i n d u c t i v e l i m i t s . d e f i n i t i o n f o r the  W - t e n s o r product  of  showed how  W - a l g e b r a s and  von Neumann's incomplete  tensor p r o d u c t s ,  of a Takesaki  tensor construc-  f a m i l y of  Bures i n [4]  an  given  W - t e n s o r product  i n [19]  a  Misonou  t h a t f o r a f i n i t e f a m i l y - o f von Neumann a l g e -  a l g e b r a , of the p a r t i c u l a r r e p r e s e n t a t i o n s of the  of  .  i  t h e r e f o r e the same a l g e b r a (More d e t a i l s are  case however t h e r e are many such a l g e b r a s , and  ted  € H  satisfying  they determine the same  w i t h the tensor product  lem  i  by  1  i n Chapter 0). o n l y one  (x )  determined  W -alge-  gave a g e n e r a l  of an a r b i t r a r y f a m i l y  t o o b t a i n the  tensor p r o d u c t s ,  W -analogue called  local  by u s i n g f a m i l i e s of s t a t e s i n p l a c e of  f a m i l i e s of v e c t o r s . Let  (G..).,  T  be a f a m i l y of  W -algebras.  For  every f a m i l y one  (^ )^ j i  6  where  i s a normal s t a t e on  o b t a i n s a l o c a l tensor product  denoted by  ® iel  (G. 1  )  of the f a m i l y  G  i  (G^) ,  A fundamental problem i s then t o  .  1  determine when two such f a m i l i e s of s t a t e s produce t e n s o r products which are e s s e n t i a l l y the same ( i . e . product phic).  isomor-  The s o l u t i o n , obtained p a r t i a l l y i n [3] and completed  i n t h i s t h e s i s , shows t h a t t h i s happens i f and o n l y i f the r e s p e c t i v e components o f the f a m i l i e s are s u f f i c i e n t l y " c l o s e t o each other".  The m e t r i c  d  i s used t o make t h i s i d e a  precise. When we a p p l y t h i s r e s u l t t o von Neumann a l g e b r a s we see t h a t an incomplete  tensor product does not r e a l l y de-  pend on the f a m i l y of v e c t o r s d e t e r m i n i n g  i t , but r a t h e r on  the f a m i l y o f s t a t e s induced by these v e c t o r s . von Neumann's c o n s i d e r a t i o n o f weak equivalence, 6.1.1), s i n c e the v e c t o r s  e  x. and  x  This explains ([12] def.  always induce the  "same s t a t e . The  c r i t e r i a which we- develop  f o r weak product  isomorphism can be i l l u s t r a t e d by l o o k i n g a t countable sor products o f type I f a c t o r s .  ten-  T h i s i s a s i t u a t i o n which  has been c o n s i d e r e d by many authors, [ 1 ] , [ 2 ] , [ 8 ] ,  [13],  [16] • and [18]. In p a r t i c u l a r we can extend in  the r e s u l t s d i s c u s s e d  ([12] Chapt. V I I ) concerning t e n s o r products  factors.  Let  G  be a f i x e d  d i m e n s i o n a l H i l b e r t space ,(x,y)  in H  . L e t iu  x  H and  I .  of l  2  f a c t o r r e p r e s e n t e d on a two  2  F i x any orthonormal  ur  basis  denote the s t a t e s on  G  .  8 induced by the v e c t o r s - i < a < 1 - let <-  each  x  and'  y  uu = auu c t x  +  r e s p e c t i v e l y , and f o r uu y  (1-a)  .  set of a l l sequences of. numbers b e l o n g i n g t o the [-£ i ]  , arid f o r  f = (a-^cxg. .. ) e F  tensor product  (G. ,to  ® 1=1,2,...  all  i  .  ) >  3"  f = (a ) i  and  Gi  ' where  = G  6 F  product isomorphic i f and p h i c , which occurs i f and  for  1  isomorphic t o  g =  the  denote the  F  i  a  be  interval  1^  Then every countable t e n s o r product of  t o r s i s weakly product If  1  let  P  Let  f o r some  , 3"  and  f  3"  fac-  f e F  are weakly  o n l y i f they are product only i f E [ (VaT i=l .  isomor+ Ul-a,  .- yr^) ] < « . • 2  Next, c o n s i d e r the more g e n e r a l case. G  that  i s a f a c t o r of type  I  , with trace  T  •,  n  i  normalized  , n < »  Suppose  so that, i t s v a l u e on a minimal  I t i s w e l l known t h a t any  state  u  projection = 1  G  on  i s g i v e n by  u.(A) = T(SAS) where St' is, a unique p o s i t i v e element of G . If 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 . .. > a are the e i g e n 0  values of of  T  S  , and  b  i 2 . ^2  ' *' — ^n  a  r  e  t  h  e  e i  Senvalues  ( i n both cases counted' w i t h r e s p e c t i v e m u t i p l i c i t i e s ) . Now suppose we have a f a m i l y ( G ) . _ where 1 1 JL y « • • « s  n  0  —  G  i  i s a f a c t o r of type ' I  set of a l l sequences operator i n  G^  ,  ('S-^Sg. . . )  1 < n  < «  i  where  such t h a t the sum  S  I  .  Let  X = the  is a positive  of the squares of i t s  e i g e n v a l u e s (counted w i t h r e s p e c t i v e m u t i p l i c i t i e s ) = 1 .. Then,'by the remark above t h e r e is_..a ^between f a m i l i e s o f s t a t e s  1-1  where  correspondence is a  9. . normal s t a t e on  G^  , and the elements of  an e q u i v a l e n c e r e l a t i o n  ~  on  X  X  .  by s e t t i n g  We  define  S = (S-^Sg...) •  09  ~ T = (T-,To...) i f and o n l y i f E [l-o(S.,T.] < » . Then . . i * l '•• i t t u r n s out t h a t the l o c a l tensor p r o d u c t s determined by 1  S  and  T  S ~ T  are weakly product  isomorphic i f and  tor  only i f  . Using t h i s approach  of  x  [2], [8], [16] . and  [18],  we  can rephrase the  results  c o n c e r n i n g the type of the  which a r i s e s from the t e n s o r p r o d u c t . o f f i n i t e type  facI  f a c t o r s , i n the f o l l o w i n g form. The  ~  equivalence c l a s s e s pf X  which produce a  s e m i - f i n i t e f a c t o r are p r e c i s e l y those which c o n t a i n a sequence c o n s i s t i n g of s c a l a r m u l t i p l i e s of p r o j e c t i o n s . unique  P  .  •  v  e q u i v a l e n c e c l a s s f o r which these p r o j e c t i o n s are a l l  e q u a l to the i d e n t i t y producesthe '  The  The  unique  hyperfinite  P ® I  factor  e q u i v a l e n c e c l a s s f o r which these p r o j e c -  t i o n s are a l l minimal produces the produce  11^  I  , a f a c t o r of type  factor.  A l l the r e s t  II  I t f o l l o w s then t h a t each of the  ~  equivalent  c l a s s e s which does not c o n t a i n a sequence of s c a l a r m u l t i p l e s of  p r o j e c t i o n s produces  a type  III  are weakly product isomorphic.  f a c t o r , and no two  However t h e r e may  t h a t are a l g e b r a i c a l l y isomorphic.  i s a much more d i f f i c u l t one.  3^,  below).  isomorphism  Powers i n [1J>] c o n s i d e r e d  e s s e n t i a l l y those t e n s o r p r o d u c t s of our n o t a t i o n above by  be some  ( D e f i n i t i o n 9.1  The problem of d e t e r m i n i n g c r i t e r i a f o r a l g e b r a i c  of these  where  f  I  0  f a c t o r s given i n  i s a constant sequence.  He showed t h a t f o r d i f f e r e n t  c h o i c e s o f the constant the t e n -  sor p r o d u c t s a r e a l l n o n - a l g e b r a i c a l l y isomorphic and . t h e r e b y produced III factors.  an uncountable  f a m i l y o f non-isomorphic  type  H i s r e s u l t s have been e x t e n s i v e l y g e n e r a l i z e d  by A r a k i and Woods i n [ l ] . I f we take i n f i n i t e gebras the problem measure spaces.  t e n s o r products o f A b e l i a n a l -  reduces t o the study o f i n f i n i t e  Suppose we have a f a m i l y  product  (G^) where each  i s an a b e l i a n von Neumann a l g e b r a and t h e r e f o r e can be c o n s i d e r e d as  L*(n^)  f o r a s u i t a b l e measure space  The t o t a l t e n s o r p r o d u c t o f t h i s f a m i l y i s j u s t fl  0 = (X,IB,H) i .  of subsets H(X  L (fi)  i s the i n f i n i t e p r o d u c t measure space of the  pose  a  (X )  n Xp) = 0  Then the s e t X  w  .  where Sup-  s p l i t s up i n t o a union  which are e s s e n t i a l l y d i s j o i n t ( i . e . f o r a ^ P)  to a product isomorphism  > and each of which  d  corresponds  c l a s s of incomplete t e n s o r p r o d u c t s .  In t h i s case, the c r i t e r i a f o r product isomorphism of the. m e t r i c  0^  was shown i n [33 t o y i e l d  theorem on i n f i n i t e product measures [63.  i n terms  the Kakutani  11.  ' •/".'''..'•  '.  '.CHAPTER 0  PRELIMINARIES ON •  W*-Algebras  •  von-Neumann Algebras.  Let w i t h norm,  H  be a H i l b e r t space over the complex numbers  || || , and i n n e r product,  t o the u s u a l norm t o p o l o g y on whereby a n e t o f v e c t o r s (x |y)  converges t o  (  x a  )  H  (|)  .  In a d d i t i o n  we have the weak topology, converges t o a v e c t o r  (x|y) f o r a l l y € H  .  This  x i f induces  j  Ct  on the s e t o f a l l bounded l i n e a r o p e r a t o r s on operator topology whereby a n e t o f o p e r a t o r s t o the operator  A  i f (A x) v  H  the weak  (A )  converges weakly t o  converges Ax  -:' f o r  a  all  x € H  ' An a l g e b r a  G  of bounded l i n e a r o p e r a t o r s on  i s c a l l e d a von > , Neumann a l g e b r a on ';.  (a)  The i d e n t i t y o p e r a t o r  i  (b)  G  (c)  G  A  e G  H  H i f 1 e G  i s self adjoint ( i . e .  i t s adjoint  ;•  A e G  implies that  .)  i s c l o s e d i n the weak o p e r a t o r  topology.  P a r t i c u l a r examples o f von Neumann a l g e b r a s a r e £(H) (j; (H)  , the s e t o f a l l bounded l i n e a r o p e r a t o r s on  H  and  , t h e ^ s e t o f a l l s c a l a r m u l t i p l e s of the i d e n t i t y  opera-  tor. v  F o r an a r b i t r a r y subset , §  of  x(H)  , the commu-  1  12. t a n t of  §  , denoted by  TS = ST  for a l l  S e S}  §' .  , i s defined  as  (T e £ ( H ) :  The fundamental double commutant  theorem o f von'Neumann s t a t e s t h a t a s e l f - a d j o i n t a l g e b r a G G  w i t h i d e n t i t y i s a von,'Neumann a l g e b r a =  G"  .  For [Sx] If  i f and o n l y i f  any subset  §  of  £(H)  and any  x e H  w i l l ' , denote the norm c l o s e d l i n e a r span of G  i s a von Neumann a l g e b r a  on  H  , then  ,'  (Sx : S e S} .  [G'X] € G " = G  .  (We w i l l f o l l o w throughout the convention employed i n the sentence above o f u s i n g the same symbol f o r a subspace and f o r the orthogonal  p r o j e c t i o n onto t h a t subspace whenever the  meaning i s c l e a r from the context.) The Lemma 0.1 .  f o l l o w i n g i s a w e l l known and important r e s u l t . Let  G  and  be a von Neumann a l g e b r a  H on  be H i l b e r t spaces and l e t H  a l g e b r a homomorphism of (F i n t o =  for a l l  (<j>(A))*  A e  i s a von;Neumann a l g e b r a Proof.  .  X(G)  , and  G  on  G  Suppose  <j> i s a  satisfying  d>( 1) = 1  .  G  1-1  (|>(A )  Then  (}>(G)  ..  See e.g. ([53 Chapt. 1 , § 4 ,  Cor. 2 ) .  /  W  -algebras By a  bra  G  over  - a l g e b r a we mean a l i n e a r a s s o c i a t i v e a l g e with i d e n t i t y  (always denoted by 1 ) ,  possesses an i n v o l u t i o n ; t h a t i s a,map from s e n d i n g ;'-A.7 i n t o /A  G  , which s a t i s f i e s f o r a l l  which  to i t s e l f , A,B € G  and  »  a.  (XA + B)  •  = XA  denotes complex  conjugation).  5  b.  A  <= A  c.  (AB)  ,  = B A  By a homomorphism from are  , (X  + B  G  to  ©  where . G  and B  - a l g e b r a s we w i l l always mean an a l g e b r a homomorphism  satisfying  •X-  a  #  a ( l ) = 1 , and a(A ) = (a(A)) fora l l A e G . * * A -algebra G i s c a l l e d a W - a l g e b r a i f f o r some  H i l b e r t space i n t o the  H  there i s a  -algebra  a l g e b r a on  H  .  £(H)  1-1  homomorphism  so t h a t  4>(G)  G.  i s a von Neumann  By a r e p r e s e n t a t i o n o f a  we w i l l always mean a homomorphism  <j> from  W -algebra  G  d) s a t i s f y i n g the above  conditions. Consider  now a f i x e d  W -algebra  G  * If selves bra. the ;  , •  H and  ^ are sub - a l g e b r a s  W - a l g e b r a s , Lemma 0.1 shows t h a t 1  Consequently,for  any subset  W - a l g e b r a generated by s  An element  a.  self-adjoint  A e G  of  which a r e them-  B fl 3 G  isa  ,  i s s e l f a d j o i n t and  2  di -• a p a r t i a l isometry tiori ( n e c c e s s a r i l y j e c t i o n )i  f o r some  i f A,A AA  W -alge-  we may speak o f  i s called •  i f A =/A*  b.' a p r o j e c t i o n i f A '' A = A 3  §  G  S ..  • ,:c/ p o s i t i v e i f A = B B .  of  B € G  ,  i s a projec-  i s a l s o a pro-  e.  a u n i t a r y i f A A = AA Two p r o j e c t i o n s  E  ;  =1  and  F -  .j  are s a i d t o  ••  •  •'(  ...  •  '•'  '  '  (  be o r t h o g o n a l i f EP ? 0 I t i s e a s i l y v e r i f i e d t h a t any r e p r e s e n t a t i o n o f G  takes elements of the v a r i o u s types mentioned i n t o opera•  i  t o r s of the same type i n the u s u a l sense. We w i l l denote the s e t o f a l l p o s i t i v e elements o f G. by  G  .We  +  w i l l denote by  _<  the u s u a l p a r t i a l order  d e f i n e d on the s e t o f s e l f a d j o i n t elements o f A _< B  if B - A € G . Any-element + square r o o t € G which we denote by +  A  h A  of  G  G  +  whereby has a xonique  E  From the w e l l known p o l a r decomposition theorem any U  A e G  can b e ; w r i t t e n as  i s a p a r t i a l isometry  e G  We w i l l denote by morphisms of  G  U|A|  satisfying Aut(G)  and by *' I n t ( G )  |A|e G  AU U = A  ' and  +  .  > the group of a l l auto-  U € G  satisfying  a(A)  for a l l A e G Any r e p r e s e n t a t i o n o f  G  i n d u c e s a norm on  G  (the u s u a l operator norm) which makes i t i n t o a Banach bra.  ; , -  alge-  I t i s easy t o v e r i f y t h a t t h i s norm i s independent o f  the p a r t i c u l a r r e p r e s e n t a t i o n and we may speak u n i q u e l y o f the norm of an element  A  which we denote by ||A||  5  For any p r o j e c t i o n A = EAE]  . ^Then  G-c  E e G  we l e t Gg = (A e G :  i s - a *-algebra with i d e n t i t y  E  .  W - a l g e b r a s i n c e any r e p r e s e n t a t i o n  <J>  * Moreover of  G  t  the subgroup o f a l l a e Aut(G)  f o r which there e x i s t s some u n i t a r y = UAU*  , where  Gg  is a  on a H i l b e r t space  H  induces a r e p r e s e n t a t i o n o f  15. G  E  as a von-Neumann a l g e b r a  on,the subspace  ([.5]  <f)(E)  Chapt. 1, §2, Prop. 1).  States  and T r a c e s .  A linear functional i f | j ( A ) J> 0  called a state  n(A)  It follows that Moreover  y  n  bounded  is  u(l) = 1  and  +  i s r e a l f o r any s e l f a d j o i n t (considering  A  .  G  as a  i s s a i d t o be f a i t h f u l i f M(A)' > 0  A € G  f o r a l l non-zero  G  ||u|l = 1  with  A state  W -algebra  for a l l A e G  |i i s a u t o m a t i c a l l y  Banach space)  on a  .  +  (A  any upwardly d i r e c t e d s e t  Q  ) o f s e l f a d j o i n t elements  , ( i ( A ) = sup{|j(A )3  A  w i t h l e a s t upper bound  I t i s s a i d t o be normal i f f o r  a  We w i l l d e a l e x c l u s i v e l y w i t h normal s t a t e s and w i l l denote by  Eg  the s e t orf a l l normal s t a t e s of  A t r a c e on closed  interval "(a)  G  i s a mapping  from  G  +  t o the  [0,«] s a t i s f y i n g  T(kA + B) = k T ( A ) and  A,B e G  T(UAU*)  (b)  T  G  and a l l  +  € G  T(B)  /(we d e f i n e  = T(A) A  +  , f o r a l l k _> 0 0-<»> = 0  , for a l l unitary  ). U € G  +  We d e f i n e a t r a c e t o be f a i t h f u l or normal i n an analogous way t o the d e f i n i t i o n f o r s t a t e s . We^say t h a t a t r a c e for a l l A e G there  exists  +  T  is finite  ;  if  T(A) < »  , s e m i - f i n i t e i f f o r every non-zero  T _< S  such t h a t  (o < T(T) < «  .  S e G  +  .  16. We  say t h a t the  i f f o r e v e r y non-zero f i n i t e ) trace  T  D e f i n i t i o n 0.2 For any  algebra  T € G  +  be a l i n e a r f u n c t i o n a l on  we  define  define  .  =1  (i 6 E  G  Then\ \x  a  , for a l l  a functional  = n(TAT*)  the  then  T  any  T  and  T  q  JQ  T e G  with  (T)  s  T  t  G  if  the  The  define G+  /from  S e G  i s f i n i t e valued  S = 1 /.  we  +  to  i n an  analogous  [0,»]  „  i s a l s o a normal t r a c e f o r a l l  i s normal and T  on  G  and  i s such t h a t may  be  , which we  In p a r t i c u l a r f o r a f i n i t e - t r a c e  has  y  a  and  A  t o a normal s t a t e of  take  A e G  and  trace., T  mappings  i s normal then If  by  , and  T  by  a  Obvious c a l c u l a t i o n s . For  way  |a^,  , a e Aut(G) |i €  G : .  A e G  , for a l l  1  Proof.  \x  a functional  (semi-  .  H  T  U (TT*)  0 < T(T)  Let  U (A)  Let  i s a normal f i n i t e  such t h a t  we  Lemma 0.5  there  (semi-flnlte) .  G  Li°(A) = u ( a ( A ) )  T e G  Is f i n i t e  on  o e Aut(G)  For any  G  T  .) =  linearity  T(1)  = 1  we  r e s u l t i n g s t a t e (denoted simply by  property.that  1  w i l l denote a l s o by with  T  a € Aut(G)  T(SS  extended by  If  T  C  can T  )  17.  T(AB)  = T(BA)  ,for a l l  .'.'\-V  A,B e G .  \ \  For any s t a t e  (0.1)  y> on G  |M(AB) I < M ( A A* ")^  |i(B*B)*  we have the i n e q u a l i t y  , fora l l A , B e G i  ,  which we o b t a i n by a p p l y i n g the Cauchy-Schwartz i n e q u a l i t y t o the p o s i t i v e conjugate b i l i n e a r form,  ( A , B ) - u ( BA )  We use t h i s t o prove some f a c t s about a f i n i t e i  Lemma 0.4. T  .  Let G  L e t A e. G  be a  *  W -algebra with a f i n i t e  and l e t E  T(|A|)  (a)  = max  (b)  |T(EA)|  trace  be a p r o j e c t i o n i n G  (|T(AV)|  e GI such t h a t ,._  trace.  : V a partial  A W * = A]  < |lA|| T ( E )  .  Then  isometry  .  .  Proof. (a)  L e t A = U|A|  be t h e p o l a r decomposition o f  T  . A  Then f o r any V  s a t i s f y i n g the s t a t e d  c o n d i t i o n s we o b t a i n from (0.1)  |T(AV)| = |T(U|A|* \A\*V[ < [T(U|A|U*)3* [T(|A|U*U)]* [T(|-A|W*)]* = T|A| Then t a k i n g  V = U  N  '  [T(V*|A|V)]* .  • .  we o b t a i n  |T(AU )| -..T(U|A|U ) = T(|A|)  (b). U s i n g (0.1) :. we have  and the f a c t t h a t  EA*AE _< HA*A|| E  'V:  ...  '  |T(E A) I  |T(EA)|  =  .  < [ T ( E A * A E ) ] * [T(E.)]*  '  2  I  |T(EAE)  =  .  < ||A*A||* [ T ( E ) ] * [ T ( E ) ] * = ||A||T(E)  We next c o n s i d e r and  vectors.  on a H i l b e r t U  relative  \x e T,^  If space  D e f i n i t i o n 0.5. T,  ments of  .  n  and  H, we  cj>  to  If  d)  i s a representation  say t h a t a v e c t o r  u ( A ) = (<J)(A)x|x)  Let We  r e l a t i o n s h i p s between s t a t e s  (Hj^iei  b  e  a n  x e H  , for a l l  of  G  induces A e G  .  y nonvoid f a m i l y o f e l e -  define i  (x ) i  i € ] [  t a t i o n of  ]  : d)  i s a represen- .  G ,on a H i l b e r t  H  and  H  such t h a t  (x^) "' i s a f a m i l y of v e c t o r s i n  t£ve._tb  x^  induces  (J)--for a l l  Lemma 0.6. .  /  Let  \A c T,^  G  Suppose  and  o n l y i f y = u'x  (d)(G))'  such t h a t  Proof.  Suppose  A e  G  .  •  and l e t <j> ..  be any  Then  '  -  \  Th. 1,  representation  [<f>,y] e Q(u) u'  if  in  .  y = U'x - f o r such a :  i n [5] o f  f o r some p a r t i a l isometry U'*U'x = x  ,  .  [d),x] e Q(u)  of  rela-  i e 1}  T h i s s e t i s never empty (see e.g. the proof Chapt. 1, § 4 ) .  space  U'  .  Then if or any  "  (<l>(A)y|y) = (<|>(A)U x|U'x) -.'(•.(.A)U**u'x|x) -. (*(A)x|x) = u(A) ;  Therefore  [<j>,y] e Q(u) . Conversely, i f both  it  x  and  y  induce  u  i s a w e l l known r e s u l t t h a t we can d e f i n e an isometry  from the subspace  [<}>(G)x]  U { ( < K A ) X ) = <f>(\A)'y  and e x t e n d i n g t o the c l o s u r e  Chapt. 1, §4, Lemma 3).  t o the subspace  Extending  on the whole space by s e t t i n g complement of  [<j)(G.)x]  [<}>(G)y]  by t a k i n g  (see [5]j  .U-£ t o an operator  U' = 0  U'  on the orthogonal  g i v e s us the d e s i r e d p a r t i a l  isome-  try. If the By  M € E  the support o f  G  smallest p r o j e c t i o n the n o r m a l i t y  of  E  u  of  G  support o f  L e t |U> e £ \x = 4>  .  G  f o r which  t o be  u ( l - E) = ;0- . The  (see [5] p. 58).  F o r any  [(<j)(G))'x]  -1  i s defined  the support always e x i s t s .  f o l l o w i n g i s a well! known r e s u l t Lemma 0.7.  ^  [<l>,x] e  .  Q(^)  , the  .,  v  Factors  AW (i.e.  ;  -algebra  {T e G : TA = AT  G  i s c a l l e d a f a c t o r i f i t s center  fora l l  A e G}  ) consists only of ,  s c a l a r m u l t i p l e s of the i d e n t i t y . A factor  G  can be e f f e c t i v e l y analyzed  at i t s s e t o f p r o j e c t i o n s .  Two p r o j e c t i o n s  s a i d t o be e q u i v a l e n t , w r i t t e n  E ~,F  isometry  F = UU •.  if  U e^G i E = U U - and  E ~ G  f o r some  be made f o r any property E ^  F  G < F  W -algebra,  .  F ^ E  .  and  F  are  , i f f o r some p a r t i a l We say t h a t  E ^ F  While these d e f i n i t i o n s can  f a c t o r s have the important  t h a t f o r any two p r o j e c t i o n s  or  E  by l o o k i n g  A projection  E  E  and  F  either  i s s a i d t o be f i n i t e  if  E ~ F  and  F _< E  which i s not f i n i t e A factor  imply that  F = E  .  A projection  i s s a i d t o be i n f i n i t e . i s s a i d t o be o f type I i f i t c o n t a i n s a  minimal non-zero p r o j e c t i o n .  Any such f a c t o r  i s isomorphic  t o the a l g e b r a of a l l bounded l i n e a r o p e r a t o r s on some H i l 1  l  bert  space and we then say i t i s o f type I  where  n  i s the  dimension of the space. A f a c t o r w i t h no minimal p r o j e c t i o n  i s s a i d t o be  of type; H-^ •» i f the i d e n t i t y i s f i n i t e ,  a. 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 p r o j e c t i o n c.  is finite,  I l l !, i f a l l non-zero p r o j e c t i o n s  F a c t o r s of type  I ' , I I , and  II  1  In f a c t they each have a s e m i - f i n i t e up t o a p o s i t i v e faithful.  »  If G  , then  G  E  t r a c e , which i s unique  on a type I I I f a c t o r  i s a projection i s a semi-finite  and o n l y i f E  i s t h a t which  elements.  i n a semi-finite factor.  G  E  factor  is finite i f  i s a f i n i t e ' . p r o j e c t i o n and t h i s occurs i f  and o n l y i f the t r a c e the f i n i t e f a c t o r s n a l , and o f type A factor  X  CO  on a l l non-zero p o s i t i v e  E  are s e m i - f i n i t e .  s c a l a r m u l t i p l e , ' and which i s normal and  The o n l y t r a c e  takes value  are i n f i n i t e .  of  E  i s finite.  are those of type II  1  I  I t follows , n  that  a finite  cardi-  .  of type  I I , i s s a i d t o be h y p e r f i n i t e 1  i f ; i t i s generated by an i n c r e a s i n g sequence of subalgebras ( i^i=l 2 • where G^ i s a f a c t o r of type I . Any G  i <  .21. two h y p e r f i n i t e  11^  f a c t o r s are isomorphic./  §7, Th. 2 ) .  ( [ 5 ] , Chapt. 3,  • '  ; \  Tensor Products o f von Neumann a l g e b r a s  Let (G ) i  I  be an a r b i t r a r y i n d e x i n g s e t and l e t  be a f a m i l y o f von Neumann a l g e b r a s a c t i n g on the 1  i e I  H i l b e r t spaces  ( i)i j H  €  A f a m i l y of v e c t o r s  (x^  x  with  e H  ±  i s called  i  a  C -sequence i f ' £ |l-||x. ||| < » . An e q u i v a l e n c e r e l a t i o n ° iel i s d e f i n e d on the s e t of a l l C -sequences by t a k i n g ( x ^ 1  0  equivalent to  (y.) i f f £ | l - ( x . | y . ) | < » . : ' iel (x^) be a C -sequence and l e t r 1  Let  1  0  e q u i v a l e n c e c l a s s . \ For each  (y.) e r  we l e t  ® (y.) iel t o $ /defined  1  1  denote the m u l t i - a n t i l i n e a r mapping'from v  be i t s  T  v  oy  ®  (y )[(z )] = T T ( y | z ) „ 1  1  i  "  i e l •-.  1  i e l  These mappings generate  , fora l l  a l i n e a r subset  H  o f a l l m u l t i - a n t i l i n e a r f u n c t i o n a l s on  r  (z )  e r  ±  .  o f the space  Q  , and  H  be-  o (®(y.» ) |&>(z. )) = | 1 (y.. |z.\,) i e l and e x t e n d i n g t h i s i n n e r product t o a l l of H by l i n e a r i t y . comes a p r e - H i l b e r t space by d e f i n i n g  1  1  1  Q  The completion o f  H  Q  i s known as the t e n s o r product of the  family  ( H ^ ) ^ w i t h r e s p e c t t o the  denoted  by  «  (H., ,x. ) ••. -A"  C -sequence 0  ^"'"'-'v  (x.^) and  1  22. An orthonormal b a s i s f o r chosen i n the f o l l o w i n g manner. ||x^|| = 1  assume t h a t  ® i€l  First  for a l l i  (H.,x.)  can be  note t h a t we  can  s i n c e r e p l a c i n g each  . x^ by .'x^/llx^||' y i e l d s an e q u i v a l e n t C -sequence. For' 'each i e l , l e t ( i ) j j ( i ) orthonormal b a s i s Q  x  t  e  a  n  €  for  x^  which i n c l u d e s  .  Then the set of a l l elements  of the form ® (y.) where f o r each i € I y. = x^ f o r • iel some j € J ( i ) , and f o r a l l but a f i n i t e number of i € I 1  y^  1  i n p a r t i c u l a r = x^  1  , c o n s t i t u t e s an orthonormal b a s i s  ® (H. , x. ) . T h i s shows t h a t ® (H. ,'x. ) i s seiel i€l p a r a b l e i f each H i s separable and I i s countable. for  1  1  1  1  i  ,  , Now,  operator  A^  A  fc  \ ( ® ( y  ±  for, any on  ) )  k e I  • i .  ®  (y[)  € G  k  k  we d e f i n e the  by t a k i n g *  (U^,X^)  =  and . A  w  n  e  r  "^1  e  = ^1  l f  1  +  k  i s then extended t o the whole space by l i n e a r i t y  and  continuity.  ' Let generated by  G  be the von Neumann a l g e b r a on  {A^  : i e I  product of the f a m i l y (x.)  , denoted by  and  , (G^)  G  1  into  G  i  Then the tensor  , Is  C -sequence Q  d e f i n e d t o be the von-  1  G. . , together  given by  .  w i t h r e s p e c t t o the  <59 ' (C,*x.) iel 1  Neumann a l g e b r a  A^ e G l  ® (H^,x^)  A. .-• X  w i t h the i n j e c t i o n s o f each . . ' . :;  .:\/ f . ' . . • .  23. Note t h a t i f (x^) and  (y^) are e q u i v a l e n t  :  C -sequences i t f o l l o w s immediately from the d e f i n i t i o n s t h a t 0  (H ,x ) = <S5 ( H , y ) and i  jL  1  ® (G ,x ) = ® (G^y.^  1  1  .  i  Complete d e t a i l s and p r o o f s o f t h i s m a t e r i a l can he found i n [12] or [ 2 ] .  Products and Tensor Products of  Let t. ( G ) i  l €  j  I  -algebras  be an a r b i t r a r y i n d e x i n g  be a f a m i l y o f  :  W  s e t and l e t ,  W -algebras.  A product f o r t h i s f a m i l y i s an o b j e c t  (ct^)^ j) €  * where 1 •'•  \ 1-1  G  isa  W,-algebra, and f o r each  homomorphism from  G.»  1  . "'  (a) a ( G ) i  v;i" .'  and  1  i + j  into aj(Gj)  e  l  isa  satisfying commute p o i n t w i s e f o r  •  {a^G^t': i e l ]  (b)  G  i  generates  G  as a  W -al-  gebra. .:.  We say t h a t the p r o d u c t s  (G, ( a ^ ) )  and  (ft, (3^))  si are product isomorphic i f -there exists-an isomorphism ,G ;  onto  B  such t h a t  We l e t A = A ( ( G ) ) i  where 7  (f>a^ =/P^  e £  n  u  e  l  l  .  G  •ii ( F f . . o ( A ) . ) ^ T T M A . ) 1  1  e  i s a product s t a t e o f the product  i s a normal, s t a t e o f  ieF  i  denote the s e t of a l l f a m i l i e s  fora l l i  jWe say t h a t (G, (a., )) *•> i f  fora l l  <j> from  1  1  ieF  . •  such  ,  th&i f o r some  1  24. •for a l l f i n i t e  subsets  F  of  I  and  all  A^ e G^  •  We  will  ® (u. ) . I t f o l l o w s from the norma- \ iel l i t y of u that i f ® ((i, ) e x i s t s i t i s unique. iel D e f i n i t i o n 0.8. For any e A we d e f i n e a p r o d u c t , denote such a  |i  by  [c|>.,x. ]  ® (G.,u.) , as f o l l o w s . For each i e l choose iel i € Q(u.) . Then ® (G.,u ) i s d e f i n e d to be the .; i e l ® (<j). (G. ),x.,)-' w i t h i n j e c t i o n s a. g i v e n by iel 1  1  1  1  1  1  1  x  = T]TA7T f o r a l l It  1  algebra  i s shown i n [3] or [4] t h a t  p r o d u c t isomorphism.  A eG . ±  ±  (G.,u )  ® iel  i s unique up  1  In [4] a p u r e l y a l g e b r a i c d e f i n i t i o n of  I  i s given.  '  to  ® (G.,u.) iel 1  * I t i s easy t o see  1  t h a t the product s t a t e  ® (y.) iel c o n s t r u c t i o n above 1  e x i s t s on  ® (G.,,u.,) In f a c t i n the iel i s induced by the v e c t o r ® ' (x.) • iel 1  it •  1  1  Following products,  the  constructed  usage i n [ 4 ] we  w i l l r e f e r t o these  from' an element of  sor products.  A  , as l o c a l  /  Suppose t h a t the ease i n n o t a t i o n l e t  indexing  I = {1,2}  set  .  I  is finite.  speak of a unique tensor product (G^Gg)  . ^ I t i s u s u a l to denote  element  a-j^A-^) a ( A )  (y^,^ )  of  ^1  ® 2 w  2  '  A  2  by'' -A^ ® A  (G^a-^ag)) G 2  by .  G^  we  °f the & G  2  and  Every element  determines a p r o d u c t s t a t e , denoted  ••  For  Then a l l l o c a l tensor  c u c t s of a given f a m i l y are product isomorphic and  2  ten-  by  pro-  may family the  25. '  Lemma 0.9. and  Let  v e r  fi  .  G  a  and  be  W - a l g e b r a s , and l e t u e £  Then f o r a l l S e G  +  and  T € i B with +  u(SS*) - v(TT*) - 1  * )(S*T) V  Proof.  Choose any  A e G  H  =  and  v  T  B e B ..  (U ® v) (S®T) [A » B] = (u ® v ) [ ( S ® T)  Then  (A ® B) (S ® T)*]  = (w « v)[(SAS*) £ (TAT*)] = u(SAS*) v(TAT*) = U ( A ) v ( B ) s  U  g  ® v  T  T  '^  [A ® B]  The lemma.now f o l l o w s from the uniqueness p r o p e r t y of pro-  •t>  duct  states.  (  26.  CHAPTER I  THE METRIC  1.  Basic properties of  d  The q u a n t i t i e s  and  d  d  p  and p were Introduced i n [ 3 ] .  In t h i s s e c t i o n we review some b a s i c p r o p e r t i e s .  * D e f i n i t i o n 1.1. - Let v € t~ . We d e f i n e . G  G  be a  W -algebra  and l e t u  and  V  'd(u,v) = i n f {||x-y|i : [4>,(x,y)] € Q(n,v)}  , '  P(u,v) = sup {|(x|y)| : [(j>,(x,y)] e Q(u,v)} . • i  I t i s proved i n [-3]' t h a t and  that  (1.1)  d  and  •  The  p  are c o n n e c t e d j by the f o r m u l a  [d(u,v)]  number  d . i s a m e t r i c on  2  =  d(u,v)  '2[1-P(LI,V)]-  :  .  can v a r y from  0  to  J2  . To'  g a i n some i n s i g h t i n t o the d e f i n i t i o n we can c o n s i d e r the extreme cases.  •.'••!  Lemma 1.2.  ' ;  (a)  d(|i,v) = 0 ' i f and o n l y i f u = v  (b)  d(n,v) «  .  \  i f and o n l y i f the support o f u.  and the support o f  v  are orthogonal.  Proof.  v  (a)  T h i s f o l l o w s from t h e f a c t t h a t  (b)  L e t E = the support o f u  and s u p p o s e t h a t  FE = 0  .  ||o>(E)x|| = n ( l ) = ||x||  Since  2  and s i m i l a r l y  2  <|>(F)y = y  .  d  i s a metric.  , F = the support o f  Choose any  [<t>,(x,yy] e QXu,v)  , we have t h a t  <|>(E)x = x  Then  1 (x|y) == j (<j>(E)x|o>(F)y) = (o)(FE)x|y) = 0.'  p(u,v) = 0 .  Therefore  = 72*  and from (1.1) we have t h a t  .  v Conversely suppose t h a t  p(u,v) = 0 unitary  .  U'  e Q(u,v)  d(u,v)  d(ui,v) =  Choose any ' [<|>, (x,y) ]  e  in  .  F o r any [$£\j'x y)] 3  0\</| (U'x|y) | _< p(u,v) = 0  (A'x|y) = 0  have then t h a t  Then  ' Lemma 0 . 6 shows t h a t  (<|>(G))'  and t h e r e f o r e  Q(u>v)  .  for a l l  .  We  A' e (<J>(G))"' by  w r i t i n g each such element as a l i n e a r combination o f u n i t a r i e s . By Lemma 0 . 7 , space  N  that  i s o r t h o g o n a l t o a dense subset o f the sub<|>(E)y = 0 j .  4>(E) and t h e r e f o r e  A'd)(E)y = 0 again  y  , for all"  A ' e (d)(G))'  d)(E) <j>(F) =.0 ; .  Since  .'•  •  EF = 0  . ,'  Then,  <j>(E)A'y  , and by Lemma 0 . 7  <j) i s a . ; i - l v  mapping we have  28. Lemma 1.3  Let G  be a  W - a l g e b r a and l e t u*v and  Then 0 <_ a <_ 1  , d(u.', ( l - a ) u + av j, _<-'VSa  (a)  F o r any  (b)  |p(u),u) - p(uu,v)| < J2  (a)  Let u'  d(u,v)  Proof.  t<j>*(x,y)] e  Choose any 'that Here  denote the s t a t e  Q(M>,V)  .  ( l - a ) ( i + av  A direct calculation  [$ © (j), (-/1-a x e ,/a x , ./I-a x e <J) © (j)  with i t s e l f : space o f  y ) ] e Q(u,u')  shows .  denotes the d i r e c t sum r e p r e s e n t a t i o n o f <J> on the space  (J>  .  H• © H  where  H  i s the u n d e r l y i n g  We have then  d(w,M )  WJT^a x © Ja x - " y i = a x © Ja y||  ;  \  = Va ||0 © (x-y)|| ± J2a (b) lity, for  d  Using  (1.1) and the backwards t r i a n g l e Inequa-  we o b t a i n ,  -  |p(«J,u) - p(iu,v)| = ^|[d(u),u)] / =  2  2 - [d(uu,v)] |  i(d(«J*u) + d(uj,v)) |d(«j,u) - d(uj,v) |  < VS d(u,v)  :2.  The product formula f o r  p  In t h i s s e c t i o n we w i l l prove duct formula f o r  p  i n g e n e r a l the p r o -  which was obtained i n ( [ 3 ] , Theorem 2. 5),  :  for s e m i - f i n i t e algebras.  We do t h i s by means of Lemma 2.1  which i s s i m i l a r i n statement  and p r o o f t o ( [ 3 ] , Lemma 1.6).  .However by d e a l i n g w i t h o n l y one element of  G  we a r e a b l e t o  a v o i d the use o f a t r a c e . Lemma 2.1. be such t h a t  Let G  *  be a  2 LA(T ) '= 1  W -algebra, .  LA e  and l e t  T e G  Then,  P(U,U ) T  = H(T)  I i  Proof.  [<|>,(x,y)] e Q ( u , u )  Choose any  .  T  A direct  [(f), (x,d)(T)x) ] e Q(u,(j^,)  c u l a t i o n shows t h a t  cal-  ., Therefore,,  v  (2.1)  P(H,H ) T  > |(x|4.(T)x| = u ( T )  On the other hand, s i n c e state r e l a t i v e t o  d) ,  y  and  <})(T)x  .  induce the same  Lemma 0.6 shows t h a t  f o r some p a r t i a l isometry  U'  in  (4>( ))' G  .  y = U <J>(T)x /  Therefore  |(.x|y)| = |(x|u'o> (T)x| - .|(U' + ( T ) * x K ( T ) * x ) | ;  •I'  < |i<J>(T)*x|| - l i ( T ) 2  T a k i n g the supremum over a l l - [<j>, (x,y) ]  Q(LI,V)  we o b t a i n  4  30. p(|iiM )  < H(T) .  T  T h i s f o r m u l a and (2.1) complete the p r o o f .  In the remainder o f t h i s s e c t i o n we c o n s i d e r two W -algebras  G^  and  G  and t h e i r tensor product  2  We a l s o c o n s i d e r s t a t e s _ v  2  € E  for  p  .  G  €  u  and  2  and  2  We want t o prove the f o l l o w i n g product  formula  .  (2.2)  Pd^ ® u  Lemma 2.2. f o r some Proof.  and  G^ ® G  2  !, v  2  L e t j = 1 or 2  T\j € G j  .  pfw-^v^-pfn^Vg)  ® v ) =  x  .  Suppose t h a t  Uj = ( v j ) 0 T  Then (2.2) h o l d s .  From Lemma 6.9, U  x  ® W  = ( ^ 1 ^ ® ^ 2^T V  2  = 2  (  v  l ®-^T^-S'Tg.  and by s u c e s s i v e a p p l i c a t i o n s of. Lemma 2.1  P(U  1  ® ( J , v -.® v ) = v 2  1  2  a  ® v  ® T )  2  2  =,v (T ) v ( T ) = p(u ,v ) p ( u , v ) 1  Lemma 2.5.  F o r any  6  such t h a t  v j = ( 1 - 6 ) v j + , Wj :'V where 6  d(v^ ® v  ..  ....  2  1  2  2  1  0 <_ 6 £ 1 l e t  j = 1 or 2  .  ,.-..vj ® V ) <, 2756 2  Then  1  2  2  .  Proof.  By the t r i a n g l e  d(v  1  ® v  2  inequality,  , • v£ ® > V g )  dCvjL  _<  ®  Vg  ,  v  1  ®v ) 2  - + d'(v-j_ ® v£ * v-[ V g )  From Lemma 1.3 ( a ) ,  . . d(v  1  ® Vg  ,  ' • ' ..  s>  1  ®  Vg)  = d ( v ® vg , [ (1-6 )v1 x  Vg  ^ ( V ^ ® Ivg » "v-{ ® V g ) <  Similarly, the  ®  + 6(  v  ±  e* Ug) ])  , which completes  proof.  - We now remove the r e s t r i c t i o n s o f Lemma 2 . 3 Theorem 2 . 4 . Proof. Let n  The product f o r m u l a ( 2 . 2 ) holds i n g e n e r a l .  ^  Given any  «' such t h a t  .3 where  V j = (l-6)vj + Mj  > 7j  6  j = 1 or 2  » U j ( A ) < 'n V j (A) ; f o r a l l  Radon-Nikodym Theorem . ( [ 5 ] , (2.3)  0 < e < 1  N  U j = (vj)  A e Gj  Chapt. 1,  .  ', f o r some T^ e G"J  c  2  T"4T  Then f o r  .  From S a k a i ' s  § 4 , Th; 5 ) , , N  T  , let 6 =  .  -..-j;:-,  y  , ^  •  .  • .•  , .  ,'.. 32.  ... ./•  :  For ease i n n o t a t i o n l e t  r;  i-V"  a  -  b  • • •y  and  let  a  Vj  v  '  1  u  1  2  '  v  1  2  .  2  and  c'  be d e f i n e d  similarly!with (a-bc) = 0"  We want to show t h a t  2.1,  •  2.3 •  and  formula (2.3)  •  •  (a'-b'c') = o)  •;r' • '/  V  l® 2^  v  p(u ,v ),  , b'  7  Lemmas 1.3, '  9  • i. .  P(u ,v ),  c =  placing l  w  =  ;  i-,'.. •  P( i  =  ,  we  •  re-  .  From  have .  ' •  ' •  1  /  '  -'  |a-a'| < 46* |'b-b'| < 26*  ;  y and  |c-c/| < 26*  i  6 , b i and  Then, s i n c e  1  . .|a-bc| =  -  r  "  l  y  , :;  x  <.  j  <  c  I a-a'  ^  Remark 2.5  was  126*  I +  ' •  Ibc-b'c'l  <  c  y.  conclude t h a t  can of course d e f i n e  d  , we have t h a t  i f and  and  p  (a-bc) = f o r any  done i n [3],  r e s t r i c t i n g the d e f i n i t i o n to s t a t e s . duces a f u n c t i o n a l - / j i  |b-b'||c-c'|  < 26*(4+b+c)  chosen a r b i t r a r i l y we We  have  . t'  t i v e normal l i n e a r f u n c t i o n a l s as was  ku.  we  '''"•'•_<• |a-a'| + b|c-c'| + c|b-b,'j +  1 c  •  are a l l _< 1  _< 2[6*(2+b+c) +2b]  Since  •  | (a-bc) - (a'-b'c'-) |  ' •; . .  •  ."  p(ku,v) = k* p(u,v)  for a l l  posi-  i n s t e a d of  Since a v e c t o r  only i f the v e c t o r  0  k x k  x  in-  induces Oi  .  .  33. •Moreover • ( M J V )  to  \x ® v  u ® v  i s defined  f o r f u n c t i o n a l s and the mapping  is bilinear.  I t follows  from these remarks  t h a t the product formula w i l l s t i l l h o l d f o r p o s i t i v e linear  functionals.  •v-  /  normal  ,i \  34,  • • • '  v •  CHAPTER I I  .'  THE SEMI-METRICS  3.  D e f i n i t i o n s of  D e f i n i t i o n 3.1.  d  Let  any subgroup of  and  u  G  fine V  .  G  p"  be a  Aut(G)  d  W - a l g e b r a and l e t  For any  u  and  v € EQ  G  P  G  a  (3.1)  P  = sup { p ( u , v ) : a , 0 e G} a  ( M , V )  From (1.1)  Proof.  be  we  de-  !  .. d (M,v) = i n f {d(n ,v ) : a,p e G}  Lemma 3.2.  G  we  , .  P  , . ..,  obtain  [d (u,v)] G  2  = £ [ l - p ^ ( u , v ) ] •  < ;  ^ (a)  d ( ^ i , v ) = i n f { d ( u , v ) : a e G}  (b)  p ( u , v ) = s u p { p ( , ) : a e G}  G  a  G  a  u  Let\ a e G  .  F o r any  so a r e p r e s e n t a t i o n of  G  [4>a,(x,y)] e Q ( | i , v ) .  Therefore  a  a  v  [<j> , ( x , y ) ] e Q(u,v) <j>a  is a l -  and a d i r e c t c a l c u l a t i o n shows d ( u , v ) < ||x-y||  i n g the infimum over a l l elements of  a  a  Q(uiv)  we  and  obtain  tak-  d(  \i  Then r e p l a c i n g replacing  a  by  a  and _1  a M  , v ) < d(u,v)  v  by  |i  a  35.  ;  a  and  v  a  r e s p e c t i v e l y and  we have A  d ( n , v ) = d((a,v) a  d(y ,v^) = d(u,v^  Therefore  , for a l l a € G  a  a  a  )  •  .  which proves the lemma. c  It follows e a s i l y that f o r a l l G m e t r i c on ^Notation. denote !>. 'and  E~ . If G = a l lo f  p.. and .If  d  d  * Aut(G)_ we w i l l l e t  respectively.  G =jIht(iG)  ,  we w i l l l e t .*p and' cf  respectively.  •v  • /  V  , d  i s a semip  and  '  "  d  ; '•_ denote  ^ p  G  N  .4.  An I n e q u a l i t y .  In t h i s s e c t i o n we prove a m a t r i x i n e q u a l i t y which 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 Notation.  F o r any i n t e g e r  n  let  R n  tor  space c o n s i s t i n g of. a l l sequences  of r e a l numbers. D* = [a e R  n  Note t h a t  D*  c  d  .  denote the r e a l vec| a = (a^,a ,...a ) 2  n  We l e t  : &  > a .  1  for  i < J • and  a  jL  2  f o r a l l i}  0  i s c l o s e d under a d d i t i o n , and m u l t i p l i c a t i o n  by a non-negative s c a l a r . For a p o s i t i v e i n t e g e r  r _< n  , let  6  denote  '  i  the element r  < i _< n  a .  of\  with  Then any element of 16 ' s  ' l i n e a r combination of In f a c t f o r '; ~  a e  Z  a  D*  ;a^ = 6r  can be w r i t t e n as a  w i t h non-negative c o e f f i c i e n t s .  +  {  i" i+l  a  a  x  .'  , l X i _ < r  D n  J  =  (  .  a^ = l  •  }  6  i Vn +  •  .  /  f  r  .  F i n a l l y we  l e t M C<|) denote the s e t of a l l n  n m a t r i c e s k = ( k ^ j ) over the complex numbers. D e f i n i t i o n ; 4 . 1. :" For a,b e D and k' € M (<£:) we . n n\ +  k  (  a  ,  b  )  define  ,n  1  T  by  ' -i,j=i =  a  i  b  J  k  i  J  '  Lemma 4.2. L e t c be an element of R such t h a t c. > o . •> . . n i — for 1 < i < n , and l e t . : k M {$) be such t h a t ' f o r a l l €  N  r  with  1 < r < n r  Then f o r a l l a.b, e D n  +  (4.1)  |T (a,b)| < R  Let  i  i  i  1  4.  _  Proof.  i  n 2 a b c  a,a,b,D e D  n  and l e t X  be a p o s i t i v e number.  Suppose t h a t (4.1) h o l d s f o r the p a i r s  (a,b) and  (a,b)  Then  ' y^-  y  :  |T.(Xa+a,b) | = | Z (Xa+a), b,k fa:; { i , j=i 1  J  1  |  J  - ' f a  < X|T (*,b)| + |T (a,b)| R  .;-  < ,  Therefore holds f o r  =  x  J  /  k  ¥ 1 = 1  +  x  J  a x  i V i  n _ T, (Xa+a).b,c. i=l 1 1 1  (4.1) h o l d s f o r (a,b) and  ((xa+a),b)  (a,"b")  .  S i m i l a r l y , i f (4.1)  i t w i l l hold f o r  I t f o l l o w s then, from the remarks above c o n c e r n i n g it  i s s u f f i c i e n t t o prove (4.1) f o r a = 6  r  and  Moreoyer by the symmetry of the c o n d i t i o n s on sume t h a t , r < s  We have i n t h i s  case;  .  (a,(b+xl)))  k  that b = 6  g  .  we may a s -  38." •" |T (a,b)|  r s = | £ T k i = l j=l  k  K  1 J  <  £ c. =  ~ }=1 and  1  .  Let S  Let  G  and  T  n £ a.b.c. i=l 1  1  ,  \  be a  W -algebra  be elements of  G  with a f i n i t e  2  and  the E^s n ; T = £ b.F. , i=l .  n  1  We  T(E^)  s i m i l a r c o n d i t i o n s ; and  = T(F )  Then, f o r any  V e G  c^  1  i = 1,2,...n  .  ||v|| < 1  with  |T(STV)I  for  1  w i l l denote t h i s common value by  trace  such t h a t :  +  E  1  with  •  1  n S = £ i i » where a = ( a , a . . . a ) e i=l are 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 i n G a  \  . :  1 J  the lemma i s proved.  Theorem 4.3. T  r s £ | £ k. . | i = l j=l  | <  < -  £ a.b.c. 1 1 x  .  1 = 1  Proof. STV  n £ a,b,E,F,V i,j=l 0 J  =  1  1  Therefore =  T(STV)  The  n £  a ^ j k ^ ^ j : /where  r e s u l t w i l l now  k^  = r(E FjV) ±  f o l l o w from Lemma 4.'2  a f t e r we  .  verify  the  a p p r o p r i a t e c o n d i t i o n s on ^(k. ..j . Choose any. 1 j< r < n r • • .• r ... Since £ F, i s a p r o j e c t i o n , || £ F-.V|| < 1 . . Then, u s i n g ' j = l ^ -;V j=l ~ '• ' •Lemma 0 . 4 ( b ) ; J  :  J  J  I.r. j=i  k VI  | T [ E ( £ F.V)] j=i  =  1 J  1  i .  v | | '"'<: c.  )!| £ F  < T(E :  I  3  1  j=i  4  .-  1  •  „  39.; T ^ E J F J V ) = T(PJVE .)  ..Similarly, since  j  r  I;  1=1  E  k. . 1 J  we  have  5.  Monotone F u n c t i o n s  , ,  Notation Let -let  R  ( R ) ' = (R +  Let to  (R ) '  denote the non-negative r e a l numbers  +  U +»)  +  D e f i n i t i o n 5.1. on  (R )'  •  M = the  such t h a t  +  set of a l l f u n c t i o n s x _< y  (Note t h a t  from  f (x) > f (y)  we  (R )' +  :  .  define a function  •%)•''. "• •  = i n f [x € ' ( R ) ' : f ( x ) < a) +  inf{^} = +« The  , sup{/zJ] = 0  ) 7 - are e i t h e r w e l l known  f o l l o w i n g f a c t s .about  or e a s i l y v e r i f i e d ;  " •  5.2  Lemma  7  (a) (i.e.  e M  a"nd moreover i t i s r i g h t  f ( i n f (S)) = s u p ( f ( S ) )  X  (b)  (c)  = x  (i)  f o r every  Let  g(x) = k f ( x ) = (f) f  o  r  (d)  a  1  1  Let  such t h a t  , i f and  S c ( R ) '. ) +  only i f , f(y) > a  f o r every/ y > x  , f ( y ) _< a  k e R  .  +  x  implies  (f-Ln  M  by  f(x) = sup(f (x))  Then  f, < f . .  n  \' ->;;^f:€. M :  g(x) -= f ( ^ ) k'  g(x) = k7(x)  implies  x  in  ;;  continuous  y < x  for a l l f  , for a l l  7(a)  (ii)  x  f  by  +  ( )  implies  For any • f € M  7(a)  f  and  o  for a l l  be a sequence of  for  i < j } .  Then, and  for a l l  :  Define  " . •!  (T )f n  7  . :  f  x x  functions on  (R )' +  ' ,  '  •  41. (e) the  L e t cf> be a continuous f u n c t i o n d e f i n e d  interval  interval. ,f(b).  J  [a,b] and l e t f € M  <K )  df(x) =  X  J  <• +(7(o))da. .  ,  on t h i s V  be f i n i t e valued  Then the S t i e l t j e s i n t e g r a l  on  a  •f(a)  We need ah e x t e n s i o n Lemma 5.3 that  Let  be a continuous f u n c t i o n on  (j>(xj >Z0' f o r a l l  be such t h a t  o f (e) t o improper i n t e g r a l s i '  f(x) < »  x  and  c}>(.0) = 0  f o r x > '0; and  .  f ( x ) -• 0  such  as  x —•» <»  '  • - J; cj»(x)df(x) = f f ( 7 ( a ) ) d a  o  J  denotes  Q  J  l i m f| } m-~ 1  Proof. that,  0.'-  .  -  ... » -  !  and i s allowed t o take the v a l u e ,  J  n-»oo  and  +  Let f e M  Then  ([  R  n  L e t f ( 0) = sup (f(x) : x e ( R ) '} +  y e f ( 0) +  and n o t e ' f i r s t  +  i m p l i e s by Lemma 5.2(b) t h a t <J>("f(y).) = cj)( 0) = 0  therefore that  .  f ( y ) = .0. >  Then u s i n g Lemma  5.2(d)  •  '', , ,  m - J c|>(x)df(x) = - . l i m ( f <f)(x)df(x)) /• V 0 m~« 1 ' .:-n-*« n !  J  •''  f ( - ) • ' ' •  ;- = l i m ( J : :  •  v  -; ;;• " >  m-» •  n  •  f (0) +  i(T( ))da = f  f(m)  a  . i ,^ '  '" 4>(T(a))da  ,0  J  .  0'* . . .  +» )*  .  •  : \ - ^ > f v ' .:•>':,••••' ;  i. , ' ' L : V ; - ' " v ;  •  "  ,.'  '  '  .  The main r e s u l t of t h i s s e c t i o n i s the f o l l o w i n g .  \[ V  Theorem 5.4. ;to  R  +  .  Let  Let  <{>  f,g  be any continuous f u n c t i o n from  and  h e M  be such t h a t f o r a l l . .  l e t [,a,b]  and  b  7(a)  denote the c l o s e d  even i f and  a > b  g(a) e  R  ).  7(a)* = x  and  choose, by the c o n t i n u i t y of  ;(5..l) •  .  +  .  a  a € (R )'  such t h a t  Given any  e > 0.  +  —  g(a) = y <|>  .  .  , 6 > 0  '•" i  '••'  so t h a t  <t>(x,y)-c  cj)(x+6,y+6) _< 4>(x,y)+s  '. ,'  : v  By our h y p o t h e s i s  .  h(<j>(x-6,y-6)) e [ f (x-6 ), g(y-6) ]  and from Lemma 5 . 2 ( b ) b o t h ' f ( x - 6 )  and  U s i n g ( 5 . l)^we' o b t a i n  .  . (5.3)  :• >  .  Then f o r a l l .  d>(x-6„y-6) 2  (5.2)  +;  +  v  Let  x,y e R  i n t e r v a l w i t h end-points  I h(a).= <J)(7(a)/g(a))  Proof.  x R  i  h(cj>(x,y)) e [ f ( x ) , g ( y ) ]  (We  R  +  g(y-6)  h(<|>(x,y)-e). _> h(<J)(x-6,y-6 )) > a  are •  '"•;.[.  ;  \;  > a  .  :  43. Prom Lemma 5 . 2 ( b ) a g a i n both  f(x+6)  and  g(y+6)  a r e _< a  and u s i n g ( 5 . 2 ) we o b t a i n s i m i l a r l y t h a t .  .(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 o f Lemma 5 . 2 ( b )  we have  TT(a) = <|>(x,y) = <f>(f(a),g(a))  • X  .  6.  D i s t r i b u t i o n F u n c t i o n s and  q  ,•['" j  . .Spectral  A  be  a s e l f - a d j o i n t operator  bounded) oh a H i l b e r t space A  ••• •  v  Resolutions.  Let  of  • '  .  H  .  J}^  Let  (not n e c e s s a r i l y denote the  domain  (For t e r m i n o l o g y c o n c e r n i n g unbounded o p e r a t o r s  to [20], )  By a s p e c t r a l r e s o l u t i o n of  the unique; f a m i l y of p r o j e c t i o n s  in  A  H  we  ,  refer  w i l l always mean , X  ( E ( x ) }  a real  number, which s a t i s f i e s the f o l l o w i n g p r o p e r t i e s :  '"'Ii-  - 1.  ;••  2.  X <. u;  U,E(X!) =  '  A.  • .  3-  E ( x )  E ( x ) <  implies  E(u) ;  i • , 0,E(X).= 0  ••  '. t  K  \^  .  commutes w i t h any  commutes w i t h  A  r  operator  in  £(H)  ,  ,'  .  which  : ,  |'/  4.  0  X > X  . , 5. • For that  E(X) = E ( X )  , for« a l l  Q  -oo < x _< u'' < +» x. e J&  and  A  X  j '  Q  Integrable  Operators and  Suppose t h a t space  H  .  An  if-,||x||= 1  ,  x < (Ax|x) < u ".. '  -  •  Radon-* Nikodym D e r i v a t i v e s .  G  operator  i s a von S  s a i d to be; af f i l a t e d w i t h  U S U  '  , x e E ( y ) - E(Xi); . i m p l i e s  •  is  -  on  H' :  G,  = S ' f o r every/unitary  bounded and  everywhere d e f i n e d ,  shows t h a t  S n G..; i f and  Neumann a l g e b r a  "  "  ' \  '.'.'J  <•  on a H i l b e r t  (not n e c e s s a r i l y bounded)  , written/  U e G'  •  If  S n G S  , if  '  -  is in fact  the double' commutaht theorem  only i f  S e G > V  For  any s e l f - a d j o i n t 1  S S'n  with s p e c t r a l r e s o l u t i o n C.  i f and Now  shown i n ([5] i d e a l of  G  'E(x) e G  only i f  suppose t h a t  normal t r a c e on  G  T  §6,  for a l l real  We and  ' a norm on'  7\  trace.  Then T'  T  on  .  It i s  i s a two-sided.  71  \\S\\  2  given  §6,  by  It .satisfies  e 71,  .  by t a k i n g . (s|T)  7}  = T '(T  Th.  .;.  ,  8) t h a t we  can  also  define  v  •  1 ! % = T'(|SD . In [15.], Segal d e f i n e s two  a f f i l i a t e d with  G  to  T  as completions of .71 • and  7h  summarize a few  f o l l o w i n g lemma.,  subclasses  of the  , and which c o n t a i n  Moreover he extends,  c l a s s of i n t e g r a b l e o p e r a t o r s .  We  : operators;  which are 'known as square i n t e g r a b l e  integrable, with respect respectively.  S)  by  =.[T'(S*S)]*-  Chapt. 1,  can be extended  7!\ .  = T '(.TS) , f o r a l l S,T  TJi. by  ;'.'  .norms.  T ( A A ) < «}  l i n e a r combinations of  t h i s produces a norm on  I t i s shown i n [[5]*  \  which are f i n i t e  can d e f i n e an inner product, on  . '[  ..  c o n s i s t s p r e c i s e l y , of  ideal  l i n e a r i t y to a l i n e a r f u n c t i o n a l T '(ST)  .  i s a semi-finite, faithful,  Prop. 1) t h a t 2  p o s i t i v e elements of f i n i t e  (6.1)  X  that,'.  7h = 7\  t h a t the  those elements of . G  , i t i s , easy to see  \7\ = [A e G;:  Let  Chapt. 1, and  (E(\)}  Proofs  T'  71  and and 7f[  to a ' f u n c t i o n a l on  These c l a s s e s can be with respect  the  considered  to the above mentioned  b a s i c f a c t s , that, we  need i n the  can be found i n [15]. .'  Lemma 6 . 1  • \ '' (a)  Let S  be a p o s i t i v e s e l f - a d j o i n t i n t e g r a b l e  .operator w i t h s p e c t r a l r e s o l u t i o n (i)>(i-E(x))  -;-  (ii)  T'(S)  (b)  Let S  < «  , for \ > o  X d T(1-E(\))  .  ,  < »  (E(x)}  T(1-E(\) ) < «  for  .  .  X > 0  v  '  ,  •  1  be a p o s i t i v e s e l f - a d j o i n t  operator w i t h s p e c t r a l r e s o l u t i o n (i)  "... Then  (E(X)}  -\ .  square-integrable  Then  , i  (ii)  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 f o l l o w i n g theorem  of S e g a l ( [ 1 5 ] ,  Thebrem 1 4 . 1 ) .  I t i s a generalization  of both  the Radon-Nikodym theorem i n measure t h e o r y and of a w e l l known r e s u l t concerning  Lemma 6.2.  Let G  X(H) .  be a von Neumann a l g e b r a  f i n i t e , f a i t h f u l , normal t r a c e  x  , and l e t  there e x i s t s a unique square i n t e g r a b l e , operator  S  w i t h a semiu e. £_  .  Then . •<•;•  positive self-adjoint  such t h a t u(A)  = T '(S. A) 2  , for a l l A e G  .  2 The  operator  Nikodym d e r i v a t i v e o f y  S  i s commonly known as the Radonwith'respect to  T  .  We w i l l denote  t h i s by w r i t i n g  x Note t h a t  if  ;;  ' w = s ..  ".' • .  T  S e G  > f o r m u l a ( 6 . 1 ) shows t h a t t h i s agrees w i t h  our n o t a t i o n of Chapter 0.''•;,.;; ' V x X s :  :  " : ' X . ' '  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  a von Neumann a l g e b r a  G  on a H i l b e r t space.  .  •  y  considered  However a  square i n t e g r a b l e , p o s i t i v e s e l f - a d j o i n t operator  can be  • V  i  viewed as simply an a b s t r a c t a family of projections  s p e c t r a l r e s o l u t i o n ; that i s  in G  s a t i s f y i n g the a p p r o p r i a t e  c o n d i t i o n s , a l l of which r e l a t e o n l y t o the a l g e b r a s e l f and the t r a c e .  Accordingly  to a  without regard  W" - a l g e b r a  sentation  of  G  Distribution  G  D e f i n i t i o n 6. 3. T  we can a p p l y these r e s u l t s t o any p a r t i c u l a r r e p r e -  Let  •  G  '  be a von Neumann a l g e b r a  be a f a i t h f u l , normal s e m i - f i n i t e t r a c e on  positive self-adjoint  (R V +  AnG  we d e f i n e  by  f (X)  = T(1-E(X)/)  G  and l e t .  the f u n c t i o n  .  A  Gi t -  .  Functions  ••;  F o r any f ^ on  '  , f o r 0 _< X < »  f (») = 0 . A  where f  A  (E(x)}  i s the s p e c t r a l r e s o l u t i o n of  o f course depends on the 'trace We^call  respect  to  T  f  A  A  .  T; .  the d i s t r i b u t i o n f u n c t i o n of  A  with  , by analogy t o the p r o b a b i l i t y d e f i n i t i o n .  Note t h a t the d e f i n i t i o n above i s p r e f e r a b l e t o t a k i n g the increasing function, . X - T ( E ( X ) )  i .  , since i f A  i s square  48> integrable  f"  A  i s f i n i t e valued,  except perhaps a t  Obviously, i n the n o t a t i o n o f S e c t i o n 5> r i g h t continuous f u n c t i o n 7  A  e M  0 f  A  . is a  and we can form the f u n c t i o n  • ,••  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  (A^n-i 2  n G, and l e t  muting elements o f  A  (1) (ii)  such t h a t  +  < Aj f o r 1 < j A = sup ( A j n  Then,  ±  L e t {E ( x ) } n  Proof.  be the s p e c t r a l r e s o l u t i o n o f  v  (E(x)}  let  G  be the s p e c t r a l r e s o l u t i o n o f E (iX) jrE(x)  hard t o v e r i f y t h a t  n  (1-E (x)) t (l-E(x))  .  n  f  Lemma 6.5 such t h a t T  fora l l  By the n o r m a l i t y  A  .  A and n  I t i s not  (  X  .  Therefore  of  T  , we have t h a t  X > 0  for "all  of  be a sequence of mutually com-  and  - ( X ) = T(l-E (x))'1> T ( 1 - E ( X ) ) = f ( x ) n  Let T e G ,TP = T T  .  +  A  and l e t  P  L e t T' and  T'  t o the a l g e b r a  e q u a l t o f ^ computed w i t h r e s p e c t P r o o f L e t  be a p r o j e c t i o n i n G denote the r e s t r i c t i o n s  r e s p e c t i v e l y t o the a l g e b r a  computed w i t h r e s p e c t  (E(x)}  .  G to  G  p  .  and t r a c e  p  T  Then  f^, ,  T' i s  .  be t h e s p e c t r a l r e s o l u t i o n o f T . - ' .  Then the s p e c t r a l r e s o l u t i o n of H i l b e r t space X  .  P  i s {E'(x)}  vector  x  all  X > 0  .  .  f >(X)  -  and  =  T  (P(1-E(X))  by r i g h t c o n t i n u i t y  D e f i n i t i o n s of  = T(1-E(X))  T  < P for  = f (X) , T  .  ' '•  Let G v  be a  • .  operators  a f f i l i a t e d with  and  T  grable with respect t o  .  W - a l g e b r a and  G  T  o(S,T) = J  T  a normal  F o r any p o s i t i v e s e l f a d j o i n t G  and square i n t e -  we d e f i n e  o  7 ( x ) 7 ( x ) dx s  T  .  By Lemma 5.'3> and Lemma' 6.1(b) we have . t h a t are square i n t e g r a b l e f u n c t i o n s on  .q(S,T)  .  = t'(P(l-E(x))  s e m i - f i n i t e t r a c e on  Trp'  .  q '  D e f i n i t i o n o. 6.  S  .:  ,  frp/(0) = f ( 0 )  i  , (Tx|x) > X  l-E(x)  or  t  X > 0  T'(P-E(X)P)  =  T  for a l l  , (Tx|x) = (TPx|x) = 0  flV(l-P) =0  Then f o r any  on t h e  E ' ( X ) = E(x)P  in l-E(x)  x  i n 1-P  T h i s shows t h a t ( l - E ( X ) )  as an operator  where  Now f o r any u n i t v e c t o r  For any.unit  T'  i s always  < »  Note t h a t ; a  .  [0,»)" .  "f g  and  Therefore_-  ' ' .  depends on the t r a c e  T  .  When  G  i s a f a c t o r we can e l i m i n a t e t h i s dependence by d e f i n i n g  o  .for states.'/ .. ;'•/;.•;  50."  IJDefinition 6.7.  Let  G  be a f a c t o r and l e t  Suppose t h a t f o r some s e m i - f i n i t e v = Tip  and  .  We then  trace  T  u.  on  and  G  v e E^ .  , u = TI'V" V  define  /  cj(ia,v) = a(S,T.)  computed w i t h r e s p e c t t o  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 t o show t h a t it  i s independent of the p a r t i c u l a r t r a c e  be another s e m i - f i n i t e T' = k T , f o r some  trace  k > 0  on  G  .  Since  , and i t f o l l o w s  If; the s p e c t r a l r e s o l u t i o n o f  S  chosen. G  that  (E(X)}  is  L e t T' i s a factor, u = s / yj^ T  /  the  E " ( \ ) = E(^/k\)  fora l l \  .  .denote calculation^ with respect t o  Prom Lemma 5 . 2 ( c )  .  ••' ;  After  '  s / J z  T  M  a'(u,v)  = o'(S/JZ  '  symbols  we have  ,  .  .  •  , TA/k")  we  obtain  V  ;  i  :  0 =  7 '(!y) t ( y ) ;  s  ./; '  ~ p s &  a similar calculation for v  /•  3  "'  /  {E'(\)}  L e t t i n g primed  r'  T  •-. .1,.•C,—  p  :  s p e c t r a l r e s o l u t i o n o f /S/Vk" i s e a s i l y seen t o be where  j  dy - a(S,T) = , a ( , v ) U  .  '  ...  St.  ••  5i. • D e f i n i t i o n 6.8. —  —  —  —  —  —  '•. -trace • v;\  —  —  T  —  '  ;  (Note t h a t trace).  be a f a c t o r w i t h a s e m i - f i n i t e  '  ".-\  "  •  •-.  •  :  • \  ••  P(G) = {a € Aut(G) : T ° = T }  P(G)  i s o b v i o u s l y independent o f any p a r t i c u l a r  Now l e t  G  be any subgroup o f  . the homomorphism from  G  of  Aut(G)  .  Let  %  be  onto the m u l t i p l i c a t i v e group o f  .. p o s i t i v e r e a l numbers d e f i n e d by The k e r n e l  .  We l e t  .  ;r;' C ;  G  : I  .  •  Let  —  4 • i s obviously  f(a) = k G 0 P(G)  if  T  . ' Then  a  =kT  .  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 r e a l s which we denote by Definition 6.9. trace and  T  Let  , let G  v e  .  Suppose  that  Aut(G)  u. = Trp .  .  4  , and l e t y. - 1  .' .  .  k  v  >v  ;.  be a f a c t o r w i t h a s e m i - f i n i t e  be a subgroup o f  We d e f i n e ,  '  G  r(G) .  r  «  - a ( u , v ) = J% \ k  • ,  7 ( x ) 7 ( k x ) dx g  T  '  , for a l l k > 0  ,  ;  . .'.  a ( u , v ) = sup ( a ( u , v ) :/k e r(G)} 1  G  k  As above we can show t h a t p a r t i c u l a r trace.  a  i s independent o f the  :•  y'.'.-)  The m o t i v a t i o n f o r these d e f i n i t i o n s i s shown by • the f o l l o w i n g . Lemma 6 . 1 0 .  ., Let  semi-finite  G  trace  '  v € £~ = T  1  T ..  f o r some  m  T e G  +  and some  • JL  Let  :• '\  a e Aut(G)  with  T ° = kj  .  '  V  Then  o{\X,\> ) =  a ( ,v)  a  Proof.  By  lemma 0.3,  T . -i ^(a" (T))  =  . L e t  T  v ,  =  a  k  (T^)  0  = J^(a  . . .  u  = T  = kT  a  (ot~ (T)), . If T  (T))  n  (a^T)). has spectral  1  resolution  (E(X)}  , T'  has  E ' ( x ) V =..a (\A/K))  where  spectral  for  _1  ;  all  (E'(x)}  resolution  X  •/  Then f /(\)  T(1 - a^iEiX/JZ))  =  T  cf^EfxA/E) ) .  = -£<r (l a  = -| f ( \ A / k ) T  !  By  lemma 5.2(c),  lemma f o l l o w s  ^/(X)  from the  Remark 6.11.  If  P(G)  .  o  G  = Aut(G) = a  .  The  with factors example o f  G  fo  a factor  type  G c  any  r  purpose of  of  , for a l l  d e f i n i t i o n s of 1$  So  7k f " ( k x )  =  of  and  type  o  G  p.  188)  and  =  , {1}'  i s therefore  shows t h a t  the if  the'  k  II-j^  , r(G)  where t h i s i s n o t  Suzuki  a  I or  Aut(c)  defining  II  a  X  and  to  case. G = P ®  :  deal An I  v  where the  P  i s the  entire  hyperfinite  1^  factor, then  r(Aut(G))  group o f m u l t i p l i c a t i v e r e a l numbers.  r e a s o n i n g employed  i n t h i s example shows t h a t  if  The  =  same  G = fil ®  I 00  for  a  11^  of  0  .  factor (See  [10]  ft  then  r ( A u t ( G ) ) = the  Theorem V I I I f o r  the  fundamental  d e f i n i t i o n of  group  funda-  m e n t a l group"). In the nite  factor  G  rest  of  this section  with normalized  trace  we  will  consider a f i -  T [ (i.e.  T(1)  =1  ).  53.' We want t o develop some c o n t i n u i t y p r o p e r t i e s  of  a  \ •  The  (See [9]  f o l l o w i n g i s a w e l l known r e s u l t .  . i  . \  lemma 14.22). Xemma 6.12. T(E) < T(F)  Let E .  P  and  Then  (1-E)  in G  be p r o j e c t i o n s n F + 0  with  .  next lemma i s e s s e n t i a l l y proved i n ([91  The  Lemma  15.21). Lemma 6.13. ;  A e G  and  +  .(6.2)  Consider any r e p r e s e n t a t i o n 0 _< a _< «  G  of  .  ,  f . ( a ) = i n f ( s u p { ( A x | x ) : ||x|| = 1 ,E  , Ex = x ) ) -  H  as  E  runs over the p r o j e c t i o n s  of  G  , so assume  a < »  r i g h t s i d e o f (6.2) ' r e s o l u t i o n of Then  A  On  > a  .  Q  [E(\)} b'  b'  . Let G  and l e t t h e  be t h e s p e c t r a l /  and  b"  with  by Lemma 5.2(b)  contains  .  I t follows  a unit vector  that  b  Q  and; b''  b  Q  So  for  ;  :  x  by •  ;  such  > b' . A  _< b"  .  b ' < b <,b"  T ( l - E ( b " ) ) = f ( b " ) _< a  So f o r any u n i t v e c t o r  Since  = b  T ( 1 - E ) _< a, ( l - E ( b ' ) ) fl E ^ 0 E  I t follows that  that b = b Lemma 6.14.  Let  the other hand,  Lemma 5.2(b). .  with  Therefore  (Ax|x) > b '  L e t "f^(a)  Choose any A  ••Lemma 6.12.  _< b".  .  E  .  = b* • . o  <r(l-E(b')) = f ( b ' )  any p r o j e c t i o n  that  .  •  -  I f a = »' b o t h s i d e s o f ( 6 . 2 ) a r e e a s i l y seen t o  Proof. 0  T ( 1 - E ) _< G  with  !  be  F o r any  by •-.  i n E ( b " ) , (Ax|x)  .  were chosen a r i b i t a r i l y we have,  :; be a f i n i t e f a c t o r . . Suppose t h a t  S,s',T  and  T' e G  ||S-S' I!  <  6  satisfy  ,  II <  UT-T*  6  ,  where  0 < 6 < 1  .  Then  .  where  |o(S,T) - a ( S ' , T ' ) | < 6(|lS|| + ||T|| + 1)  a  ,  >[  i s c a l c u l a t e d w i t h r e s p e c t t o the normalized  trace  T  Proof. f  A  _< 1  Note f i r s t  that since  , and t h e r e f o r e  T(1) = 1  7 (a) = 0  , f o r any  fora l l  A  f o r e the i n t e g r a l ! i n the d e f i n i t i o n o f _ a the  interval  .  +  There  can be taken over  [ 0 , 1 ] . .  Since  • (6.3.)  a > 1  A e G  fg(!|S||) = f (||T||) = 0 T  we have  f ( a ) < ||S|| , 7 ( a ) < ||T|| , f o r 0 < a < s  T  Now c o n s i d e r any r e p r e s e n t a t i o n o f  '  G. and any u n i t v e c t o r  Then-. |(Sx|x) - (s'x| x) | = . |^ (S-S' )x 1 x) | _< |1S-S' || /j 6  x  .  From Lemma 6.13 we can conclude  (6.4:)'  Using  |7 (a)-7 /(a)J < s  s  6  and  JT (a)-T * (a) | < T  (6.3) and (6.4) we. have f o r a l l  T  a > 0  6 , for  0 <  55.  |fg(a)f (a)-f ,(a)T ,(a)| T  .  s  T  < 7 ( a ) |7 (.a)-7 , (a) | + 7 ( a ) |7 (a)-7 , (a) | s  T  T  T  + <  6(||S||  +  ||T||  +. 6.)  <  s  g  |f (a)-f /(a)||f (a)-f (a s  6(||SH  +  s  ||T||  +  T  1)  T  .  Therefore !a(S,T) - a ( s ' , T ) | < J- |7 (a)7 (a) g  <  6(||S|| +  v  <r  T  ||T||  +  ~"7g, (a)7 , (a) |da T  1)  .  1  56. 7.  Compatibility,  . D e f i n i t i o n 7.1. (not  Let  necessarily  that  (E(X)}  .'•  S  and  T  be s e l f - a d j o i n t o p e r a t o r s  b o u n d e d ) on a H i l b e r t s p a c e  i s the s p e c t r a l r e s o l u t i o n of  i s the s p e c t r a l r e s o l u t i o n of  T  We  • •  Remarks. .-remains . tions  Suppose  •, and S  (a,|3)  {P(\')} and  T  o f r e a l num-  ( l ) or (2) h o l d s :  (1)  E(cx) < P(f3)  (2)'  F(f3) < E ( a ) . .  ,  I t i s not h a r d t o v e r i f y t h a t t h e above d e f i n i t i o n ;  u n c h a n g e d i j f we use l e f t  continuous  i n place of r i g h t continuous Note t h a t i f  viously  S  .  say that'  a r 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 bers e i t h e r  H  S  commute w i t h each  5  and  T  spectral  resolu-  ones. a r e c o m p a t i b l e t h e y ob-  other.  -  The m a i n 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 n e e d i s " the following.  .• •  Theorem 7.2.  Let  semi-finite trace  G T  .  s a t i s f y T ( S ) < <*  f  Proof.  •.  be a von-../Neumann'" a l g e b r a w i t h a  2  and  /  S T  Let ,  S ' and  T, e G  +  be  normal,  compatible,  p  T ( T ) <. *>  .  Then f o r  0 < a <_ «  ( a ) ' .= f g ( a ) 7 ( a )  We a p p l y Theorem 5 . 4 ,  T  taking  <j)(x,y) = x y  .,  I tre-  ,  mains to v e r i f y the (F(X)}  and  c o n d i t i o n s of t h a t theorem.  (G(x)}  , and  be  ST r e s p e c t i v e l y .  each other  any  (a,B)  Let  two  the  spectral  Since  resolutions  the o p e r a t o r s  of  For  any  S , T ,  a l l commute w i t h  of these s p e c t r a l p r o j e c t i o n s w i l l  be any  ordered  Assume t h a t unit vector  •  E(a) _< x  commute.  p a i r of non-negative r e a l numbers. /•  Case 1.  (E(x)}  Let  in  •  F(B)  E(a)  € E(a)  S^x  _< F ( B )  , which  shows t h a t  (STx|x) =' (TS^x|S^x) < P(Sx|x) < a B  For  any  u n i t vector  y  1 -  in  G(qB)  (STy|y) > aB *'  E ( a ) ( l - G(aB)) = 0  (7.1)  .(1 see  _< (1 - E ( a ) )  , we  (7.2).  .V./  From (7.1)  and  , so  unit vector  (STx|x) > a B  have  (7,;2)  .  G(aB)) jC (1 - E ( a ) )  t h a t f o r any  (1 -  .*  ' '  T h i s shows t h a t  S i m i l a r l y , we  . x e (1 -  , and  G(aB)) 2 . (il - F ( B ) )  we  .  we  F(B))  obtain  .  o b t a i n by the p o s i t i v i t y of  T  that  f (S)  < f  T  Case 2.  Assume t h a t  (aP) < f (a)  g T  s  E(a) j£ F ( B ) F ( 8 ) •<_ E(a)  By c o m p a t i b i l i t y we must have  and a r g u i n g as ' I•  above we o b t a i n  . f ( a ) < fg (aP) < f 0 ) s  T  We have then t h a t a  and  T  • '  ' •'  .  i n any e v e n t , - f o r a l l non-negative  B  f ( a i 3 ) e [ f ( a ) ,. f ( B ) J S T  . .  T  s  j  , '  \  Now we note from Lemma 6.1 t h a t are f i n i t e , 0  .  "fg and . 7  are f i n i t e  T  2  T(S )  since  and  '2  T(T )  valued except perhaps a t  We apply Theorem 5.'4 t o complete l  the p r o o f .  /  Corollary 7 . 3 . I f S and T € C a r e compatible, a n d / 2 p satisfy T ( S ) < » , T ( T ) < » , then +  \:'. . ' T ( S T ) = o ( S , T )  Proof.  .  Prom Lemma 6.1(a), Lemma 5.3, and Theorem 7.2, •CO  X  Remark.  T(ST) =  7  S T  (a)da  = o(S,T)  .  .  As a f u r t h e r a p p l i c a t i o n o f Theorem 5.4 we may  59. obtain by  that  taking  f (  S  +  T  ) = 7  + 7  g  <}>(x,y) = x + y  Similarly,  taking  T  f o r compatible  and  (f>(x,y) = k x  yields  f  k  g  = k7g  ,  •  7.2. k > 0  for  i n the remark f o l -  D e f i n i t i o n 6.7).  Definition 7 . 4 .  Let  G  be a  be c a l l e d s i m p l e ,  W -algebra.  An e l e m e n t  A e G  will  linear  combination of mutually orthogonal projections  Remark. n A = E 1=1  T  and p r o c e e d i n g a s i n T h e o r e m  ( 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 lowing  S  Let c.E, 1  A  i f i t c a n be w r i t t e n a s a f i n i t e  be s i m p l e and s e l f - a d j o i n t .  , f o r some r e a l numbers  (c.)  1  € G  .  Then  and m u t u a l l y  1  , , n orthogonal, non-zero p r o j e c t i o n s (E. ) w i t h T, E. = 1 . • i=l '•(We c a n a l w a y s o b t a i n t h e l a t t e r c o n d i t i o n b y i n s e r t i n g i f 1  n e c e s s a r y an a p p r o p r i a t e  projection with  1  0  coefficient.)  v It  i s e a s y 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  A  i s given  (7-3)  (E(\)}  /of  by  E(X) = E E .  , t h e 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.  , t h e sum t a k e n o v e r a l l  j  such  that  ,  6o. Theorem 7 . 5 .  Let  #  of a  W -algebra  S  G  and .  T  be simple, s e l f - a d j o i n t  Then  S  and  T  elements V  are compatible i f and. \  only i f there e x i s t .  tions i n  (i)  a s e t of non-zero,  G  (G. ),  n  0  m  mutually orthogonal p r o j e c m . with £ G, = 1 ;  xx—x^£-j.*«m  0  2  .  n o n - i n c r e a s i n g sequences of r e a l numbers,  )  and m  m  =  (d,,d ...d  )  : m T = £ d.G. i=l  0  m  £ c.G. i=l 1  Proof.  t  i  ' (ii) (c,,c ...c 1 such that  * ^ x  and  1  Suppose.that  1  S  and  T  1  are of the above form.  From ( 7 . 4 ) , any non-zero, s p e c t r a l p r o j e c t i o n of e i t h e r • . •• m . or T i s = E G . f o r some k such that 1 '< k < m . . i=k . ~ ~ • viously S and . T< are compatible. f  S . v. ;  Ob-  1  Let  Conversely, suppose t h a t S and T n k S = E a.E. , T'= E b .F . , where (E.) 1=1 jti J J 1  1  and  1  sets of non-zero, For any  are compatible,  i , j,  S =  t  a  i s a p r o j e c t i o n and, '  ^  G.. = 0  and  T =  E bjG^  and s u i t a b l y renumbering  the remain-  i n g ones, we o b t a i n . m  X  m  S =. E c.G. i=l'  and  T =  E d.G.  ,  i=l  where; (i)  '  are  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 w i t h sum >'= 1 ^ = E ^ j  D e l e t i n g those  (F .) J  2  m  i  s  a  s  e  t  of non-zero  mutually  o r t h o g o n a l p r o j e c t i o n s w i t h sum (ii) (  c  1  > c . . . > c 0  ^.  i  i+l  =1  ,  .,'  m  hen  3  T h i s of course is. a f a m i l i a r procedure which can be whenever  S  and  T  commute.  We  c o m p a t i b i l i t y i m p l i e s t h a t the Suppose t h i s i s not and  necessarily,  such that and  d  be  spectively.  c^ >  are  .  and  < F(d)  !G.  c  and  c^ > c > c.j_+l' ' '  (7.4),  we  S  L e t  and  d  i +  ^  d  t (^)3 E  T  re-  obtain  (1 - E ( c ) )  n  the  i , d^ <  Choose 'numbers  s p e c t r a l r e s o l u t i o n s of  (7.3)  that  non-decreasing.  Then f o r some  and  the  From  d^'s  case.  < d <  i  (F(X)}  the  want to show now  applied  ,  I  G  Since  G  i+1  F(d)  a c o n t r a d i c t i o n to .the i —  d  i+l  ?  ^  .We  o r  1  now  of p r o j e c t i o n s  C  )  0  (  1  " (4)) P  '•  are both .non-zero, we  E(c)  l  (  E  •t»  and  i  i  =  and  /  j^'.E(c)  compatibility. Therefore,  l ^ ^ • •• ""l 2  F(d)  necessarily have  m  and  the  we  theorem i s prove d.  need some lemmas c o n c e r n i n g the in a factor.  standard*results  must have  In the p r o o f s  which can be found i n ([93  we  use  comparison several  Chapters VI  and  •>...> Lemma 7.6. that  H  . 62.  Let E  and  H  be p r o j e c t i o n s  is infinite  and  E _< H  .  i n a f a c t o r such •'j  Then e i t h e r  E ~ H or  H-E ~ H  Proof. for  Suppose  i f not,  Therefore  E  E^H-E  + H-E-^P + Q = H-E  l a r l y that  Then  would be f i n i t e  H-E = P + Q  If  .  where  H-E  must be i n f i n i t e ,  and then  H  H-E ~ P ~ Q  .  , and i t f o l l o w s t h a t  E-^H-E  ., .then  H-E^E  would be f i n i t e . Then  H = E.  H ~ H-E  .  and we o b t a i n  simi-  H ~ E '.  D e f i n i t i o n 7.7.  Let H  be a p r o j e c t i o n i n a  W -algebra  G .  i  If  E  and • P  are p r o j e c t i o n s  E-: ~ ; F - , i f '• E ~ F  E .  F  •  (  H  we say  H-E ~ H-F  .  We say t h a t  , f o r some  P  and . Q  .  t  L e t E i F , and  such that  E < H  E ;  Proof. ~  contained i n H  )  Lemma 7.8. G  and  ,. i f E < P ~ Q < F ~ (H) ,  (H)-'  G  \  . . .  that  •  of  1  If H  and  i s the same as  ~  be p r o j e c t i o n s  F'< H '.  ^ F (H)  i sfinite,  H  i n a factor  Then e i t h e r  , or. P ,<~~ E ' (H)  .  i t i s a w e l l known r e s u l t t h a t  , and we are done.  So assume  H is  (H)  infinite. E ~ F and or  1  < F  H-F-L F^  By symmetry we may assume t h a t f o r some .  N  I f H-E ~ H-F  1  we are done. • I f n o t ,  cannot both be . ~ H  is ~ H  P^ ,  and n e c e s s a r i l y  .  H-E  From Lemma 7.6, e i t h e r  E ~ F ~ H  . B y  symmetry  E  again we may assume t h a t f o r some Then. H-G fore,  contains  F < H-G  ~ E (H)  C o r o l l a r y 7.9*  U*PU > E  , which shows t h a t  G <_ 1-H  = H-P  f o r some  and  There^  =|H-Q  .  Then,  Let S  such that  \S  Let S =  and G  and  m. £ a,E, k=l K  .  T  /  and  or ( i i )  §2, Prop. 6.)  E F . (H) V  VV = P  .  and , W  U = V + W + 1-H  Then, W  be  = Q  ',  •  i sa .  (a) and (b) ( i i ) .  be s e l f - a d j o i n t , simple  Then, there  U TU  for a l l  U*FU _< E  Q. . . . Let  G. s a t i s f y i n g  elements of a f a c t o r  Proof. -~ :.  P  which s a t i s f i e s  Theorem 7.10.  U e G  .  E  (a) U GU = G  ( c f . [ 5 ] , Chapt. I l l ,  , W*W  unitary i n G  (F^)  such t h a t :  ; (b) E i t h e r ( i )  p a r t i a l , isometries: i n W*W  F  We can suppose by Lemma 7.8 t h a t  E < P ~ Q < F  to H  (H)  U e G  holds.  Proof.  and must be e q u i v a l e n t  T  .  Assuming the c o n d i t i o n s of Lemma 7.8, there  exists a unitary projections  F  G. , H-E ~ G _< H-F  exists a unitary  are compatible.  T =  £ b.F.  1=1  K  1  , where  1  (E, ) and' K  are s e t s o f non-zero,, 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 ".with  ;  sum -= 1  and  a ^ C a ^ . .. < a „. 2  m  .. ., We d e f i n e the'pro jec.tions  and  b^ < b  2  .. . < b  p  .. .  , 64. Then any s p e c t r a l p r o j e c t i o n of ...m  .  , corresponding  B = {0,1,...p} S  f o r some  i = 0,1, ,  We d e f i n e i n a s i m i l a r way the p r o j e c t i o n s  j = 0,1,...p  If  S = E^^  and  , and order  T  to  T  .  the s e t  ,,  Now, l e t A = {0,1,.. .m}  A x B lexicographically.  are not a l r e a d y compatible  there must e x i s t i  some  ( i , j ) ,In  (*)  E ^  Let  (i *J )  and  j  E  A x B  (i ) Q  '.that  Q  0  Q  >0  b  e  t  n  ^ P^  .. ( J - l ) jf.P 0  and. p (  J )  j  )  .| E  l e a s t such element.  e  .-Then  :  such t h a t  E -  { F  °  (  i  .  )  Obviously  i  , so o b v i o u s l y ;  • , . j T h i s i m p l i e s by the m i n i m a l i t y of  • ••  •  > 0  (i ,J ) Q  0  •  (7.5)  P  U !  °- ><K J  ( i  °  ;  S i m i l a r l y we o b t a i n  (7.6)  E  Case 1. Let . E'  Suppose  o  From (7.5), viously  = E E'. o  E  < F  °  <_ F  - F  °  , and l e t H = 1 - F  i s a p r o j e c t i o n contained  F . < H •. x  ° .  Choose a u n i t a r y  U e G  the - c o n d i t i o n s of C o r o l l a r y 7-9 f o r E'.. •. .'. . ,.,,' " o 1  :  in  H  which  , F J  . o  ° . /'Gbsatisfies and H • -'  .  1  65. Consider  the element  projections, . .  P' K  T  = U TU  = U*F, U *  , and the  , and  F'( )  =  J  i F' k=l F^^  .  Let  r e p l a c e d by '  v.  From the c o n c l u s i o n s of C o r o l l a r y 7.9 F  .  w  K  '"(*'••)• denote the statement (*) above w i t h p'(J)  corresponding  (J) _ '(j)  for  F  j < j  . (a) . F' .  ,we have t h a t  ; and t h a t e i t h e r  Q  < E.'  , i n which case  J  o  . o  or (b)  E^ • \< P'. o j o  , i n which case  J  = F(Jo*- ) 1  E  F' ,<<U ° .  , + Ef .<_ o  So (*')  does not h o l d f o r Now  c o n s i d e r any  • ^ (a) m a l i t y of  If  (1 *J ) 0  0  ~ (b)  m So again,  If  (i -l) 0  (*^) /  j < j  Q  (i ,J 0 Q  O  •  (i,j)< (i ^J ) 0  , .F' ( J ) = p ( J )  •  0  , and by the m i n i -  >(*') does not h o l d f o r j = j  Q  , -we must have  (jQ-i)  does not h o l d f o r  i _< i  (V1) (i,j)  (i,j). Q  - l > and.  (1)  •'.  (i -l)  (J-l)  D  Case 2.  Suppose  E  | F  •  By the m i n i m a l i t y of We  , > we  (i *J ) 0  0  have  F  (V1)  (V1)  _< E  can then argue i n a s i m i l a r manner t o Case 1,  using  (7.6)  i n p l a c e of (7.5). In any that when we (i,j) rem  replace  can f i n d a u n i t a r y  T  f o r which (*)  by  holds  U TU  U € G  , the number of p a i r s  i s s t r i c t l y decreased.  and  T  Theorem 7.10  are h o t  bra c o n s i s t i n g of tion.  The  theo-  Let  G  Let  L [0,1]  a c t i n g on  w  be any f r e e ,  c o n s t r u c t e d from  M  w i l l not h o l d i n g e n e r a l i f  simple.  group of t r a n s f o r m a t i o n s  of  and  Chapter X I I ) .  G  M  be the von  let  T^  e r g o d i c , measure p r e s e r v i n g [0,1]  and  let  i n the standard  denote .the corresponding  ,,  "  T  on  G , which  T(T ) = I*  , Iri p a r t i c u l a r d e f i n e d by  by m u l t i p l i c a -  G  be the f a c t o r  manner (see  [9],  *  i s a f a i t h f u l trace  (7.7)  Neumann a l g e -  L [0,1]  For any bounded measurable f u n c t i o n we  such  then f o l l o w s e a s i l y by i n d u c t i o n .  Example 7.11. S  event we  1  h  h  on  element of  G  [0,1] .  There  satisfies  h(x)dx  .  o consider the;functions . '  ;  •  f  and  g  x  Letting  d e n o t e 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 0 _< X £ 1  have f o r  ,  the  spectral projection  E(x)  of  T,  the  spectral projection  F ( x ) ,of  T  and X <  E(X)  0  .  =1  F(x)  =  Appling  [l.l)  T(U*F(X)U)  (7.8)  for we  s  ='T(E(X))  X  >  •  '  =  1  T  *[o,x]  x([0,X/2] U [l-X/2,1]^  E(x)  U T U  (7.8)  and t h e f a c t t h a t  Ty  and  U*F(x)U  3  <  X  <  +»  T  f  and  e G  U  existed a unitary  U  such  w e r e c o m p a t i b l e , we w o u l d h a v e f r o m T  and  T since  3  for  0  <  X  <  +•  .  that/the  f  generates  abelian, while  h(x) = h(l-x)}  T  would have  v o n Neumann a l g e b r a s  are u n i t a r i l y equivalent. T  . y  is faithful,  E(x)  =  and t h e r e f o r e  contradiction, i s i" ; m a x i m a l  for  .  B y t h e 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 *  t e d by  = Q  F(x)  =  I  that  U T g U = T^  '  see t h a t f o r any u n i t a r y  Consequently, i f there *  T  But t h i s  {T^ : h € L [0,1]}  '• g e n e r a t e s  which i s not maximal  w  {T : n  that  generais a which  h e L [0,1]  abelian.  W  ,  68. 8...  Calculation of  The  p  G  f  o  r  semi-finite  calculations f o r  p  made i n t h i s s e c t i o n  be b a s e d on t h e f o l l o w i n g r e s u l t f o r  Lemma 8.1  1  ([3L  a normal f i n i t e that  y = Tg  P r o p . 2.3),.  Let  trace  Let  and  T  v = T  . t  factors.  G  f o r some  proved In [3].  p  be a  y  will  W -algebra  and S  v e  and  P(u,v) = T ( | S T | )  with  be  T € G  such .  +  Then  .  f  We w a n t t o t a l k a b o u t t h e t y p e o f s t a t e s  used i n t h e  'i  a b o v e lemma,  s o we (make t h e f o l l o w i n g d e f i n i t i o n . •  \  D e f i n i t i o n 8.2. finite  trace  Let  .T  elementary with  •  .  be a  An e l e m e n t  respect to  T(E') < »  s u c h that..  G  , where  T  W -algebra with  a normal, semi-  y.. o f  be c a l l e d  Z  will  Q  , i f y, = T E  to  T  will  of  a l w a y s be  .  Lemma' 8 . 3 .  for  G  +  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  S e G  i s the range p r o j e c t i o n  In the remainder of t h i s s e c t i o n a f a c t o r and we  f o r some  G  Let  G  and l e t S X a n d any  V  '' '•' ' :  R  e  G  with  ,•.•'•/'.•  be a f i n i t e ; f a c t o r w i t h T  normalized  be s i m p l e e l e m e n t s o f ; G  ||v|!'_< . l ' ; 'y  ' /  | T ( S T V ) I,•< d ( S , T )  .  trace  Then, '•-•u.  .  S  Proof.  By Theorem 7.10, choose a u n i t a r y *  and  U € G '  m  U TU  are compatible.  Let  S = £ c.G. ' i=l  and  so t h a t  S  *  U TU =  m E d.G. be the r e p r e s e n t a t i o n s given by Theorem 7.5. L e t i=l ' -.gj_. J= TCGJ^) . By the u n i t a r y i n v a r i a n c e o f T and C o r o l l a r y 1  1  7.3,  .  •  1  a ( S , T )  =  o ( S , U  T U ) =  =  m E d.g 1=1  T ( S UTU.)  <  The  .  c  i  1 1  lemma now f o l l o w s e a s i l y from Theorem 4.3.  Lemma 8.4.  L e t G; be a f a c t o r w i t h a f i n i t e t r a c e  Let  v  u. and  v = T  t  f o r some (a)  and  T  be Elements of S  and  T € G.  p(y,v) <_ d"(u,v)  E Q . such t h a t .  ti = T  T g  . .  and  Then:  , and e q u a l i t y h o l d s i f ^ S  a r e compatiblej (b)  Given any  e > 0  , there e x i s t s  a e,Int(G) .  such that  p ( U , v ) > a(u,v) - e  .  a  . Proof.  We can assume that  . 0 < e.< 1 ' • • L e t  T(1) = 1  Choose any  r =-(||s|| + ||T|| + 1) i A  I t follows  e  with  immedi-  a t e l y frpm t h e ^ s p e c t r a l theorem t h a t the simple elements of G  +1  are norm dense i n G  elements, • S '  and  T'  +  of  .Therefore, G  +  such t h a t  we can choose simple  70.  Ils-s'll < h > ll - " <' T  T/  e  Prom Lemma 6.14,  (8.1)  and  |a(S,T)  o(S',T')|  -  e/2  <  ,  a s i m i l a r type o f c a l c u l a t i o n shows t h a t  |ST  For  any V e G  ISTV  and  since  -  with' ', ||v|! <\  •- . S ' T ' V I !  the s t a t e  T  < e/2 .  S ' T ' ||  _<  we have  HST -  has norm  S'T'  1  e/2 ,  ||||vl| <  ,  v ( - | 8  |- 1 T | ( \ S T V ) J ^ J T ( S ' T J V ) |  2  .•"  : s  s  :  -  •  v-  . From (8.1),  ""'  (8.2),  •  " 1.'^  ':=  /  "IT(STV-)..-  |T(STV  < e/2  -  T(S'T'V)|  S ' T ' V ) I;  .- ' "•"'"  and Lemma 8.3,  TISTV).|<  <  |T(S'T'V) I + e/2  a{S',T')  + e/2 < a ( S , T ) + e  7 1  T h e n . f r o m Lemma 0 . " 4 ( a ) a n d Lemma 8 . ' l , ••  •  p(u,v) = T | ST | < o ( S , T ) +. e = o(M,v) +. e  Since  e  was c h o s e n a r b i t r a r i l y ,  •:  p(u,v) < o(H,v)  If  S  and  T  are compatible theequality follows  from C o r o l l a r y 7 . 3 .  and  U T U  directly  This proves ( a ) .  By T h e o r e m 7 . 1 0 , S  .  choose  a unitary  U e G  such  are compatible.  C a l c u l a t i n g a s i n ( 8 . 2 ) w e . c a n show  (8.3)  Let.  JT(SU*T.U)|  a  |T(S'U*T'U)|  be t h e i n n e r a u t o m o r p h i s m  v° = T * 7.3,  >  u  .  From  ( 8 . 1 ) ,  T U  ( 8 . 2 ) ,  -  e/2  induced by  U  Lemma 0 . 4 ( a ) ,  .  Corollary  ;  and Lemma 8 . 1 ,  p(n,v ) a  '  :' \ %  which proves  = T( | S U * T U | -2  )  ;  |T(SU*TU) |  | T ( S ' U * T ' U ) I - e/2  = .a(S',T' ) - e / 2 > a ( S , T ) - e  •/;.->'../\ = a(u,v) - e  (b). /;  2  ;  ;  Then.  ...  that  72.  v  be a s e m i - f i n i t e f a c t o r a n d l e t u. a n d  Let G  Lemma 8.5.  J.^  be e l e m e n t a r y s t a t e s o f  8.4  .  Then,  t  where  S  be t h e r a n g e p r o j e c t i o n s  of  hold.  Proof. Let  \A = T  Suppose  E  a n d .F  spectively.  Since  E  g  , v = T  and  F  are f i n i t e  r e s u l t that the projection  G = E U F  Lemma 7-35).  v'  U  ( a ) a n d ( b ) o f Lemma  and  v  L e t y,'  and  ner  automorphism  in  Gr,  G  a-, o f  induces  and  +  T  re-  i t i s a w e l l known ( s e e [9]  denote t h e r e s t r i c t i o n s o f  G^,  automorphism o f  S  i sf i n i t e  respectively t o the f i n i t e  be a n y i n n e r  T € G  and  .  We e x t e n d  as f o l l o w s .  a'' , we t a k e  a  G  factor  .  Q  a'  Let . a ' t o an i n -  I f the unitary  as t h e i n n e r  U  automorphism  I  induced by t h e u n i t a r y  U + (1-G)  r e s t r i c t i o n of  G Q .  v  C :  to  a  .  Then  (v')  Moreover,  a  i s the  •, .•  /  v (G) A  v(a(G))  =  and s i m i l a r l y From  =  v(G)  u ( G )= 1  and t a k i n g  T ( S ) 2  =  u(l)  =1  •'" /  (y,v ') = a  P  =  .  ( [ 3 ] P r o p . 1.10),  •  T(SGS)  =  (u',(v') ') a  P  .  a ' = t h e , i d e n t i t y , a u t o m o r p h i s m ^ ..•  ' • '•; V ;  •  •  : :.;;v  P(W,V) =. ,'p(u',v')  F i n a l l y , we s e e f r o m Lemma' 6.5  that  •<;••:"•)  • /.  ,  •  a(u,v) = a(u'»v')  The lemma now f o l l o w s d i r e c t l y from Lemma 8.4.  Theorem 8.6.  • Let G  a subgroup o f any  u, v e £  be a s e m i - f i n i t e f a c t o r and l e t  Aut(G)  which c o n t a i n s  Int(G)  G  be  .' Then, f o r  Q  p (u,v) = a (ti,v) • G  G  We f i r s t prove the r e s u l t i n a s p e c i a l case.  Lemma 8.7-  Proof.  Theorem 8.6 h o l d s i f u  Choose any  elementary.  a e G  .  and  v  Then o b v i o u s l y  i s also  From Lemmas* 8. 5 and 6.10 we see that 5  P(u,v ) < a(u,v ) < a (u,v) a  (8.4)  '  Now choose any  a  G  p ( u , v ) < a (u,v) G  G  e >0  Let k € r(G)  a' (y, v) > o (u,v) - e/2 ',- and choose k  f o r any t r a c e  .  3.2.,  Then from Lemma  ye. I n t ( c )  are elementary.:  T  on  such that  G/.. ...•,  .  be such that f3 e G  so t h a t  T  = kT  'Applying Lemma 8.5, there e x i s t s :>X.:'.'.'"  '; • ';•:./;•. ••• '•  • (8.5)  P(U,V  '  P y  e/2  ) > o(u,v )3  74  e/2  = o (y,v).k  r >  Since  Int(G.) c G  follows  , we  f r o m (8.4)  Remark 8.8.  and  The  (n,v)  a  -  «  .  have t h a t  ^  e G  , and  the  (8.5).  definition for  a(y,v)  can  obviously  made f o r any  p o s i t i v e , normal, l i n e a r f u n c t i o n a l s  for  If a state  states.  u = T  .  for  any  positive  k  , the  c  , and  we  as w e l l  functional  ky  a(y,v)  •  I n Remark 2.5  Jkr  p(y,v)  •  It follows  as  = T JXS  have seen t h a t  positive  Jkr  k  we  and  r  that  Lemma 8.7 we  "F  = »/K  T  , a(ky>rv) =  noted that  n o r m a l , l i n e a r f u n c t i o n a l s , and following  be  S  C o n s e q u e n t l y , f o r any  i n the  lemma  p(ky,rv) =  holds f o r p o s i t i v e ,  w i l l apply i t i n  this-form  proof.  P r o o f of-Theorem  8.6. i  ,  . '  ••  ,  Let  T  be  For S  t  any  .  Let  = SP  .  n n p r o j e c t i o n and range.  We  n > 1  Since S  define  n  on  be let  the  G.  w h i c h we  u =  p r o o f . "Suppose t h a t  {E(x)}  integer  •  a semi-finite trace  f i x e d throughout the v = T  /  and  s p e c t r a l r e s o l u t i o n of  P ,n  (E(n)  - E(i))  keep  S  , and  .  let  S  i s square,integrable, .P is a finite a ... , " 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 the  functional  u  n  = u  • .  p  n  Then  u  n  =  T  S  n  so i t i s e l e m e n t a r y . 5.2(d) a n d 6.4  ST  Since  , we o b t a i n b y Lemmas  that  '"'  TJ  (8.6)  7  S  Analogously,  S  n  .  S  we d e f i n e p r o j e c t i o n s  elementary f u n c t i o n a l s  v  f  (8.7)  , operators  corresponding  n  t V  T  ' '••  to  v  T  .  , and  rf  We. o b t a i n  '  n  .From Lemmas 6.1 a n d 5.3 we h a v e  (8.8)  Now u s i n g  ;.;  ( S ) = f ' (T  ( a ) ) d a , and  2  T  2  2  2  q  (8.6) - (8.8) a n d t h e m o n o t o n e c o n v e r g e n c e t h e o r e m  u ( P  T(S )  ) =  N  v(Q ) n  2  \  : 1  t  T ( S  .  n  n  2  ) . =  u(i)  =  n  /  n  |  f o ra l l  o (u ,v )  T  .  r i  G  n  Then g i v e n any  n  e > 0  k  a (u,v) G  /  a(S,T) = o(u,v)  ^.•a (u,v)  n  .  :  - :  a (u ,v ) k  I  r  a(|i ,v ) = a(S ,T )  Similarly  T(S ) = J"(7 (a)) da  we c a n Choose  k > 0  .  .  and t h e r e f o r e  an i n t e g e r - m  so t h a t  |i(P ) > 1 m  (YQ)  /and  2  ( 8 . 9 ) a ( M , v ) > o(u,v). - |  .  Let  .  G  m  a  • •  m  be any element  of  G  vCQj > 1 - ( ~ )  , and  2  Then  "V  I  ( n,) v  ( fo ^)  =  a  v  (  = ^  a  » by Lemma 0. J  -1  .'•(-a' (0 ))  V  ;  ,  .  1  1n  :. •  and  v (a- (Q )) = v ( Q j > 1 a  1  m  (^)  .  2  1.9(c)) to obtain  We can then apply ( [ 3 ] Prop. \  -  <_ c/2  |p(u,v ) - P ( ^ v ) | a  m  .  , for a l l a e G  "  |p (u,v) G  •  "  1 1  <  G  m  '  ,  ''  «/2  ..  /  u  v  and  v  •'  )  =  P ( V m)>^ (u G  From formulas^ ( 8 . 9 ) and s i n c e  e  v e T^r  l  v  G  l n  ,v )  .  m  ( 8 . 1 1 ) we have  |p (u,v) - a ( u , v ) | < e G  G  was chosen a r b i t r a r i l y the p r o o f i s completed.'  D e f i n i t i o n 8.Q. and.  P (%>^ )l  a p p l i e d t o the elementary f u n c t i o n a l s  gives  (8-  ....  '• '  (8.10)  '  :  -,.  »  and t h e r e f o r e ,  Lemma 8 . 7 ,  ;  ~.  Let G  be a s e m i - f i n i t e f a c t o r and l e t  We say t h a t > u  .and  v  are compatible,^  u  • ' v.  i f f o r some s e m i - f i n i t e t r a c e where  S  and  Lemma 8 . 1 0 . Then  Proof.  • 7 7 .  •••  T  T  G » u = T  on  G  and  v  = T  T  are compatible o p e r a t o r s .  Suppose t h a t  p(u,v) = a(p-,v)  p. and  v €  are compatible.  .  T h i s f o l l o w s d i r e c t l y from Lemma 8 . 4 ( a ) i f the Radon-  Nikodym d e r i v a t i v e s of  u  and  v € G  .  F o r the g e n e r a l case  we can employ the same type of approximation procedure  used i n  p r o v i n g Theorem 8 . 6 .  Theorem 8 . 1 1 . Then, f o r any  Proof.  Let G u  arid  be a f a c t o r of type I or of type 1 1 ^ . v 6 E  (a)  d(ji,v\) = V2[  (b)  If  p. and  Q  :  •' *  l-o(n,vj v)} are compatible,  d(u,v) = d(p.,v)  Immediate from Theorem 8 . 6 , Lemma 8 . 1 0 'and f o r m u l a  (3.1).  Example 8 . 1 2 . separable.  Consider the a l g e b r a  Let T  be t h e t r a c e on  i t s value = 1 on minimal p r o j e c t i o n s .  G = £(H) where G  H is  , normalized so t h a t Let S  Hilbert-Schmidt operator ( i . e . T ( S ) < »  ).  be a p o s i t i v e , I t i s w e l l known  that  S =  where  i n the sense of s t r o n g  convergence,  ( a . ) i s a n o n - i n c r e a s i n g sequence o f p o s i t i v e  real  78 E (a.)  numbers such t h a t  < «  (E.)  , and  is a  sequence'  of non-zero, m u t u a l l y o r t h o g o n a l , minimal p r o j e c t i o n s that  I E 1=1  = 1  1  determined by of  S -.  We  . S  Morever .  the sequence  (E(x)}  Let  (a.)  such  i s uniquely  be the s p e c t r a l  can a p p l y (7-4) t o see t h a t .  resolution j  k  1 - E(X) =  E E^  , where  k  i s such t h a t  a  k  > X _> a  k + 1  .  It follows that f(x)  Therefore, i f f ( X ) _>  n  +  1  7(n+r) = a  n +  X > a  = the number of  , f(x)  n + 1  a^'s > X '.  < n  ; and i f  X _< a  •  1* f o l l o w s from Lemma "5,2(b) t h a t  ^  > where  n  S = E a.E., •• i = l . t o r s we have  ,  i s any non-negative i n t e g e r and' 0 <, r < 1  es  Then, i f  n + 1  oa  and  T = E b.F., . i=l 1  are two  such opera-  .  1  ' I '\i  at  a(S,T) =  E a.b, i=l 1  .'  -v-'M  1  A s i m i l a r r e s u l t of course holds when  H  Is f i n i t e d i m e n s i o n a l .  Since any s t a t e y On G = T f o r some H i l b e r t + S e G » t h i s g i v e s us an immediate c a l c u l a t i o n g  Schmidt  formula f o r  N  d(u,v)  i n the case of a type I f a c t o r .  .  V •''-"'". ' . v  • : •' Example 8.13. T  .  ; Let  G.  E  = / X [ o a)  .Therefore, that  ^  ^  i f E  E € G  and  F  a r e a n y two p r o j e c t i o n s  Note t h a t i f a / B e r ( A u t ( G ) )  F  onto  u = l/a(T )  p(u,v) = 1  .  E  and t h e r e f o r e  =  1  v  v = 1/B(T )  that  F  s  (o'(E,P))-  7T(E)/T(F)  .  . '  , t h e same.type o f c a l c u l a t i o n  Of c o u r s e t h i s  s i n c e we c a n i n t h i s  and  E  a n d Remark 8.8  TOTP  such  .  p ( u , v ) = ; o ( u , v ) = 1/Ja&  =  e G  , we h a v e  the states  .  diately  ,  denoting the c h a r a c t e r i s t i c function of a s e t )  we h a v e f r o m T h e o r e m 8 . 6 • " 1  shows t h a t  T(E) = a < «  such t h a t  a(E,F) = a  •V  trace  that  a = T(E) < T(F) = 3 < »  Now i f we c o n s i d e r  79..  be a s e m i - f i n i t e f a c t o r w i t h  F o r any p r o j e c t i o n  we c a n e a s i l y v e r i f y  f  .  '.  case f i n d onto  u  i s evident  an a u t o m o r p h i s m .  ,' immesending  ,  _ •?:  '  CHAPTER I I I  ;; .  APPLICATIONS TO INFINITE TENSOR PRODUCTS  9.  Isomorphisms of Products;  D e f i n i t i o n 9.1.  Let  Definitions.  (G,(c^))  ducts f o r the f a m i l y of  .  and  (fl,^))  W -algebras  be two p r o -  ( i)i i  •  G  €  W  e  s  a  that  v  they a r e : (a)  product isomorphic, I f there .exists an isomorphism  from In  G  6  onto  satisfying  <j)a = 8^  such a case we w i l l w r i t e  (b)  fora l l  i  i  (G, (a^).) ~ (©-, (6^))  e  i e l (c)  G  onto  0  satisfying  <|)(a (G )) = S ( G ) i  i  i  l  ;  weakly product isomorphic, i f t h e r e e x i s t s an  d) from  (}>  i  •>  isomorphism for a l l  ; permutably  morphism  product isomorphic, i f t h e r e e x i s t s an i s o - ;  dp from  satisfying  G  onto  fj.' and a permutation • /  <|»(a (G ).) = P ( ' ) ( G ( ) ) i  1  T  i  i r  f  o  r  a  1  1  1  6  ir o f >(  1  i  I  l n  other words they are weakly product isomorphic a f t e r a r e a r rangement o f the (d)  );  a l g e b r a i c a l l y isomorphic i f there e x i s t s an  from  G  onto It  it  G^'s  isomorphism  B i s obvious that:- (a) - (b) - ( c ) - (d)  ,  , and  i s not hard t o show.';that, a l l i m p l i c a t i o n s a r e s t r i c t .  .  N o t e t h a t i f (G, (oj.')). a n d • (fi»(^))'  Remark 9.2.  are  w e a k l y p r o d u c t i s o m o r p h i c , we c a n d e f i n e a u t o m o r p h i s m s ^ <p^ of  f o r each'  We. t h e n h a v e  is  '. . ;  1  ±  L  1  . <J>a£ =  • •  'x  by  = P5 <|)a (A ) , f o r a l l A  $ (A ) ±  i e I  1  ±  6 G^  , o r i n o t h e r words t h e diagram  1  commutative f o r a l l (  V-v  i  e  A similar^ r e s u l t  i  ^  l  .  o f course holds f o r permutable  •product isomorphisms. ...  •  .  Remark'9-3. that 'is, <f>  r  ! /  I t follows  immediately from the d e f i n i t i o n s  a product isomorphism i f v e  = .^(v.^)  f o r the product  i s a product isomorphism  then  = ^(v^  preserves product states.  f/om  (iB, (P^)')  (G,(a^))  f o r .the. p r o d u c t .'; (G, ( c ^ ) )  That  ». a n d i f  onto .  I  f <j> i s i n  * s t e a d a.-we,ak- p r o d u c t ^ i s o m o r p h i s m , ' ^ v-'' = :• ®j( v^-J'-.' f o r t h e .. '<j>^,' .defined  i n Remark 9.2.  •/'['•'i- ' : : :  .  '••.'•••.•  ••• •«'•' /  (G,(a,d),)) ~ 1  1  Proof.  •.^•%i^v !  ® (G^u/) i e l  1  .  F o r each  i  e  ' :.1 .•" .Then o b v i o u s l y  [\[rid)71,xi  :i>  morphism,  )  .  1  i  there e x i s t s  ®(R ,x^)  an isomorphism  1  . Let  where  i  ] e QCu^)  = ^ t ^ (G ) <  *i [ ^ ^ x ^ ] e Q(u^ )  l choose  ft be the von-Neumann a l g e b r a  •I •*.  i .  :  1  = ^(G^)  , and s i n c e • d)^  i s an auto-  So by the d e f i n i t i o n o f is from  G  onto  ft  . ..;  ®(G >U ) I  I  satisfying .  Mi  \i<a (A ) = i(fid)~ (A ) , f o r a l l  iel-  1  1  Then, Aj^ e G^ • i !  1  i  tya^^)  =  " ^ j ^ ±) A  lemma.  ra 1 1  ®(G ,U I  -.  1  €  )  I  1  and'•  proves the '  • •' •  '  '  ^ (G - ^ U J ) and ® (GJ,V^) are weakly product •' l v ,' i e l isomorphic, i f and o n l y i f t h e r e e x i s t s f o r each i e 1' , an i  a r e product  e  d>  1  of  G.,  Proof.  such t h a t •'  1  ® (G i e l  1 }  uJ'  and  ® (G^v., ) i e l . .• 1  1  isomorphic.  Vs.••..•'. • • • '' •//•  o  .  automorphism "/';••};.  f  ;  " A\ V ' - . -;}>'. Lemma 9- 5. ;  .  A  , which by; the d e f i n i t i o n o f  -SS;; . S$' -  = •i( i)  and A^ e G^ .  '  ..This f o l l o w s immediately  /  •'  from Lemma 9-4  and Remark  9.2.  s  N  v. •  10.  Main R e s u l t s on Tensor Products.  .. •  •  In t h i s s e c t i o n we apply the product formula for'. •p  1 \  5  which we obtained In S e c t i o n 2 t o develop n e c e s s a r y and  sufficient  c o n d i t i o n s f o r two l o c a l tensor products of a .  given f a m i l y of  W -algebras t o be product isomorphic.  same r e s u l t was proved  The  i n ( [ 3 ] , Theorem 4.1) under the assump-  t i o n s t h a t the a l g e b r a s are s e m i - f i n i t e , and i t was shown t o be a non-commutative e x t e n s i o n of Kakutani's Theorem on i n f i n i t e product measures [ 6 ] .  The o n l y need of the s e m i - f i n i t e n e s s  r e s t r i c t i o n was i n i n v o k i n g the product formula f o r  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 ] f o r a p r o o f o f t h i s p r e s e n t r e s u l t . We w i l l however p r e s e n t a complete proof here, p a r t of i t , p  :  simplifying  and at the same e x t e n d i n g the product f o r m u l a f o r -  t o the case of an i n f i n i t e number of factors,.  o b t a i n necessary and s u f f i c i e n t  F i n a l l y we  c o n d i t i o n s of a s i m i l a r  nature  f o r two. l o c a l tensor products t o be weakly product isomorphic . and t o be permutably product  isomorphic. •  D e f i n i t i o n 10.1. (G^) space i!'  .  i € l  H  Let .  Let ( | >  Let  '  /  (G,(a ))  .  be a product f o r the f a m i l y •  i  be any r e p r e s e n t a t i o n of  u e  .  we d e f i n e : ••'  J--'  •  .. i A  €  G  i  }  ' '  J  A  '  (j) = the r e s t r i c t i o n U  on a H i l b e r t  subset  J  of  ,  ';• v  .<'  G x.= the . W - a l g e b r a generated by  ;  G  For any non-void '  :  '  {a ,(A ) ; , i j  - / ' ^  cj> t o '. G:-:  = the r e s t r i c t i o n of \  1  :  :!  to, G • J  :  1  '  e J ,  Lemma 0.1 shows t h a t G  on  J  d)  H  Lemma 10.2. " ~  Suppose  (G,(a.,)) ~  .  Then t h e r e  Ji c I  v  J  exists  some s u b s e t  p(y jV^) J  with f i n i t e any  ' , and l e t  on a H i l b e r t y.  ', a n d  1  0  >  J c I 'l e t  H^  each  x' = • ® ,(xi )  y^  on  1.2 t h e s u p p o r t  = 1  , E  J  G^ of  G^  2. E  N  E. = 0 to  of  v  y  x^  i n H^ H =  I  on  i n Chapter there  compliment,  fora l l  i e J  .  1  i t I s obvious that  x'  i  i sorthogonal J  J  .  i n G Since  Ex' = 0  to  y = 0  E  .  v(E ) = v (E ) J  ... . S i n c e  i n , some o r t h o n o r m a l b a s i s o f  u  . ;  1  T h e n , b y o u r a s s u m p t i o n a n d Lemma  y 2  inducing  R e c a l l that  x' = x.  G  '.  algebra  1  , with f i n i t e 1  of  ® (H. , x . ) i e l  •  .  , and t h e r e f o r e  the fact that  for a l l  .. •  -as d e s c r i b e d  7  a n d we m u s t . h a v e  t r u e f o r a l l x'  = 0  J  a s a v o n Neumann  H  ||E x'|| = y ( E ) = 0 J  with a f i n i t e  denote t h e support  , where  1  ® (x. ) induces iel .  Therefore,  J  with a vector  J = J(x') r o f iel  induces  I  p(y , v )  be a n y e l e m e n t o f t h i s b a s i s .  e x i s t s a^subset  Since  of  a s a v o n Neumann a l g e b r a o n v .  L e t x'  such t h a t  be a n y e l e m e n t o f  .  denote t h e support  space  G  E^  -Choose a n o r t h o n o r m a l b a s i s , . f o r 0.  J  1  E  L e t u. h e  compliments.  We c a n c o n s i d e r  .  v  Suppose t o t h e c o n t r a r y t h a t  For ;  and l e t  •  1  iel  compliment such t h a t  Proof.  1  ® (y. )  G  G  ® (G.*,|i.) i e l  1  the element o f -2 = '^E .  i si nfact a representation of  J  H  J  J  this i s we h a v e  , which i s a c o n t r a d i c t i o n  i s a s t a t e . • '.;>-''  ;  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 p r o d u c t s of numbers. - I f  (TTr.)  as  F  .Suppose  . ( j_)j_ ;r/' i r  there  deleting.these  m  i  y  l  I. exists  iel  •  1  ;.  converges \ .  iel  '  1  _'  ;  a f i n i t e number of zero f a c t o r s and a f t e r  t h e ' r e s u l t i n g f a m i l y o f p o s i t i v e numbers con- .  v e r g e s ' a s above. ;  TTr, iel  In t h i s case we d e f i n e TTr,  f o l l o w s from these ( d e f i n i t i o n s t h a t '  i  j  E | l - r . | l< » iel •  only i f  a  TTr;., = r .  We then d e f i n e •  We then say t h a t TTr;  r. = 0  are-only  f  subsets o f  .... if  a  f a m i l y of non-negative numbers w i t h  s  e  l e a s t one  .  S  converges i f C' >. j  runs over the f i n i t e r  i  e  iel  as a p o s i t i v e number :  at  r  TTr.,  of p o s i t i v e numbers, we say t h a t lim F  ( j . ) i -~I "  .  e  I  = 0 . I t  converges i f and -  . : '.-M-  '  i t i s easy t o v e r i f y t h a t i f  1  TTr. iel  - r  and  1  TT s. = s iel  then  1  We now c o n s i d e r  I I (r.s. ) = rs iel1  . • ':;  1  the f o l l o w i n g s i t u a t i o n ;  the no-  t a 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  e  I  be an a r b i t r a r y i n d e x i n g  f a m i l y of  W -algebras  elements of  .  A[(G )]  set and.let  Let^ (Mj.)i j_  8 1 1 ( 1  e  ( i)i i G  b  ( i)iei» v  b  e  e  -  a  t  w  o  .  i  As a n o t a t i o n a l convenience we w i l l l e t ®'. denote ;  ® iel  (similarly for ' • ,  E  and "\~\  j.: -  1  Lemma 10.4. . The f o l l o w i n g c o n d i t i o n s on are  equivalent: i , 2.; E[d ( u , v ) ] . < G  (a)  n  x  i  i  09  (u.)  and  (v..)  X  (b)  E [ l - p ( u , V ) l - < « ', i  i  1  (c) T T P ( u - v ) .  converges]  i  i 5  for  any subgroups  Proof :  i  G^.  of  Immediate'from  Theorem 10.5. s t a t e on  Aut(G )  (3.1) and Remark 10.3.  Suppose t h a t  fl^G^y.^)  .  Let  v = ® ( )  Suppose  .  Then  , a convergent  (G,(a^)) ~ ®(G ,y ) i  !  e x i s t s as a p r o d u c t  y = ^(UJ^)  p ( l l , v ) ^["^(u^^v^^)  Proof.  .  1  .  j L  product.  We c o n s i d e r  y  i  and  v  as elements  and l e t F  of  E  be any f i n i t e  [<Mx,y)] e Q(y,v)  . • Choose any subset o f  I •.  A direct  calcula-  t i o n shows t h a t [<|> „', (x,y) j , e Q(y ,v ) and t h e r e f o r e F F i ' p ( u , ^ v ) _> l ( l y ) l • T a k i n g the supremum over a l l elements x  of  Q(y,v)  we o b t a i n  (10.1)  '  p ( u , v ) > p(u,v) P  I t i s obvious t h a t  y  F  =  .  F  ® (y. ) ieF  and  • .  "  F v =  1  ® (v., ) ieF  on  1  •pi  ® (GJ) = G ieF 1  i . T h e r e f o r e , by Theorem 2.4, which < . yS-y  extends  from two t o any f i n i t e number o f factors', by an obvious i n d u c t i o n and standard a s s o c i ' a t l v e t y arguments>  (10.'2)  P ( U  F  , V  F  )  ^ T T p ^ i V i )  .;^f'V  By  (10.1) and (10.2),  {TTp^.,  (10.3)  p(w,v) < i n f ~, ieF  1  ) : P  Now by Lemma 10.2, choose a subset p(u , v ) .>; 0,  p l i m e n t such t h a t  J  gument t o the a l g e b r a  so  j ' has f i n i t e  subset o f  I  compliment.  with f i n i t e  .  subset of  By (10.4),  Moreover  J} > 0  J c j'  j' - J  F  where  ieF  F , F'  spectively.  J  r e p l a c e d by  i s always  lim(TTp(  i s finite  J'- ;  u  v  1  1  )) = i n f ( T T . p C u ^ V i ) ) p'  ieF'  T h e r e f o r e | |p(u^,v^)  (10.3) as  subsets of converges  P(u,v)  p o s i t i v e number t i v e numbers  < 1  < [ \ P ( U  ±  , V  ±  )  '.V  .  Let k  Choose any sequence  i  i  be any  (k ) of posifi  1 X k = k , Let I = f i e I : n=l ° . T h e convergence of | | p ( u , v ) and Lemma  < 1  so t h a t  n  d ( u , v ) > 0}  i ', j ' re-  and we may w r i t e  the other d i r e c t i o n . .  ,  ''  /  We now prove  Then, s i n c e the  1',  runs over the' f i n i t e  (10.5)  ,  1  so (10.4) h o l d s w i t h p  com-  (10.3) t h a t  i  value o f  1}  By a p p l y i n g the above a r -  a finite  J'.= { i e I : ' p ( u i , v j L ) > 0}  Let  of  , we o b t a i n from  1  i n f { I 1 p(u,,v,) : F ieF  (10.4)  J  .  J  G^  a finite  1  1  i  88. 10.4 show t h a t f e c t i o n from tion  g  I I  on  i s a t most countable.  i n t o the p o s i t i v e i n t e g e r s .  Q  {  , for  \(i)  Then choose, f o r each I  3 V  I  i  >  e  f  i e I -I  o  r  1  6  X  o  '  >  e I - I  implies that  Q  u  = j_ v  i  [$,(x,y)]  g(i)[p(u ,y )} ±  c e r t a i n l y p o s s i b l e by the d e f i n i t i o n of i  Define a f unc-. •  Q  l , an element  |(xi|yi)|  satisfying  )  be an i n -  I by  1  Q ( u  Let y  •  p  ±  .  of This i s  :  and the f a c t t h a t  By m u l t i p l y i n g the vec-  t o r s by s u i t a b l e s c a l a r s i f n e c e s s a r y we can assume t h a t . 1  (10.6)  ( i l y i ) 2 g(i)[p(u v )] x  1 3  i  , for a l l i  e  l  v. I t i s easy t o see t h a t T T g ( i )  converges and t h a t i t s v a l u e  2 TT • T h e r e f o r e , (10.6) shows t h a t "^(x^^ j y ) n=l • converges and t h a t ': is  k  =  k  n  1  • /  TT , (  (10.7)  ,We have then t h a t  x  ilyi)  I'kCTTpd^iVi)]; y  E [ l - (x^|y^)j]  f o r the f a m i l y of H i l b e r t spaces underlying  space o f  C -sequences. Q  states  and  y  (H^) , where  T h e r e f o r e , the v e c t o r s  are elements of  i  v  I s the  ( y ^ ) are e q u i v a l e n t x = ®(x^) and  H = .^H^x.^)  respectively.  .  converges, which shows t h a t  dp^ , (x^) and  y = ®(y )  for a l l i e I  which induce the  Then-using  (10.7),  "  8 . 9  P(u*v)  Since  k  >  was  |(®(x )|«(y ) 1  j =| | ( x | y )  i  i  chosen a r b i t r a r i l y we  10.5  above f o r m u l a and  C o r o l l a r y 10.6  ([3]  v  and  ®(G^,u^)  Proof.  l  V  i  )J.  >  f  o  r  a  1  i e l  1  complete the p r o o f .  3.6)  Lemma  Suppose t h a t  k[^p(u  have t h a t  2]Tp(Hi*i)  p(u,v)  The  >  1  E[d(u *v )] i  are product  <•»  i  .  Then  ®(G ,v ) 1  i  isomorphic. .  10.5  In the second p a r t of the p r o o f of Theorem 2  the convergence of  E[d(y^,v^)]  to produce the H i l b e r t the tensor product  sp ace  ®(<|>^(G^),x^) = ® ( < ^ ( 6 i ) * i ) v  S^G^u^)  necessary  and  , and  which i s p r o -  ^G^v^).  .  The  corol-  immediately. '•'  We  a j l t h a t was  H = ®(H^,x^) = ®(H^,y^)  :  duct isomorphic t o both lary follows  was  now  /  •  '  '  w i l l s t a t e the t h r e e main theorems of t h i s  s e c t i o n and then g i v e the p r o o f s .  Theorem 1 0 . 7 .  The f o l l o w i n g •conditions "bh>;.(u.)  are e q u i v a l e n t :  .  /p  (a)  • '  (b) ®(G^,U^)  " 2  1  and  r(vV)>  ' '  E [ d ( u , v ) ] .< • i  and  ;  (gi(G^,v^)  V ,  xi are product  isomorphic;  90. (c)  ®(v^)  e x i s t s as a p r o d u c t s t a t e o f  ®(G^,y^)  .  M o r e o v e r , i f a n y o f t h e s e c o n d i t i o n s h o l d we h a v e  p(^(\i ),  =JXp(u ,v )  ®(v ))  ±  i  ±  , a convergent  1  •  Theorem 10.8.  product.  I  The f o l l o w i n g c o n d i t i o n s on  (u.^) a n d ( v ) i  are e q u i v a l e n t : (a)  E[d(u ,v )] i  i e l (b)  < »  i  the infimum &(G^,u.j_)  , a n d f o r a l l b u t a c o u n t a b l e number o f  i n the d e f i n i t i o n of  and  ^(G^jv^)  d  i s attained;  a r e weakly p r o d u c t . isomorphic;...  (c)  T h e r e e x i s t a u t o m o r p h i s m s (j)^ o f G such t h a t ta <%(v^ ) e x i s t s a s a| p r o d u c t s t a t e o f ( g ^ G ^ , ^ ) . 1  ! I  T h e o r e m 10.9-  (u )  The f o l l o w i n g c o n d i t i o n s o n  a n d (v^,)  i  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  , E(d(u .jv' ^ j) ]  I  i  , and f o r a l l b u t a c o u n t a b l e number o f  mum i n the. d e f i n i t i o n o f (b)  ®(G ,u ) i  isomorphisms  e  i  l the i n f i - '  i s attained;  a n d ,®(G ,v )  i  i  isomorphic; (c)  d  i  7r  ,  i  are permutably ;  product  •„-'  ; >'  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 • ta ^ of G onto ^ ( i ) such that, ^ ( v ^ ^ ^ ) i  e x i s t s as a p r o d u c t s t a t e o f  P r o o f o f Thedrem 10.7.  (S^G^u^)  .  v  By T h e o r e m 10.5 a n d Lemma 10.4, ( c )  implies  ( a ) . B y C o r o l l a r y 10.6, ( a ) i m p l i e s  mediate  that  ment.is  i m m e d i a t e f r o m T h e o r e m 10. 5. .  (b) i m p l i e s  (b).  ( c ) ( s e e Remark 9-3). '• •  I t i s im-  The l a s t  state-  91. P r o o f o f Theorem 10.8. the subset o f d(u^,y^)  I  (a)holds.  Let I  be  Q  f o r which the infimum i s n o t a t t a i n e d i n  , and t h e r e f o r e b y (3.1) t h e supremum i s n o t a t t a i n e d  in  p(u^jV^)  I  into thepositive  Q  Suppose t h a t  .  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 integers.  Define a function  "y  g  from  on  I  by  0  , for  i eT-I  0  g(l) l/2 ( ) Y  , for  i  i e l ' . o  By Lemma 3.2,, we c a n c h o o s e ' a u t o m o r p h i s m s that  •  "  such  i  ta (10.8)  ~  p(w ,v ) > p(u ,v )  - g(i)  i  1  i  i  i  V. g ( i ) < 1 , and s i n c e  .  4), S[-l - p ( u  i J  v  1 i  ,  £[ 1 - p ( u , v , ) ] < » 1  t h e h y p o t h e s i s a n d Lemma 10.4, we o b t a i n f r o m  (10.9)  G^  !.  •  Now s i n c e  <j)^ o f  j  by  (10.8) t h a t  •  )] < Z[l/-  p(u ,v )] + £ g(i) < • i  i  .  T h e n f r o m Lemma.10.4 a g a i n , . •• Vi ,  (10.10) ....  ."  • • '• \ ' •  : ••  ,  .  .  ;  n  ' '' "• '  .  '  E[d( '.'  ta  o  ,Vi )]S< «  .  1  U i  'xy-•';;.;|; • •  r V f |  •  ;  :  '  .  • '  C o n v e r s e l y , suppose,:there e x i s t automorphisms • v  G.  ?  '  •  s u c h t h a t . (10.10) holds..' • S i n c e  have .  '  :: '  ~  '•'  •  <j>. o f <}>.  d(u.,v.,) < d ( y . , v . ) 1  •  L  we  E[d(u ,v )3  2  < »  i  i  and m o r e o v e r , f o r a l l b u t a . c o u n t a b l e must h a v e  d(u^,v^) = d ( ^ , v i  t o t h e statement 4^  of  G^  .  >  number o f  ) = 0. .  i  )  we  .• ,  So ( a ) i s e q u i v a l e n t .  t h a t (10.10) h o l d s f o r some a u t o m o r p h i s m s  R e p l a c i n g (b) w i t h t h e equivalent^ statement  g i v e n b y Lemma 9.5, we c a n a p p l y Theorem (Vj_  i e I  t o prove  10.7 t o  ( y . ) and  T h e o r e m 10.8.  P r o o f o f Theorem  10.9•  T h i s f o l l o w s i n an o b v i o u s  manner  f r o m Theorem 10.8.  Lemma 10.10.  Suppose t h a t f o r each,  o f t y p e I. o r t y p e 11^, a n d t h a t states.  Then  ®(G.£,M )  and  i  y^  i e I and .  ^(G^v^)  I m m e d i a t e f r o m Theorems  i sa factor  are compatible  are weakly  i s o m o r p h i c i f and o n l y i f they a r e p r o d u c t  Proof.  ,  product  isomorphic.  10.7, Theorem  10.8, a n d Theo-r  rem 8.11.  E x a m p l e 10.11.  Let G  trace  T  in  .. S u p p o s e t h a t  G.  be a  , and l e t E  and  Ig  F  factor with normalized  be t w o o r t h o g o n a l p r o j e c t i o n s  G^ = G . f o r a l l i  E x a m p l e 8;12 a n d the' t h e o r e m s o f t h i s obtain the following. ,(a) Then  <5?(G >y ) i  i  they a r e weakly  L e _ t  -  \l±  and  =  2  T  E  e  l .  s e c t i o n we c a n e a s i l y  •• a  n  d  v  i  =  2T  F  f  o  r  a l l  $>(G.^,v^) • a r e n o t p r o d u c t  product  Using  isomorphic.  i'.e^I  isomorphic but  •; ' ' , ::  :  :  (b)  Let  I  be the p o s i t i v e i n t e g e r s .  2T  ., f o r  e  T  v  Then  ^(G^Uj.)  i  =  w  and  for  3  (i+l)  '  malized t r a c e  i  Let T  normalized t r a c e a G  Then  f o r  <^(G ,v )  G  i  .  a  even  1  1  1  ;  €  5  1  product  .  Let  Iy  w i t h nor-  uj. be a f a c t o r of type E  isomor-  isomorphic.  be.a f a c t o r of type  ,| and l e t T^'  odd  are not weakly product  i  p h i c but they are permutably  Example 10.12.  i  Suppose t h a t :  I  with  2  be a minimal p r o j e c t i o n i n  G = A3 ® (ft ® fl . . Define, the s t a t e s  u  and  v  on  by,  -  ;  : u  -  (2^)  «  (2^)  ®  T'  v = ( 2 T ) ® T'  = 2  e  Let  G  i  = G  3  Example b.13,  Uj, = u  and  = ^  v.^ = v  T  (  (  E  ^  E  ^  1  1  }  .  }  .  for a l l I e l  p(u,v) = V £ , s o ^ ( G ^ , ^ )  and  .  ®(G ,v ) 1  i  By are not  weakly (and t h e r e f o r e i n t h i s case not permutably) product •isomorphic'  However there e x i s t "refinements" of each of  these t e n s o r ^ p r o d u c t s which are permutably  product  T h e r e f o r e they are a l g e b r a i c a l l y isomorphic. pose  I  i s countable.  isomorphic.  In f a c t ,  sup-  U s i n g standard a s s o c i a t i v e t y arguments,  '.'•we c a n f i n d :  K =  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  with .J u K = I  J  so t h a t t h e r e s u l t i n g a l g e b r a  \ 'J  K  on type  ;•  h  ' ( s e e [2] e.g. .),-.=\P ® I  where  i s t h e hyper-finite'  P  factor.  .V  :;f V::'^-  / . V.-;  , '  J  E x a m p l e 10.13. '•As a t r i v i a l  .  example, C  let  such t h a t .  y  and  v  p(y,v) < 1 But  1  u  2  > v  i sa useful fact,  x  .  T h e n o n .G ® G ,  p ( y , v) p(-v,u) < 1  .  ® v ) 2 P(W >V ) P ( U , v ) 2  .  be a n y t w o s t a t e s o f a  ...  I t i s e a s y t o show h o w e v e r t h a t we a l w a y s h a v e  p(u  ' ' '•  .  The p r o d u c t f o r m u l a d o e s n o t h o l d f o r p  p(u ® v , v ® u ) = 1  This  1  IIrV  '  W -algebra  '.<•  ; , . ;\ .  [®((S,2T4) ] ' ,• w h i c h by s t a n d a r d r e s u l t s  [ ® ( I B , T ' ) ]  and  1  1  2  .  2  s i n c e i t shows t h a t i n c e r t a i n  -  .  cases  .  ;  we may be a b l e t o d e d u c e t h e a l g e b r a i c i s o m o r p h i s m 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 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 t h e r e s u l t i n g p r o d u c t s t a t e s o n the. r e s p e c t i v e  d  I  .between  pieces.  ;,;.'. .  • ' BIBLIOGRAPHY  '  • .; , . 95  .  [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 o f f a c t o r s , ; P u b l . RIMS., Kyoto Univ. S e r i e s A, 3(1968), 51-130. [2] . D. B u r e s , C e r t a i n f a c t o r s c o n s t r u c t e d as i n f i n i t e  tensor,  p r o d u c t s , Composito Math., 15(1963), 196-191. f2]  D. B u r e s , An e x t e n s i o n o f K a k u t a n i ' s theorem on i n f i n i t e p r o d u c t measures t o t h e t e n s o r p r o d u c t o f s e m i - f i n i t e W - a l g e b r a s , Trans. Amer. Math. Soc. 135(1969), 199-212. D. B u r e s , T e n s o r . p r o d u c t s o f W - a l g e b r a s , P a c i f i c n a l o f Mathematics 27(1968), 13-37.  Jour-  [5]  J . D i x m i e r , Les a l g e b r e s d ' o p e r a t e u r s dans l ' e s p a c e 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. K a k u t a n i , On e q u i v a l e n c e o f i n f i n i t e p r o d u c t measures, Ann. o f Math. 49(1948), 214-226.  ITJ  Y. Misonou, On t h e d i r e c t prodtact o f hoku Math. J . 6(1954), 189-204.  [8]  W -algebras,. To-  C.C.'Moore, I n v a r i a n t measures on p r o d u c t 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 o f o p e r a t o r s I , • A n n . o f Math. 37(1936), 116-229. . [10] .  F. Murray and J . von-Neumann, On r i n g s o f o p e r a t o r s I V , Ann. o f Math. 44(1943),. 716-808.  [11]  M. Nakamura, On t h e d i r e c t p r o d u c t o f f i n i t e Tohoku Math. J . 6(1954');, .205-207. . • N  factors,  -  •  '.  )[12]  J . v o n : N e u m a n n ,  /.'"'  M a t h .  [13]  R . T .  O ni n f i n i t e  d i r e c t  p r o d u c t s ,  P o w e r s ,  o  a s s o c i a t e d  f u n i f o r m l y  . S u z k i ,  c e n t e r  r i n g s ,  A n n .  o  .  f M a t h .  .'•  A u t o m o r p h i s m s  e l e m e n t  :  . \  h y p e r f i n i t e  86(1967), 138-171. N  a  • .'.  R e p r e s e n t a t i o n s  a n d t h e i r  f  6  C o m p o s i t o  6(1938), 1-77.  a l g e b r a s  [l4]  9  w i s e  o  f  W - a l g e b r a s  i n v a r i a n t ,  l e a v i n g t h e  T o h o k u  M a t h .  e x t e n s i o n  o  J .  7(1955),  186-191. [15]  I . E .S e g a l , i n t e g r a t i o n ,  [16]  E . S t o r m e r , o f  A n o n - c o m m u t a t i v e A n n .  o  C - a l g e b r a s ,  s t a t e s  J o u r n a l  a b s t r a c t  57(1953), 4 0 1 - 4 5 7 .  f M a t h .  S y m m e t r i c  f  o  o  f i n f i n i t e  fF u n c t i o n a l  • '.  t e n s o r  3(1969),  A n a l y s i s  48-68. [ 1 7 ]  ••  Z . T a k e d a , o f  o p e r a t o r  s t r u c t e d K y o t o  r [ 1 9 J  l i m i t  a l g e b r a s ,  U n i v .  T o h o k u  O nt y p e  a s i n f i n i t e S e r i e s  a n d i n f i n i t e M a t h .  J .  c l a s s i f i c a t i o n t e n s o r  A ,  p r o d u c t s ,  d i r e c t  /  p r o d u c t  7(1955) 67-86..''! o  f f a c t o r s  P u b l .  c o n -  R I M S ,  ,  '.  J  4(1968), 467-482.  l  *  M  .T a k e s a k i ,  T o h o k u  M a t h .  [20]  F . R i e s z  '•  N e wY o r k ,  y  I n d u c t i v e  0 . T a k e n o u c h i , .  [18]  V  p r o d u c t s  O nt h e d i r e c t J .  o  f . W  - f a c t o r s ,  10(1958), 116-119.  a n d B . S z - N a g y / 1955.  p r o d u c t  '  F u n c t i o n a l -Analysis, .  U n g a r ,  .  :  r :  '  *V'^  

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