UBC Theses and Dissertations

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UBC Theses and Dissertations

Maximal abelian subalgebras of von Neumann algebras Nielsen, Ole A. 1968

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MAXIMAL OF  ABELIAN  VON  NEUMANN  SUBALGEBRAS ALGEBRAS  "°y  OLE A. NIELSEN B.Sc.j U n i v e r s i t y o f B r i t i s h Columbia,  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  ,  i n t h e Department o f  MATHEMATICS  We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d  standard .  THE UNIVERSITY OF BRITISH COLUMBIA May, 1968.  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further  agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department nf  hflTHE  M/4T/CS  The University of British Columbia Vancouver 8, Canada  Supervisor:  D. J . Bures.  ABSTRACT  We a r e concerned w i t h c o n s t r u c t i n g examples o f maximal a b e l i a n von Neumann s u b a l g e b r a s (MA s u b a l g e b r a s ) i n h y p e r f i n i t e f a c t o r s o f type I I I .  Our r e s u l t s w i l l  show  t h a t c e r t a i n phenomena known t o h o l d f o r t h e h y p e r f i n i t e f a c t o r o f type 11^ a l s o h o l d Let  7?L and tl  f o r type I I I f a c t o r s .  be s u b a l g e b r a s o f t h e f a c t o r  We c a l l 7KL and 7 L e q u i v a l e n t i f ??L i s the image o f by some automorphism o f algebra of G  G  G .  L e t N(7)t)  G . %  denote t h e sub-  g e n e r a t e d by a l l those u n i t a r y o p e r a t o r s i n  which i n d u c e automorphisms o f %t ',. and l e t  N^(tH),...  .  N (7)t) , 2  be d e f i n e d i n the o b v i o u s i n d u c t i v e f a s h i o n .  F o l l o w i n g J . D i x m i e r and S. A n a s t a s i o , we c a l l a MA suba l g e b r a 7K. o f  G  s i n g u l a r i f N(?H) = ?VL , r e g u l a r i f  N(M) = G , s e m i ^ r e g u l a r i f N(?H) G , and  (m >_ 2)  m-semi-regular  not f a c t o r s but  N^OM)  i s a factor distinct  i f ' N(7H),. . .N " (?30 m  1  are  i s a factor.  The MA s u b a l g e b r a s o f the h y p e r f i n i t e 11^ IB  from  factor  have r e c e i v e d much a t t e n t i o n i n t h e l i t e r a t u r e , i n the  papers o f J . D i x m i e r , L. Pukanszky, S. A n a s t a s i o .  I t i s known t h a t  of  Further,  each type.  B  &  S i s t e r R. J . Tauer, and c o n t a i n s a MA s u b a l g e b r a  contains pairwise inequivalent  sequences o f s i n g u l a r , s e m i - r e g u l a r , 2 - s e m i - r e g u l a r , and 3 s e m i - r e g u l a r MA s u b a l g e b r a s .  iii.  The o n l y h i t h e r t o known example o f .a MA.subalgebra I n 1956 Pukanszky  i n a type I I I f a c t o r i s r e g u l a r .  gave a  g e n e r a l method f o r c o n s t r u c t i n g MA s u b a l g e b r a s . i n a c l a s s o f ( p r o b a b l y n o n - h y p e r f i n i t e ) type I I I f a c t o r s .  Because o f an  e r r o r i n a c a l c u l a t i o n , t h e types o f t h e s e s u b a l g e b r a s i s n o t . known. The main r e s u l t o f t h i s t h e s i s i s t h e c o n s t r u c t i o n , i n each o f t h e u n c o u n t a b l y many m u t u a l l y  non-isomorphic  h y p e r f i n i t e type I I I f a c t o r s o f R. Powers, o f : (i) (ii)  a s e m i - r e g u l a r MA subalg-ebra two sequences o f m u t u a l l y i n e q u i v a l e n t 2s e m i - r e g u l a r MA s u b a l g e b r a s  (iii)  1  two sequences o f m u t u a l l y i n e q u i v a l e n t 3s e m i - r e g u l a r MA s u b a l g e b r a s .  Let  G  denote one o f these type I I I f a c t o r s and l e t  -denote the h y p e r f i n i t e 11^ f a c t o r .  Roughly  ever a n o n - s i n g u l a r . MA s u b a l g e b r a o f  8  B  s p e a k i n g , when-'  i s c o n s t r u c t e d by  means o f group o p e r a t o r a l g e b r a s , o u r method w i l l produce a -.MA s u b a l g e b r a o f  G  o f t h e same t y p e .  H. A r a k i and J . Woods have shown t h a t  G ® B = G , _  and i t i s t h e r e f o r e o n l y n e c e s s a r y t o c o n s t r u c t MA of  G <8> B  o f t h e d e s i r e d type.  of  G ® B  b y t e n s o r i n g a MA s u b a l g e b r a i n G  8- .  G ® e  We o b t a i n MA s u b a l g e b r a s w i t h one i n  I n o r d e r t o determine t h e type o f such a MA  we r e a l i z e , B  subalgebras  subalgebra>  as a c o n s t r u c t i b l e a l g e b r a and then r e g a r d  as a c o n s t r u c t i b l e a l g e b r a ;  t h i s a l l o w s us t o c o n s i d e r  iv.  operators i n  G ® iB  as f u n c t i o n s from a group i n t o an  a b e l i a n von Neumann a l g e b r a . As a c o r o l l a r y t o our c a l c u l a t i o n s , we a r e a b l e t o c o n s t r u c t m u t u a l l y i n e q u i v a l e n t .'sequences o f and 3 - s e m i - r e g u l a r MA  subalgebras  2-semi-regular  o f the h y p e r f i n i t e  f a c t o r which d i f f e r ' f r o m those o f A n a s t a s i o .  11-^  TABLE OF CONTENTS  REVIEW OF VON NEUMANN ALGEBRAS MAXIMAL ABELIAN SUBALGEBRAS:  DEFINITIONS  AND SOME KNOWN RESULTS THE MAIN CONSTRUCTION EXAMPLES OF MAXIMAL ABELIAN SUBALGEBRAS  REFERENCES  ACKNOWLEDGMENT  I t i s a pleasure s u p e r v i s i o n o f Dr. D. Bures  t o acknowledge t h e h e l p f u l as w e l l as t h e f i n a n c i a l  a s s i s t a n c e o f b o t h the N a t i o n a l R e s e a r c h C o u n c i l o f Canada and t h e Mathematics Department o f the U n i v e r s i t y o f . B r i t i s h Columbia d u r i n g the p r e p a r a t i o n o f t h i s ' .thesis.  1  REVIEW. OF VON NEUMANN ALGEBRAS  I n g e n e r a l , o u r n o t a t i o n and t e r m i n o l o g y i s t h a t o f D i x m i e r ' s hook [ 6 ] . A H i l b e r t space the complex numbers x,y -* (x,y) norm  C  M  i s a non-zero v e c t o r space over  t o g e t h e r w i t h an i n n e r p r o d u c t  such t h a t  M  i s complete w i t h r e s p e c t t o t h e  x -\ || x || = ( x , x ) ~ .  By an o p e r a t o r on  2  W  we mean  a bounded ( e q u i v a l e n t l y : norm-continuous) l i n e a r t r a n s formation of  M  into  M .  a l g e b r a o f a l l o p e r a t o r s on  We use  £(W)  t o denote t h e  M , 1^  ( o r I , when  u n d e r s t o o d ) t o denote t h e i d e n t i t y o p e r a t o r on to denote t h e s c a l a r m u l t i p l e s o f  1^ .  If that of  M  3  ["#]  containing  p r [ ^ ] i s t h e ( o r t h o g o n a l ) p r o j e c t i o n onto t h i s subspace G c  ,  AB = BA G .  ){ i s a If  & , and  I f "tf c K  i s t h e s m a l l e s t c l o s e d l i n e a r subspace o f and  it i s •  G'  i s t h e s e t o f a l l those  for a l l A e G ;  G'  B e £(Jf) such  i s ' c a l l e d t h e commutant  A von Neumann a l g e b r a ( o r r i n g o f o p e r a t o r s ) on * - a l g e b r a o f o p e r a t o r s on  Q a £(&)  a l g e b r a on (G U G*)" .  #  satisfying  G" = G  i s a r b i t r a r y , ft(G) , t h e s m a l l e s t von Neumann^ U  containing  G , i s e a s i l y seen t o be  T h i s a l g e b r a i c d e f i n i t i o n o f a von Neumann  a l g e b r a (which i s used b y D i x m i e r 'in h i s book [ 6 ] ) i s e q u i v a l e n t t o t h e t o p o l o g i c a l one o r i g i n a l l y employed b y von Neumann:  G c £(&) i s a von Neumann a l g e b r a i f  weakly c l o s e d * - a l g e b r a c o n t a i n i n g  Ijj .  G  isa  The e q u i v a l e n c e  2.  o f t h e s e two d e f i n i t i o n s i s a p a r t o f t h e f o l l o w i n g more ' g e n e r a l r e s u l t , known as t h e Double Commutant Theorem (see [6;  p..44],  [7;  p . 8 8 5 ] , o r [14; §2]):  -x--algebra o f o p e r a t o r s on R,(G) . = G"  i f G  3i w h i c h c o n t a i n s  i s t h e c l o s u r e of  G  is a  1^ , t h e n  i n each o f t h e f o u r t o p o l -  o g i e s : . weak, s t r o n g , u l t r a w e a k , and u l t r a s t r o n g - on £(»). Let  G  and  B  H i l b e r t spaces  it and  morphism o f  onto  map • ''(J) o f all  G  G onto  A e G .  we say t h a t G = IB .  be von Neumann a l g e b r a s on the K  IB IB  , r e s p e c t i v e l y . . An  i s a l i n e a r and m u l t i p l i c a t i v e  which s a t i s f i e s  ,<J>(A*)'= (<j>(A))* i  I f t h e r e i s an isomorphism o f G  and  B  iso-  G  onto  for B  a r e i s o m o r p h i c , and we w r i t e  I t t u r n s out t h a t an isomorphism o f  G  onto  B  i s n e c e s s a r i l y u l t r a w e a k l y and u l t r a s t r o n g l y b i c o n t i n u o u s [6;  p.57].  An isomorphism  <j> o f  G  s p a t i a l i f there i s a l i n e a r isometry 'such t h a t  <{>(A) = T A T Let  on  G  +  G  onto  "f  B  of  i s called W  onto  X  f o r a l l ' A e G' .  1  be a von Neumann a l g e b r a on  = {A e G : A >_ 0}  i s a mapping  uu : G  U . +  A trace  - [0 «°)u{* } p  J  which s a t i s f i e s the f o l l o w i n g : (i) (ii)  f o r a l l S,T e G for a l l  S e G  +  , uu(S + T) = u).(S) + u>(T) and  +  (where t h e c o n v e n t i o n (iii)  for a l l S e G uu(USU*) = u>(S)  all  X >_ 0  0-oo = 0  ,  i s used)  and a l l u n i t a r y .  w(XS)  U e G ,  = Xuj(S)  3.  The t r a c e \  (a) ' (b)  u) on  • (d)  i s called .  f i n i t e , i f u>(l) <  • T e G  w i t h - 0 <' S <_ T  +  f a i t h f u l , i f S.e G  and  and  +  •  - {0} , t h e r e i s an  +  u)(S) < °°  iu(S) = 0  imply"  S = 0 n o r m a l , i f , whenever set i n G then  3? i s an u p w a r d l y - d i r e c t e d  w i t h l e a s t upper bound  T e G •,  cu(T) = sup {m(S) : S e 3} '. 1  A f a c t o r on . M with  •  serai-finite, i f , given S e G  (c)  G  GA G' =  .  i s a von Neumann a l g e b r a  G  on  H  I t i s t h e f a c t o r s t h a t have r e c e i v e d t h e  most a t t e n t i o n I n t h e l i t e r a t u r e . . T h e i r extreme non. c o m m u t a t i v i t y a c t u a l l y makes them r e l a t i v e l y easy t o s t u d y ; moreover, e v e r y v o n Neumann a l g e b r a l o o k s l o c a l l y l i k e a f a c t o r , :  'and  i n f a c t i s b u i l t up from f a c t o r s ' by means o f t h e d i r e c t  i n t e g r a l [17].  The comparison theorem ( [ 6 ; p. 338] o r  Theorem V I ] ) i m p l i e s  t h a t i f uu - i s a normal trace.-on  [12;  G, +  where  >.,G .is. a f a c t o r , t h e n one o f t h e f o l l o w i n g - must be t h e c a s e : •  . ( i ) ' u)(A) = 0 (ii) (iii)  tu(A) w  f o r a l l A e G- ' +  for a l l  A e G.' - {0}  -  +  i s f a i t h f u l and s e m i - f i n i t e .  Moreover, t o w i t h i n a p o s i t i v e m u l t i p l e , n o n - t r i v i a l normal t r a c e on  G  +  t h e r e i s a t most one  . ' A factor  G  t h e r e i s no normal non-zero s e m i - f i n i t e t r a c e on t o b e of type I I I .  I f a factor  such t h a t G  +  i s said  G. " i s n o t o f t y p e I I I  4.  t h e r e i s a normal f a i t h f u l s e m i - f i n i t e t r a c e  uo  on  G  which, i n some n o r m a l i z a t i o n , must s a t i s f y one o f : * ( c ) = {0,1,...,n} uj(G )  (ii)  P  n >_ 1  = {0,1, . . . ,<*>}  . ( C ) = [0,1]  (iii)  P  (iv) where  f o r some i n t e g e r  p  (i)  = [0,OP)u{-} ,  j(G ) P  U  dP  i s t h e s e t o f p r o j e c t i o n s i n G ".  G • i s s a i d t o be o f t y p e I ; t o t h e a l g e b r a o f a l l nxn In. case ( i i ) ,  i n t h i s case  G  (i),  i s isomorphic  m a t r i c e s w i t h complex e n t r i e s .  G . i s s a i d t o be o f t y p e 1^;  unique i n f i n i t e C a r d i n a l  In-case  a  such t h a t  G  there i s a i s isomorphic to"  t h e algebra-, o f a l l bounded l i n e a r o p e r a t o r s on an a - d i m e n s i o n a l H i l b e r t space. . I f ( i i i ) h o l d s , ( i v ) h o l d s , o f t y p e 11^.  G . i s o f t y p e 11^, and i f .  I t i s c l e a r that the n o t i o n of a  f a c t o r and i t s t y p e a r e . i n v a r i a n t under isomorphisms.  Given  t h a t f a c t o r s o f each t y p e e x i s t , o n s e p a r a b l e H i l b e r t  spaces,'  t h e .tensor p r o d u c t enables one 'to c o n s t r u c t f a c t o r s o f each t y p e on a r b i t r a r y i n f i n i t e - d i m e n s i o n a l H i l b e r t  spaces.  i'  Let space.  51- be s e p a r a b l e i n f i n i t e - d i m e n s i o n a l  .At p r e s e n t , t h r e e [two] n o n - i s o m o r p h i c  & of type 1 ^  [11^] a r e known ([23; .p. ' 3-85],  Hilbert"  f a c t o r s on [24]]).  In  • t h i s t h e s i s , t h e ' o n l y f a c t o r - o f t y p e 11^ ,which i s o f . i n t e r e s t i s t h e h y p e r f i n i t e one.  I n general, a f a c t o r  G  on  W  is  c a l l e d h y p e r f i n i t e i f i t i s g e n e r a t e d b y an i n c r e a s i n g sequence (G' ) n  w i t h each  G  n  a f a c t o r of type I  n  . • Murray and  von Neumann showed t h a t a l l h y p e r f i n i t e f a c t o r s o f t y p e I I ,  on  M  a r e Isomorphic [13; Theorem XIV] ( s e e a l s o [6;  hence one can speak o f t h e h y p e r f i n i t e 11^ f a c t o r on  p.291]); & .  R e c e n t l y , Powers [19] announced t h e e x i s t e n c e o f an uncounta b l e number o f p a i r w i s e n o n - i s o m o r p h i c h y p e r f i n i t e f a c t o r s of type I I I on  ( f o r the proof, s e e [ l 8 ] ;  M  i n [2] A r a k i  and Woods g i v e a d i f f e r e n t p r o o f o f t h i s r e s u l t ) .  It is  these f a c t o r s t h a t .we s h a l l be p r i m a r i l y c o n c e r n e d w i t h i n this thesis.  Two n o n - i s o m o r p h i c n o n - h y p e r f i n i t e f a c t o r s o f  type I I I have been c o n s t r u c t e d on [20]  M , one b y Pukanszky  and one b y Schwartz [ 2 5 ] The  remainder o f t h i s s e c t i o n d i s c u s s e s t h e t h r e e  methods w h i c h we employ t o o b t a i n von Neumann a l g e b r a s . These c o n s t r u c t i o n s - t h e group o p e r a t o r a l g e b r a , t h e cons t r u c t i b l e a l g e b r a , and t h e i n f i n i t e t e n s o r p r o d u c t  - are  a l l due t o Murray and von Neumann. Let  G  be a group w i t h i d e n t i t y  e .  We u s e  t o denote t h e H i l b e r t space w i t h o r t h o n o r m a l b a s i s notice that  G  able.  F o r each  V  G  (1.1)  on  i s separable'whenever g £ G  G  G  (g\) „ ;  i s ' a t most count-  there .is a unique u n i t a r y operator  satisfying V n = (gh) '  for a l l h e G .  A  This defines a u n i t a r y r e p r e s e n t a t i o n  g - V  of  G  on  G .  g The  group o p e r a t o r a l g e b r a over t h e group  Neumann a l g e b r a . Qr  = B(V  : g e G)  on  G G  i s t h e von ( f o r a complete  d i s c u s s i o n o f t h e group o p e r a t o r a l g e b r a , see. e i t h e r [6; pp.  301-303] o r [13;  §5.3]).  Alternatively,  £  r  can be  d e s c r i b e d as t h e s e t o f a l l t h o s e o p e r a t o r s (T g, n) = (T e, ( h g  _ 1  ) )  for all  A  i s a. f a c t o r i f and o n l y i f  G  T  on  g,h e G .  G  with  The a l g e b r a  has t h e i n f i n i t e  con-  j u g a t e c l a s s p r o p e r t y , i . e . , whenever (1.2)  {hgh~"  i n t h i s case, £  i s n e c e s s a r i l y o f t y p e 11^ .  G  s e p a r a b l e and i f 6q whenever  G  : h e G} - i s i n f i n i t e whenever  L  i s a f a c t o r , then  g ^ e  If G is  i s hyperfinite  i s hyperfinite, i . e . , G - tj G„ , where n=l n  G, c G ±  c • • • and  d  '(1.3) each  G  i s a f i n i t e .subgroup; o f G .  n  B e f o r e p r o c e e d i n g t o t h e c o n s t r u c t i b l e a l g e b r a , we w i l l b r i e f l y c o n s i d e r t h e t e n s o r p r o d u c t o f two H i l b e r t spaces. basis (cp  L e t 2i and K ( ?i)i i c  n  (^j)jeJ >  d  e  j) ixj  i  each  a  be H i l b e r t spaces xvith o r t h o n o r m a l  i  s  a  n  or  €  j e J  x -> x ® t - i (j>*. A V J  f  M  i  n  t  u  o  M  @  K  Pectively.  Then.  * h o n o r m a l b a s i s f o r U®K .  we denote b y o  r e s  For  <j). t h e c a n o n i c a l , embedding J • G i v e n A e £(M®K) , t h e  ( w h i c h a r e o p e r a t o r s on  W) a r e c a l l e d t h e m a t r i x .  elements o f A  r e l a t i v e t o t h e orthonormal b a s i s  an o p e r a t o r on  £®K  (f.) .  T  i  i s c o m p l e t e l y determined b y i t s m a t r i x  elements. Lemma 1.1  With the notation" of the preceding paragraph  7.  f o r each  (i)  j,k e J  A,B e Z(ll®U) ,  *t  ffl  the  and  V < * !  V  B  •  sum c o n v e r g i n g i n t h e s t r o n g t o p o l o g y on.  X (M) (ii)  If  (A ) D a  a  weakly t o an (^J  a  a e  ^  ^(^® 0 w h i c h converges 3  A e £(3i©K) , t h e n f o r each  D  c  o  n  v  e  r  S  e  s  weakly t o  j,k e J ,  «j)* A ^ '.  Simple c a l c u l a t i o n s ( s e e [6; pp. 23-24] o r  Proof: '[12;  ^)  A  n e  a G  §2.4] )..• C o n s t r u c t i b l e a l g e b r a s were f i r s t c o n s i d e r e d b y  Murray and von Neumann i n [12] and [ 1 6 ] , and f u r t h e r d e v e l o p e d by D i x m i e r i n [6; pp. 1 2 7 - 1 3 7 ] ; i s t a k e n f r o m Bures [ 3 ] c a l l e d a C-system  i f  The system  U , i f G  i s a unitary representation of  U..7YI TJ* =7tl f o r a l l g C-system. 0  fc>  g e G .'- L e t to  i s a group, and i f 1  G  on  M  with  [TfV,W,G,gU ] g  be a  F i n i t e l i n e a r combinations of the operators  (M©IA)(U .®'V ) , "  [?7l,y,G,g -> U ] i s g  i s a maximal a b e l i a n von Neumann  a l g e b r a on t h e H i l b e r t space g -> U  o u r n o t a t i o n and t e r m i n o l o g y  M e 1U  and  g e G , form a ^ - a l g e b r a on  to'  Jf©G (V as i n ( l , l ) ) j . . . w e u s e G[?7l,M,G,g - U 1 t o denote to g the von Neumann a l g e b r a on M@G g e n e r a t e d by t h i s * — a l g e b r a . • I f A e G[TU,^,G,g - U ] , t h e m a t r i x elements o f A s r e l a t i v e t o the orthonormal b a s i s that f o r a l l  g,h e G , <j> * A <J>  h  (g) >, f o r G = <l> * _i h  A <{>  e  and  a r e such  8.  i * A <b U* <s Tfi, . S e t t i n g M_ = <j>* A <b U* , we o b t a i n a g ' e. & g g xe g family (M ) i n lYl which c o m p l e t e l y determines. A , v  &  y  5  r  and we w r i t e  A ~ [M  G[?)I,M,G,g -» U ] T e  : g e G] .  A l t e r n a t i v e l y , the algebra  c a n be d e s c r i b e d as t h e s e t o f a l l t h o s e  such t h a t f o r some f a m i l y  <|>* T <L = M _nU _ gh gh b  1  Lemma 1.2 let  A  (M ) i n% , g g£^ f.or a l l g,h e G .  q  r  L e t [??l,W,G,g -+ U ] be a C-system and o be o p e r a t o r s i n G[?H ,M,G,g -+ U .] w i t h  and B  §>  A ~ [M : g € G] o and M £ 7ft : (i)  (J)* AB <J) U* = E M _ i U keG Sk gk g  sum (  and B ~ [N : g e G] . g  e  g  x  converges  F o r a l l g,h e G  _ i N. U* _ ' , where t h e gk  1  k  x  ±  i n t h e • s t r o n g t o p o l o g y on 7U  ii)  (iii)  <f>g(M8I)(U ®V ) $ h  h  (iv)  ^  A(U ®V )  (v)  Vs  (U OT ) A *  • Proof.  h  h  (ii)  h  Simple  D e f i n i t i o n 1.3 (i)  h  ^ e  e  U* = 6 ^ M g  h  U* = M ^ . i " U* = U M . h  h  l g  y* .  calculations. The C-system  [ftt,tt,G,g - U ]  i s called:  f r e e , i f 771 n U ' W = {0} f o r a l l g e G - {e} e r g o d i c , i f % n {U^ : g e G}' =  .  A von Neumann a l g e b r a i s c a l l e d c o n s t r u c t i b l e i f i t i s s p a t i a l l y i s o m o r p h i c t o G[#t ,3i,G,g -» U ] g system [#t,M,G,g - U ] .  f o r some f r e e C-  9.  P r o - p o s i t i o n 1.4 [?U,W,G,g - U ]  ( [ 3 ; §4] and [4; § 7 3 ) .  The C-system  i s f r e e i f and o n l y i f , f o r each  there  exists a family  that  EE EE. , =1  and  ( j_)i i E  o  f  €  P r o p o s i t i o n 1.5  ([6]).  f r e e C-system, and l e t  P r o j e c t i o n s i n 7f{  E, U* = 0  E, U  g e G - {e},  Let  f o ra l l  such  i 6 I  [?ft,W,G,g - U ]  G = G[7?[,K,G,g - U ] .  be a  . Then  to'  7H.®<£A  i s a maximal a b e l i a n sub algebra- o f  a f a c t o r i f and o n l y i f  [?72,34,G,g -* U ]  G , and  G is  i s ergodic.  If  to  G  i s a f a c t o r , then: (i)  G  i s o f t y p e I i f and o n l y i f ?U  . minimal p r o j e c t i o n ;  (Ii)  If n  contains  i s the c a r d i n a l i t y of  a maximal f a m i l y o f p a i r w i s e  orthogonal minimal  p r o j e c t i o n s i n Kfl  i s of type I  G  3  then  G  ( i . e . , o f t y p e 11^ o r I  i sfinite  , n <° ) a  i f and o n l y i f t h e r e i s a normal f i n i t e w  trace all .. ( i i i )  a  G  g e G  on  %  +  w i t h . u>(U M U*) = u>(M) f o r  and a l l M e  %  +  i s o f t y p e . I l l i f and o n l y i f t h e r e  e x i s t a normal s e m i - f i n i t e f a i t h f u l W  with  +  P r o p o s i t i o n 1.6 [M,3i,G,g. - U_]  trace  for a l l  u) on  g e G  •  ( [ 8 ] , [13; Lemma 5 . 2 . 3 ] ) .  Let  be a f r e e C-system, and suppose that'  G[ftl,!t G,g - U .] 0  to  a.belian, t h e n  does,not  •  to  3  u)(U_ M U*) = to(M)  M e • frC'  all  faithful  I s a f a c t o r o f t y p e I I . ..  If." G i s  -1-  G[7?l,M,G,g - U ] i s h y p e r f i n i t e . g  and  10.  [15]  In  a complete d i s c u s s i o n o f t h e i n f i n i t e  t e n s o r p r o d u c t o f von Neumann a l g e b r a s can be found. I  be an i n f i n i t e i n d e x i n g s e t , and l e t  f a m i l y o f H i l b e r t spaces. a C -sequence i f each  (^jJiei  Q  2 iel  if  a  n  ( i^iel  d  g  1  Let  | 1 - || f.. ||  r  e  i  | <oo.  1  c  a  l  l  e  equivalent  d  1  C -sequences. Q  (^^iei  ^e  f i x e d C - s e q u e n c e , and l e t £ •  a  Q  denote t h e e q u i v a l e n c e c l a s s d e t e r m i n e d b y  (g )  a  I 1 - ( f . ,g. )| <°° 1 t h i s i s an e q u i v a l e n c e r e l a t i o n . •  on t h e s e t o f a l l  ea,ch  e  w i l l be c a l l e d  T  L iel a  b  €  (f-)-  f . e W. . and i f  0  Two C -sequences  A family  (^i)j_ j  Let  (f-).  e $  T  •  F o r  , l e t ® f.. denote t h e map iel  -*TT ( i^§i) iel' f  i € l  ( f j ^ i g j ••  @  o f  i  n  t  o  •  c  0  l i n e a r combinations of the ® f. iel  Defining  finite  i n t h e o b v i o u s manner,  1  we o b t a i n a v e c t o r space  V .'  The map  (f.). 1  <§ ( © iel  into  V  i s clearly multi-linear.""  f . , @ g. ) -»TT ( f - j g - ) i e l i e l -  1  form on  1  V  1  n 1  n  1  €  -> ® f . o f _iel  T 1  1  The .form  extends t o a s e s q u i - l i n e a r  w h i c h c a n be s h o r n t o be an i n n e r p r o d u c t .  w i l l r e f e r t o the completion of p r o d u c t , w h i c h we denote b y t e n s o r p r o d u c t o f t h e M.  We  "V r e l a t i v e t o t h i s i n n e r ,0  ® (M^,f?) , as t h e i n f i n i t e iel relative to  (f?).  ;  x  von Neumann,  r e s e r v i n g t h e p h r a s e i n f i n i t e t e n s o r ( d i r e c t ) product' f o r a much l a r g e r H i l b e r t . s p a c e , c a l l e d t h i s space t h e  £ -adic  incomplete d i r e c t product.  really  Note t h a t ® (M.,f?) • i e l 1  1  Q  11.  depends on @ selected. the  Q  , and n o t on t h e p a r t i c u l a r  f  ^  G  e  0  The f o l l o w i n g r e s u l t f a c i l i t a t e s w o r k i n g w i t h  i n f i n i t e t e n s o r p r o d u c t space. P r o p o s i t i o n 1.7  Let  ( °)j[ i  I  ( [ 1 5 ; Lemma 4 . 1 . 4 and Theorem V I I ] )  be a n ^ i n f i n i t e i n d e x i n g s e t , l e t  f a m i l y .of H i l b e r t s p a c e s , and f o r each be a u n i t v e c t o r i n (i)  F o r each (:£"?)J  I e l ,  be a  let f?  . i e l ,  with  T  (K.). ^  choose an o r t h o n o r m a l b a s i s  0 e J . f o r M. .  :he s e t o f a l l t h o s e  L e t J be  j e TT J . w i t h iel !  j(i) = 0  x  for  a l l b u t f i n i t e l y many  i e l ,  j ( i ) ." j e J , l e t f , = ® .f p^ J ia eT l x  and f o r each  Then  )  1  (f.).. 0 <J  T  is  £<J  r  an o r t h o n o r m a l b a s i s f o r ® (34','. , f ? ) iel 1  (ii)  1  L e t I = U I, be a d i s j o i n t Linion. keK  Then t h e r e  K  i s a unique l i n e a r isometry ( c a l l e d t h e a s s o c i a t i v i t y isomorphism) o f ® (34.,f.) iel 1  onto  x  ( © (34. ,f°), ® f ? ) which c a r r i e s ® f . keK i e l , i e l , i e l k  into  L  ® (® f . ) f o r each keK i e l . ' k  C -sequence (f.). o ^ i'iel T  1  equivalent t o  Let  \  (" i)i j i  a  e  and l e t J! = ® (M ,f,°) . iel T  x  1  ( f ? ) -' T  n  d  (^i^iei  '  o ea  s  i  n  Proposition  1.7>  I f T e £(M, ) , t h e r e i s a u n i q u e o x  12.  a. (T) £ £(3f) o  which  satisfi?  1  1  ( T ) ] [ ®^ f ] = (_®_^ _ ^ f ) ® (T f ) i e l 'iel-Ci } ~ ~o O" ±  f  ±  ±  x  C ~sequence  f o r each _ •  o  (^j^igj  Q  .  i t i s e a s i l y seen t h a t  equivalent to  (^i)j_ i €  r  a.. i s a- i s o m o r p h i s m ; o i n g t h e u s u a l n o t a t i o n , we w r i t e T f o r a. (T) . o G^ i s a v o n Neumann algebra, on 34 , then o o  followIf  G^  = {T : T e G. } i s a von Neumann a l g e b r a on 34 . o o f o r each 1 e I , G. i s a von Neumann on 3-4. , t h e n  If,  1  ® (G_. , f ? ) iel  denotes t h e v o n Neumann a l g e b r a ft(G. : i e I ) _  x  on  34 ;  1  we c a l l  ® (G ,f°) t h e i n f i n i t e t e n s o r p r o d u c t o f iel o• r e l a t i v e t o ( ±)± j 1  the  G  f  i  £  P r o p o s i t i o n 1.8 (G, ). j  ([3; §3]).  Let  )  >• ( i )  i  f  i  e  be as above, and l e t 34 = ® (34. , f ? ) ,  I  G.= ®  i  e  a  n  (G.,f?)  Then: i  (i).G  i s , m a x i m a l a b e l i a n on  34 i f each  G^ i s  maximal a b e l i a n on 34. i (ii) (iii)  G  i s a f a c t o r i f and o n l y i f each  G = £(34) Let  i e l , Q  1  let G  I 1  i f each  G  ±  = £-(M )  G^  be an i n f i n i t e i n d e x i n g s e t . be a group w i t h i d e n t i t y  = GCfll^aV^g  - u j ] , where  e  •[n ,V G , 1  ±  9  i s a factor  .'  ±  1  S  1  d  F o r each , let - uj] '  i  s  a  13.  f r e e C-system,  and l e t  U. ~ © ( M V ? ) , iel and l e t each  G  -  f ? be a u n i t v e c t o r i n M  Ik = ® ( W . , ^ ) , i e l  1  i  be t h e weak d i r e c t p r o d u c t o f t h e -e G , l e t U  1  T l e l  = TT 11% i e lg  s  i n w h i c h t h e f a c t o r s commute).  Set  G = ® ( G , ^ . ®(e ) ) , i e l  1  g = (g ).  .  1  G  .  1  (a f i n i t e  A  For produce  From P r o p o s i t i o n 1.7 i t  follows that there i s a l i n e a r isometry  Y  of  ,i\A „o i\A• ® ( M ® ( G ) , f ? ®(e ).) iel ' 1  X  A  1  1  onto  34 ® G  with >( ®'(f ® ( g ) " ) ) = ( ® f ) ® ( . ( g ) iel ' i e l < 1  1  whenever and  (^jJigj  (g ) 1  i e I  i  sa  0  i  ~sequence equivalent  i  to  e  l  i e I  )  A  -  (^i^iel  ( [ 3 ; P r o p o s i t i o n 4.1]  and P r o p o s i t i o n ^  • With the n o t a t i o n of the previous paragraph,  [7FT,34,G,g -» U ] o n l y i f each A -» 7'AT~" :  1  € G . '  P r o p o s i t i o n 1.9 1.4).  G  1  ±  L  i s a.' f r e e C-system w h i c h i s e r g o d i c  [Wl ,34 ,G ,g -* U^] 1  1  1  i s an isomorphism o f  i s ergodic. G  onto  i f and  The map  G[W2,34,G,g - Ii ]. .  2  MAXIMAL ABELIAN SUBALGEBRAS:  DEFINITIONS AND SOME KNOWN RESULTS  Only s e p a r a b l e H i l b e r t spaces w i l l be c o n s i d e r e d i n the' remainder o f t h i s  thesis.  The f i r s t p a r t o f t h i s s e c t i o n c o n s i s t s o f t h e b a s i c d e f i n i t i o n s which, t o some e x t e n t , serve t o c l a s s i f y t h e maximal a b e l i a n (MA) s u b a i g e b r a s o f a f a c t o r . p  Next, a  summary'of t h e known r e s u l t s c o n c e r n i n g MA s u b a l g e b r a s h y p e r f i n i t e 11^ f a c t o r i s g i v e n .  of the  We conclude t h i s s e c t i o n  w i t h a complete c l a s s i f i c a t i o n o f t h e MA s u b a l g e b r a s o f  £(3i) ;  a l t h o u g h t h i s r e s u l t xvas known t o von Neumann, i t s p r o o f does not seem t o appear e x p l i c i t l y i n t h e l i t e r a t u r e . If algebra (or  7H  and %  a r e s u b a l g e b r a s .of a von Neumann  G , we say t h a t tt[ and .71 a r e e q u i v a l e n t i n G  simply equivalent, i f G  automorphism o f  G  i s u n d e r s t o o d ) i f t h e r e i s an  which c a r r i e s  IK . onto %  .  This defines  an e q u i v a l e n c e r e l a t i o n on t h e c o l l e c t i o n o f a l l s u b a l g e b r a s of  G  . . One problem i n t h e s t r u c t u r e t h e o r y o f von Neumann  a l g e b r a s i s t o c l a s s i f y up t o e q u i v a l e n c e a l l o f t h e suba l g e b r a s o f a g i v e n von Neumann a l g e b r a , i . e . , t h e d e t e r m i n a t i o n of a l l equivalence c l a s s e s of subalgebras. of course, extremely d i f f i c u l t .  T h i s problem i s ,  • The m u l t i p l i c i t y t h e o r i e s  o f Halmos'[lO] and o f S e g a l [2?] g i v e s o l u t i o n s t o t h e c l a s s i f i c a t i o n up t o e q u i v a l e n c e o f t h e a b e l i a n s u b a l g e b r a s  15.  of a f a c t o r o f t y p e I a c t i n g on a H i l b e r t space o f a r b i t r a r y dimension.  . F o r f a c t o r s o f t y p e I L ^ , t h e analogous problem has  been examined and some r e s u l t s have been o b t a i n e d b y Bures [ 4 ] , R e c a l l that a subalgebra G  i s MA i n G  i f and o n l y i f 711* Let W  D e f i n i t i o n 2.1 Neumann a l g e b r a  ^  G .  o f a von Neumann G =.  be a s u b a l g e b r a o f t h e von  F o r each i n t e g e r  i n d u c t i v e l y define subalgebras N°(7U)  =  algebra  N m (7ft)  m >_ 0 , we of  G  by:  7K  N (?7i) = &(U e G : U u n i t a r y and m  UN m _ 1 (7n) U* = N (T)l)' } . m-±  We w i l l w r i t e  N(fll)  i n s t e a d of  N ^) 1  m > 1 .  , and we c a l l , t h i s t h e  o f 7?l ( i n G) .  normalizer  ( ^ ( W ) ^ 1  Notice that  sequence o f s u b a l g e b r a s o f D e f i n i t i o n 2.2 •  I f Vl  G .  i s  a  n  e x  P  a n d i n  S  . . .  i s a MA s u b a l g e b r a o f f a c t o r  G ,  we c a l l Vfi : (i) (ii) (iii) (iv)  regular i f  N-(1U) = G  semi-regular,  i f • N(Tt)  singular, i f m-semi-regular Wl.  = %, (m >_ 1 and a n ' i n t e g e r ) , , i f  N(7K),.; • . j N ^ f y p a r e  a factor.  i s a f a c t o r d i s t i n c t from  not factors but N ™ ^ ) i s  G  16.  .' D e f i n i t i o n 2 . 5  L e t 7^1 be a MA s u b a l g e b r a o f a v o n  G , and l e t m >_ 1  Neumann a l g e b r a  be an i n t e g e r .  We say  7H has:  that  (i)  proper  (ii)  length  m , i f N " (?n) ^ G  improper l e n g t h N^CW)  m  1  b u t N (??0 = G  m+1  (?n) f G  .  m  m , i f  f N (m) = N m  The d e f i n i t i o n s o f r e g u l a r , s e m i - r e g u l a r and s i n g u l a r MA s u b a l g e b r a s were f i r s t g i v e n by D i x m i e r [ 5 ] , w h i l e t h e n o t i o n o f m - s e m i - r e g u l a r i t y i s due t o A n a s t a s i o [ 1 ] . D e f i n i t i o n 2 . 3 i s a r e f i n e m e n t o f Tauer's l e n g t h o f a MA subalgebra [28]. I t i s easy t o see t h a t i f Tfi s u b a l g e b r a s o f a von Neumann a l g e b r a and  N(7l)  .  Consequently,  and U  are equivalent  G , then so "are  N(7W)  each o f t h e p r o p e r t i e s o f  D e f i n i t i o n s 2.2 and 2 . 3 i s an i n v a r i a n t o f t h e e q u i v a l e n c e c l a s s determined by a MA s u b a l g e b r a . 11^  The study o f MA s u b a l g e b r a s o f t h e ' h y p e r f i n i t e f a c t o r was i n i t i a t e d by D i x m i e r i n h i s s e m i n a l paper Let  G  £Q  on  be a group, and c o n s i d e r t h e group o p e r a t o r a l g e b r a G .  If G  the n o r m a l i z e r o f G ) = R(V Q  [5].  g  . i s a subgroup o f  Q  G  G  : g e G ) c f Q  and  N.(G )' be  i n G , and l e t  Q  c e r t a i n c o n d i t i o n s on algebra of.-'d  G , let  G  G  and  .  D i x m i e r showed t h a t , under G  ^(G )  i s a MA sub-  N(ft?(G )) = 7fl(N(G )) .  Using these  Q  r e s u l t s and c h o o s i n g s u i t a b l e groups  Q  G  and subgroups  G  Q  he c o n s t r u c t e d examples o f a r e g u l a r , a s e m i - r e g u l a r and a  17.  s i n g u l a r MA s u b a l g e b r a o f t h e h y p e r f i n i t e 13^ f a c t o r . The groups used by D i x m i e r i n these c o n s t r u c t i o n s may be d e s c r i b e d as f o l l o w s .  Let  P  be a c o u n t a b l y  infinite  f i e l d w h i c h i s t h e i n c r e a s i n g u n i o n o f a sequence o f f i n i t e subfields  ( i n p a r t i c u l a r , we may t a k e f o r F  c o m p l e t i o n o f a f i n i t e f i e l d ) , and l e t K group o f n o n - z e r o elements o f  F . •'  the a l g e b r a i c  be t h e m u l t i p l i c a t i v e  The* s e t  K x F  becomes  a group under t h e o p e r a t i o n ( a , b ) ( c , d ) = (ac,ad + b ) The group  K x F  i s h y p e r f i n i t e and has t h e i n f i n i t e con-  j u g a t e c l a s s p r o p e r t y (see t h e p r o o f o f Theorem 4.1). subgroup  K x {0}  77l(K x {0}) {1} x F  of  K x F  i s i t s own n o r m a l i z e r and  i s a s i n g u l a r MA s u b a l g e b r a o f  i s a normal subgroup  r e g u l a r MA s u b a l g e b r a .  2 x 2  normal subgroup o f the i d e n t i t y m a t r i x . subgroups o f ; H  x  7^l({l} x F)  and  • Let  m a t r i c e s over H  ^ j ^ p > while is a  , I t i s a b i t more d i f f i c u l t t o o b t a i n  a s e m i - r e g u l a r MA s u b a l g e b r a . non-singular  The  H -be t h e group o f a l l F  and l e t L  be t h e  c o n s i s t i n g of a l l s c a l a r multiples of Let  G = H/L , l e t H  Q  and  be t h e  w i t h t y p i c a l elements  b ^ 0  r e s p e c t i v e l y , and l e t G  = H /L .  ,  Then t h e n o r m a l i z e r o f -  13.  G  Q  in  G  is  H^/L  subalgebra of Let  , and  %(G )  i s a s e m i - r e g u l a r MA- •  o  (?Q . F  and  K  be as above.  t h a t f o r some subgroups  K  s i n g u l a r MA s u b a l g e b r a o f  of  Q  £  K  K ,  x F  Pukanszky has shown 7A (K  [21].  X {0})  is a  By v a r y i n g  F  and  o K  Q  a p p r o p r i a t e l y , he c o n s t r u c t e d a sequence o f p a i r w i s e  i n e q u i v a l e n t s i n g u l a r MA s u b a l g e b r a s o f the h y p e r f i n i t e I L ^ • factor.  The mutual i n e q u i v a l e n c e o f t h e s e s u b a l g e b r a s \ia.s  e s t a b l i s h e d by means o f t h e m u l t i p l i c i t y t h e o r y o f S e g a l . U s i n g group o p e r a t o r a l g e b r a s o v e r groups o f m a t r i c e s , A n a s t a s i o c o n s t r u c t e d i n f i n i t e sequences o f p a i r w i s e i n e q u i v a l e n t 2 - s e m i - r e g u l a r and J>-semi-regular a l g e b r a s o f t h e h y p e r f i n i t e 11^ f a c t o r [ l ] .  MA  sub-  The i n v a r i a n t  o f p r o p e r l e n g t h was used t o e s t a b l i s h t h e m u t u a l i n e q u i v a l e n c e of these subalgebras.  I n t h e p r o o f s o f Theorems 4.2 and  4.3  t h e groups, used w i l l be d e s c r i b e d . Tauer's c o n s t r u c t i o n s o f MA s u b a l g e b r a s o f t h e h y p e r f i n i t e 1 1 ^ . f a c t o r a r e based on a d i f f e r e n t method. F o r each i n t e g e r 2^ x 2 M_ V i each  p >_ 1 , l e t M^  denote t h e a l g e b r a o f a l l  m a t r i c e s w i t h complex e n t r i e s .  P  Embedding  Mp  in  i n a s u i t a b l e manner and u s i n g t h e n o r m a l i z e d t r a c e on M^ , p  M =  U M p=l  becom'es a p r e - H i l b e r t space.;' l e t p  34. denote i t s c o m p l e t i o n .  We can r e g a r d  M  as a s e t o f  19.  operators  on  M  by l e t t i n g each element o f  by l e f t m u l t i p l i c a t i o n . generated b y  M  M  a c t on  The von Neumann a l g e b r a  G  i s t h e h y p e r f i n i t e 11^ f a c t o r .  M on  U  Tauer  . c o n s t r u c t s examples to' show t h a t : (i)  f o r each i n t e g e r  .  (ii)  m >_ 2 ,  G  contains  pairwise inequivalent semi-regular of proper l e n g t h  m  f o r each i n t e g e r  m >_ 2 ,  ([28],  m  MA s u b a l g e b r a s  .[29])  G  c o n t a i n s an m-semi-  r e g u l a r MA s u b a l g e b r a [ 3 0 ] . The remainder o f t h i s s e c t i o n i s t a k e n up w i t h t h e c l a s s i f i c a t i o n o f t h e MA s u b a l g e b r a s o f  £(#). '  ••Lermiia-2; 4—'—Let- -—fl~--b-e~a--Hilbert- space o f d i m e n s i o n a t l e a s t two, and l e t VI there i s a family with-, 2  one. for . M I  ( i)j_ j E  As  o  €  E.' = I ..  Proof.  be a MA s u b a l g e b r a o f f  £(31)  minimal p r o j e c t i o n s i n  7h - i s MA on ' W , each  E^  must be o f rank  E^cp^ = cp^ f o r each  I e l .  (cp.).  Suppose t h a t an E^  A e £(W)  T  In particular,  must c o n t a i n a t l e a s t two elements.  As each  ?t  Then 7*1 i s r e g u l a r .  Hence we can s e l e c t an o r t h o n o r m a l b a s i s such t h a t  such t h a t  .  commutes w i t h each  E^ .  i s a minimal p r o j e c t i o n , a simple c a l c u l a t i o n  •shows t h a t each  E^ A E^ e ?ty . - And as • A = E iel  E. A 1  =  E E. A E. i e l 1  1  20.  i n t h e weak t o p o l o g y on  £(34)  A e TA  F o r d i s t i n c t elements unitary'operators  U. . and  r  i  and  V. . on  j  of  I  3  define  34 by s e t t i n g  k  cpk  /  k = i  <  k *j .  k ^i , j k = i k = J )• for a l l k e I . each  .. G i v e n an  U. . A(U. . ) * and each  A e Vl, i t i s easy t o v e r i f y  V. . A(V. . ) * commute w i t h e v e r y  ,• and hence u «1«l(y l  i;I  )*.-v ,^(v l  lil  )«.  M  .Therefore ft(U e £(34) P  U u n i t a r y and ±j  and so i t s u f f i c e s t o show t h a t i f an w i t h each • \  U. . and each i j Suppose t h a t  and each  V.  U ? f t u * = Wl) t=>  R(U ., V . : i , j e I ±J  V. ." , t h e n IJ  A e £(34)  F o r each  that  k e I  and  i ^ j) ,  A,e X(34) commutes A e C,, . «  commutes w i t h each we can w r i t e  U.  21.  Acp  =  £ a,, cp. . where t h e a,, -f e l "  Fix  i , je I £  with  On comparing  A  Then  a. . cp, = Acp. = U. .Acp . =  Jj. V  that  i ^ j .  a r e complex numbers,  *k "  A t p  x  E  a, . U. . cp.  - i j *>J -'  " k j i j *k •  V  V  c o e f f i c i e n t s i n t h e s e two e x p a n s i o n s , we see  a.. = a . . , a. . = a., and a. . = - a . . , and t h e r e f o r e n 33 13 3~^~ ^J J % •  e  Lemma 2.5 where  E  L e t (X,£,u)  be a f i n i t e measure space,  is. a a-algebra o f subsets o f X .  F o r each  cp e L** (X,£,|i)' , t h e r e l a t i o n 0  ( \ ) ( x ) = c p ( x ) f (x) f  f e L (X,£,n) 2  2 e £(L (X,£,n)) , and cp -  d e f i n e s an ^-isomorphism  o f L°*(X,£,u)  and x e X  ~~ i s an i s o m e t r i c  onto a von Neumann a l g e b r a  which  i s MA i n £(L (X,Z,g)) . 2  Proof. [11;.pp.  Easy c a l c u l a t i o n s ( s e e e.g. [6; pp. 117-118] o r  6-14]).  Lemma 2.6  L e t X = [0,1] ,  s e t s o f X , and l e t M = L (X,E fc) 2  a  ^  and l e t  be Lebesgue measure on  £ .  = [u^..: cp e L~(X,E,\)}  sub-  Let Then  i s a r e g u l a r MA s u b a l g e b r a o f Z(tt)... Proof.  Let  \  l e t . I. be t h e B o r e l  By Lemma 2.5, Tfi i s a MA s u b a l g e b r a o f £(l!) .  r e (0,1) be a f i x e d i r r a t i o n a l number, and l e t  T : X  X . be a d d i t i o n by  .the map o f of  H .  f - f»T  1 .  modulo  I t i s clear that  i s a unitary transformation, U fllu* = IK  Moreover,  cp € L ° * ( X , £ , X )  r  , fori f . f € M  say and  are a r b i t r a r y ,  .  U M^ U* f = U M ^ f o T " ) = U ( c p . ( f . T " ) ) 1  •>}••,  ;  =  1  = (cp.T).f = M ^  U ,  f .  To show t h a t  7*1 i s r e g u l a r , i t x v i l l s u f f i c e t o show t h a t i f  an  commutes w i t h  in  A € tH  then  A € C  n e 2 , l e t cp (x) = e  simple  ( * C  )  N  )  N E  2  i s  3 X 1  o r  ^  n o n o r m a  l  basis f o r  Lto .= e n each i n t e g e r n . Now suppose t h a t an o p e r a t o r V commutes w i t h U and w i t h each u n i t a r y i n 7k.. .  n e.Z  c a l c u l a t i o n shows t h a t  we can w r i t e  complex numbers.  . S •a m=  _*»  Acp = n  Then  r  i s irrational,  v 27 e  for  F o r each ,  are  u A cp  wxnz  .  • cp n A e  cp^ , where t h e  m n  ^n  '  As  Z a m=-«° :.  cp = Acp = e. *  mn m v  operator  , x e X ;  2 7 r i n X  n  i t i s well-known t h a t -M ' . A  and w i t h each u n i t a r y  .  y  •'• F o r each  U  ^n  = -27rinr m=-oo e  • m=-o»  mn  m  mn  ^-(m-n)r ^  ±  ^m  u  n  l  e  s  s  m  =  n  .  comparing c o e f f i c i e n t s and u s i n g t h i s remark, we see t h a t = 0  unless  m = n .  Consequently, there i s a f a m i l y "  ((CL.V,, o f complex numbers such t h a t a iic ci  Ao) n = a n cp_n f o r each  23.  n e Z .  Now f o r each i n t e g e r  op e r a t o r i n  n , M  and t h e r e f o r e  i sa unitary  n  acp=Acp=M Acp = a M cp = a cp . o o ^o cp -n -n cp^ -n -n o n  Thus  a  Q  = a  +  1  = a  +  2  = •••  3  and so A e  L e t H i . be a MA von Neumann a l g e b r a on 3i  Lemma 2.7  which p o s s e s s e s no m i n i m a l p r o j e c t i o n s . Proof.  As  34 i s s e p a r a b l e ,  Then  there i s a u n i t  x e 34 w h i c h i s s e p a r a t i n g , f o r 1 H , i . e . , imply  M = 0  [27; Lemma 2.5].  [6; p.6] shows t h a t  x  i s regular.  A simple  i scyclic for  vector  M e 7fy and M x = 0 calculation [ I H x ] =34  = W[ , i . e .  A p p l y i n g now [27; Lemma 1.2],  t h e r e i s a compact H a u s d o r f f  space  \i -on t h e a - f i e l d  X , a r e g u l a r measure  by t h e compact s u b s e t s o f X isometry {  Mcp  o f J! onto  : cp e i f ^ X ^ u ) } As  with  L (X,E,|i;) 2  £  generated  M-(X) = 1 , and a l i n e a r  carrying 7 H  .  onto -  D t does n o t p o s s e s s m i n i m a l p r o j e c t i o n s , t h e  measure a l g e b r a o f  (X,S,u) i s non-atomic.  Let'  (f ) R  everywhere-dense sequence i n L (X,Z,)i) , a n d . f o r . e a c h let .E  E  n  = ( x e X :' | f (x) - 11 <_  e 2 .  such t h a t  Given  E e E  ..."  be a n . n. ,  C e r t a i n l y each  and e > 0 , t h e r e i s an i n t e g e r . n '  24.  -  .'.  > J  |f (K) - 1|  E-E  , where  |f (x)|  n  ,  n  denotes symmetric d i f f e r e n c e .  algebra of  (X,E,|i)  i s separable.  Hence t h e measure  By a c l a s s i f i c a t i o n  theorem o f Halmos and von'Neumann ( c f . [9; t h e r e i s an isomorphism .(X,E,u)  onto t h a t o f  B o r e l subsets of  $  of A  ; i  S  §  173]),  A  ( [ 0 , 1 ] , ^ , X) , where  X  [0,1] and  as such,  p.  o f t h e measure a l g e b r a o f i s the  i s Lebesgue measure on A  I n an o b v i o u s manner, we can r e g a r d into A  dM(x)  2  n  E -E  n  > i n(E A E )  A  dji(x)'+ J  2  n  §  i s n o t onto  .  as a mapping from  Z  , - b u t e v e r y member  e q u i v a l e n t t o a member o f A  i n t h e range o f  I t i s r o u t i n e t o check t h a t modulo t h e e q u i v a l e n c e  § . .'  relation  "equal almost everywhere", t h e map n . ^  n i E. "".^  a  X  . - • a  i *(E.).  ~  X  a  i. €  C  a n d  E  i  6  E  2  i s w e l l - d e f i n e d , l i n e a r , , and i s an L - i s o m e t r y simple f u n c t i o n s on or  (X,E,u)  of the set of  onto t h e s e t o f s i m p l e  functions  l _ - (C9A],-df , X)_;_ hence t h e map e x t e n d s ' t o a l i n e a r  isometry  of  L ( X , E , u ) 'onto  L ( [0,1],-cf,- X) .  2  2  readily', seen t h a t t h i s i s o m e t r y [M^ .: cp e L°*(X,E,|a)} Therefore to  onto  carries  {M^ : cp e L ~ ( [ 0 , 1 ] , A ,X)}  a c t i n g on  [VL ,: cp e L ( [ 0 , 1 ] , tf ,X)3 ee  I ti s  M  .  i s s p a t i a l l y isomorphic  a c t i n g on  L ( [0,1] ,4 , X) . 2  25-  0  'As t h e l a t t e r i s r e g u l a r (Lemma 2 . 6 ) , so i s t h e former. Remarks (1)  Our p r o o f o f Lemma 2.4 does n o t make u s e o f t h e assumption  (2)  that  W  i s a separable H i l b e r t  space.  S e g a l has shown t h a t Lemma 2.5 h o l d s p r o v i d e d o n l y t h a t the measure space- i s s e m i - f i n i t e ( i n t h e sense t h a t e v e r y s e t o f i n f i n i t e measure c o n t a i n s s e t s o f a r b i t r a r i l y l a r g e f i n i t e measure) and l o c a l i z a b l e ( i . e . t h e measure a l g e b r a i s complete  as a p a r t i a l l y  • ordered set) [26].. (j5) Lemma 2.7 c o n s i s t s e s s e n t i a l l y i n showing t h a t a MA von Neumann a l g e b r a w i t h o u t m i n i m a l p r o j e c t i o n s on a s e p a r a b l e H i l b e r t space i s s p a t i a l l y i s o m o r p h i c t o {M^  : cp e L°°( [0,1],-$,  .  X)}  This -is e s s e n t i a l l y  ...  due t o von Neumann, and i s w e l l - k n o w n , . a l t h o u g h an' e x p l i c i t p r o o f does n o t seem t o appear i n : t h e literature.  I t c a n be deduced from t h e g e n e r a l  Maharam c l a s s i f i c a t i o n t h e o r y o f measure a l g e b r a s (-cf. [26; C o r o l l a r y 5 . 1 ] ) .  Our p r o o f - a v o i d s t h i s  ' deep theorem, u s i n g i n s t e a d a weaker c l a s s i f i c a t i o n ••. theorem. I f 7YI  L e t 1*1 be a MA s u b a l g e b r a o f £(«) .  s a t i s f i e s t h e h y p o t h e s i s o f Lemma 2 . 4 , s e t e(7ty) = 0 ; o t h e r w i s e , s e t c(fl[) = 1 .  L e t n($l)  be the maximal number •'  of p a i r w i s e orthogonal minimal p r o j e c t i o n s i n (0 <_ n(7)1) < _ o » ) .  The c o m b i n a t i o n  T^L  c(fl) = 0 , n(%) = 0  i s i m p o s s i b l e , whil-e examples o f a l l o t h e r c o m b i n a t i o n s c a n be r e a l i z e d as  L° (X,E,u) D  a c t i n g on  L (X,E,n) 2  w i s e m u l t i p l i c a t i o n f o r some f i n i t e measure space  under p o i n t (X,E,|a) .  \  Theorem 2.8  Let ^  be a MA'von Neumann, a l g e b r a on Jt  771 i s r e g u l a r i f c(flt) = 0 o r i f c 0 H ) = 1 and n(JM) = 0 ; f o r a l l o t h e r p o s s i b l e c o m b i n a t i o n s , does n o t f a l l i n t o any o f t h e c l a s s e s o f D e f i n i t i o n 2.2. Lemma 2.4 [Lemma 2.7] shows t h a t tK i s  Proof. regular i f that  c(tt[) = 0  c(k) = 1  [c(TH) = 1 and ri(?ft) = 0 ] .  and n(M) >. 1 •  f a m i l y o f pairwise orthogonal  Let ( E ) ±  j  i  Now suppose be a maximal  m i n i m a l p r o j e c t i o n s i ntyk.,  and l e t E = E E. iel Then b o t h  E  Notice that  ,. F = I - E .  a r e non-zero p r o j e c t i o n s i n 7 f t .  and F  i s a MA von Neumann a l g e b r a on  %E[WF]  [F(JI)'] ' s a t i s f y i n g t h e h y p o t h e s i s and  o f Lemma 2.4 [Lemma 2.7]  therefore N(W ) = X(E(3*)) , E  The  suffices  t o show, t h a t T ^ ^ ^ p  £(E(W)© P(M)) Let  F(M)  N ( W ) = £(F(M)> . p  c a n o n i c a l isomorphism o f 14 onto  an isomorphism o f 7H onto T ^ © ? ) ^ -  of  E(Jt)  U  E(M,)© F(14)  induces  [6; p. 2 2 ] , and so i t  i s a semi-regular  subalgebra  . and V  be u n i t a r y o p e r a t o r s  on  E(14) and  , r e s p e c t i v e l y , such t h a t . U % ^ U* = % ^ and  vTJlp V* =7flp . E(«)«P(M)  Then  U© V  " i s a u n i t a r y o p e r a t o r on  with ( U © V ) (THg©^) (U© V ) * = W E © 7 ? l  F  .  27.  Therefore that  W  N(ffl ) © N ^ ) E  <= N ^ © ^ )  i s a u n i t a r y o p e r a t o r on  wflYlgeTRjJW* = f/[  .  /  .  C o n v e r s e l y , suppose  E(3f) © F ( X ) w i t h  As automorphisms o f a von Neumann  a l g e b r a map m i n i m a l p r o j e c t i o n s i n t o m i n i m a l p r o j e c t i o n s , . W(E$0)W* = E # 0 Therefore on  E(W)  and  vW  p  W = U$ V , where and  and.'W(0«F)W* = 0®-F . U  and  V  are u n i t a r y operator  F ( H ) , r e s p e c t i v e l y , such t h a t  V* = 7 ^  .  •  U*  =7^  on  li  T h i s shows t h a t  = £(E(M))e£(F(M)) ,  which i s not a f a c t o r . Theorem 2.9 are e q u i v a l e n t i n n(7Kl)  = nCTU)  Proof.  Two MA s u b a l g e b r a s W l and "TL  i f and o n l y i f c(1H) = c ( 7 l )  and  .  The p r o o f o f t h i s theorem i s c o n t a i n e d i n t h e  .proofs o f t h e p r e c e d i n g r e s u l t s .  3  THE MAIN CONSTRUCTION  Throughout t h i s s e c t i o n , point i n  (0,^)  and  G  p  w i l l denote a f i x e d  w i l l denote a f i x e d c o u n t a b l y  . i n f i n i t e group w i t h i d e n t i t y  e .  We b e g i n w i t h a summary o f t h i s s e c t i o n . . f i r s t task i s t o c o n s t r u c t a type I I I f a c t o r a regular  MA  subalgebra  and, f o r each subgroup of  iB(p,G) .  G  ', a t y p e 1 ^  G  of  F o r a subgroup  containing  factor  G , a subalgebra G  of  Q  Our  <B(p,G) ,  7l(PiG,G ) Q  G , we w i l l  use  'Dl(G.>G )  t o - d e n o t e t h e s u b a l g e b r a o f t h e group o p e r a t o r  algebra  £  0  g e n e r a t e d by  r  : g e G } .• g O  {V  \a  N(G )  Recall that  denotes t h e n o r m a l i z e r o f a subgroup  Q  G  of  Q  G .  Our second t a s k i s t o p r o v e t h e f o l l o w i n g s i x theorems, w h i c h c o n s t i t u t e t h e main r e s u l t s o f t h i s Theorem 3»1 /  W  p  ® 3l(p,0,G ) /  o  Let  G  section:  be a subgroup- o f  Q  i s a MA s u b a l g e b r a o f  G  G .  Then  ® R(p,G)  p  i f and  only i f (a) v  : G  o  ' i s a b e l i a n and  i n f i n i t e whenever Theorem 3-2  {g^-'g g o o  - 1  g e G -. G  Suppose t h a t  G  : g^ e G } o o  is  .  Q  i s a subgroup o f ' G  F  of  satisfying (3)  : gi'ven a f i n i t e  subset  t h e r e a r e i n f i n i t e l y many  g  Q  G  and a  e G  Q  g e G ,  such t h a t :  29-  (i)  h,k e F  (ii)  and h g k " = g  imply' h = k  L  Q  i f ' , g $ N ( G ) , then  Q  g g  Q  g  Q  _ 1  k %  • "  Then .  N(73l. ® ^ ( p , G , G ) ) = G  p  ®  ft(p,G,N(G ))  N(G  p  ®  7l(p,G,N(G ))  O  p  ®ft(p,G,G )) = G  p  Q  Theorem 5-3  0  F o r a subgroup  i s a f a c t o r i f and o n l y i f G  Q  .  0  G  of  C -®7l(p,G,G  G ,  has t h e i n f i n i t e  )  conjugate  c l a s s p r o p e r t y (see (1.2)). Let G  Theorem 5-.4 % G  .®fll.(G,G) 0  Q  q  G .  be a subgroup o f  i s a MA s u b a l g e b r a o f G. ® 6Q-.  satisfies condition Theorem 3-5-  If G  Then  i f and o n l y i f  ( a ) o f Theorem 3«1. i s a subgroup o f  Q  G  satisfying  c o n d i t i o n • ((3) o f Theorem 3*2, then N(7R N(G  p  p  ® m(G,G )) = G Q  p  ® 7n(G,N(G-))"  ® 7H.(G,G )) = G ® tH(G,N(.G )) »^ ' o " p Q  Theorem 5-6  0  For. a subgroup  i s a f a c t o r i f and o n l y i f G  Q  G  Q  of  G  G  p  has t h e i n f i n i t e  ® 7ll(G ) Q  conjugate  class property. The a l g e b r a • G  and i t s s u b a l g e b r a  d e f i n e d i n t h e t e x t p r e c e d i n g Lemma 3.10 w h i l e the  7l(p,G,G )  7ty  are  B(pyG) and  a r e d e f i n e d a f t e r Lemma 3-13 and i n D e f i n i t i o n  3.15, r e s p e c t i v e l y . a t t h e end o f t h i s  The p r o o f s o f t h e s i x theorems a r e g i v e n section.  30.  B e f o r e p r o c e e d i n g t o the a c t u a l c o n s t r u c t i o n s , we f i r s t establish a technical result.  [Jf[ ,W,G,g - U ] be a C-system, and g G = G[7Yl,H,G-,g - U ] . L e t TV) be a s u b a l g e b r a o f VK, g o Lemma 3-7  let  let. G  Q  Let  be a subgroup o f  for a l l g e G  .  Q  G , and suppose t h a t  U  g?^  U 0  g = 7K  Q  Then  R( (M ® I & ) ( U  ® V ) : M e THQ .and  c o n s i s t s o f a l l those operators  A e G  with  g € G ) Q  A ~ [M  : g e G]  satisfying: (i) (ii)  M e § M  g  Proof.  "Pi O = 0  whenever  g e G  whenever  g e G - G  o  Let  (?  u  = { E (IM ® I A ) ( U ® V j geF e> « & e> F c G o  9  (?  g  2  Observe t h a t  g  : Me  = [A e G : A s a t i s f i e s Q  Q  c (J  : each  M  e>  e  7)1  o  and  finite }  = fc( (M ® I ^ ) ( U ® V )  1  of m a t r i x  o  VL  Q  and  g e G )  ( i ) and ( i i ) } .  (Lemma 1.2) and t h a t , b y t h e c o n t i n u i t y  elements (Lemma l . l ) ' ,  l?  2  i s a von Neumann a l g e b r a .  A s i m p l e c a l c u l a t i o n t o g e t h e r w i t h an a p p l i c a t i o n o f t h e double commutant theorem shows t h a t  <P  Q  sub«*-algebra o f 6^ , and t h e r e f o r e t h a t  i s a w e a k l y dense <P  c S? . 2  To -  31.  show t h a t  ^  a  2  Suppose t h a t M ® G  T e  , i.e., T  w h i c h commutes w i t h each  and . g e G  .-  F o r any  M U <p* k K  _ 1  g  Q'  w i l l - s u f f i c e t o show t h a t  i i :  k e G  Q  i s an o p e r a t o r on  (M ® I ) ( U ® V ) , M e 7K g g , g,h e G , and M e 7 R , o  Q  T <|> = <f>*(M 9 I ) ( U ® V )<j> , cp* k g k g h  c  g  K  K  1  1  = cp*(M ® l ) ( U ® V ) T cp k  k  = <J>* T(M ® l ) ( U ® V ) (p k  =*g *kh  ^  T  M  0  k  ) (  I  U  k ®  T cp n  h  h  V  *h  = <b* T <J>. . M U. g kh k Y  Let  A e (?  g,h e G  p  with  Y  A ~ [M- : g e G]  he g i v e n .  3  4>* A T cj>, = E <p* A <p _ • keG k g g  n  s  T cj> k ^g ••  x  o 9  =  E k  • =  e  G  Y  o  <j>* T <k . Ms U, g kh k k Y  S <p* T k , <j>* A ck keG ' s  y  g  k  Y  h ••'  h  h  1  For a l l  32.  where t h e sums a l l converge i n the. weak t o p o l o g y .  As an  o p e r a t o r i s c o m p l e t e l y determined b y i t s m a t r i x elements, T e (Pg •  C o r o l l a r y 3.8  Let  G,g - U J be a C-system, l e t  G = G[T^t,M,G,g - U ] , and l e t  G  be a subgroup  of G .  Then R(U  ® V  g  :g e G )  g  Q  c o n s i s t s o f a l l those o p e r a t o r s  A e G  with  A ~, [M  : g e G]  satisfying: ( i ) M„. e <D ' g  A  V  (ii)  M  §>  = 0  F o r each  whenever ft  g e G ° o  whenever  g e G - G o  g e G , let M  s  ..space with.'orthonormal b a s i s • (tp^)^-™ ii l i t T o  =  +  -  a/ 1  be 2 - d i m e n s i o n a l H i l b e r t •  The v e c t o r s  p  form a second o r t h o n o r m a l b a s i s f o r #^ . L e t Fg =  p  •  TUg  =  |]  r  { a  F  g  +  b  F  n e Z . 2  S .  a  j  b  e  Define a u n i t a r y representation setting  US?§  = ?  n  ^  m  e  }  • •  t  n -»  of  f o r a l l n,m e Z  g  .  on Then  by'  33-  (3.1).  F|<U )* U  Lemma 5-9  Proof.  ^ (U )*  g  g  If  =  A e  (7t  )'  g  i s a C-systc-  U ] g  P r o p o s i t i o n 1.4,  i s MA on  g  g  z  g  is  g  2J  M  A^  g  .  F  and  g  - U ] g  2  + b F  g  ,  g  n e Zg  3  Hence  s  a F  = F  g  .  g  [ftt ,M ,Z ,n  show e r g o d i c i t y , suppose t h a t U  g  the p r o j e c t i o n s  i m p l y t h a t t h e C-system  commutes w i t h  2  [7*l ,M ,Z n - U ]  , then  [1H )i ,Z ,n-.2  6  Z  C-system  971  g  n  g e G ,  wtfhich i m p l i e s t h a t J  n,me  'ftg  g  F o r each  a f r e e and e r g o d i c  s  ••"  g  g  F  g  (3.1)  , and  i s free.  To  (a,b e l )  Then  a F^ + b F = U^(a F + b P^)(U^) o 1 1 o 1' 1' g  6  v  = a F which i m p l i e s that As with  F  F  g  f a c t o r of type I ^ tt  g  ®  %  2  A  2  + b F  and  F  g  o  ,  are minimal p r o j e c t i o n s i n  g  G  g  = G[$} , X , Z , n - U ] g  g  is a  g  2  on t h e 4 - d i m e n s i o n a l H i l b e r t space  .  '  Let. A Z  s  1  a = b .  + F f = I , each  g  v  *  be t h e s e t o f a l l f u n c t i o n s from  w h i c h have f i n i t e s u p p o r t . i s an' a b e l i a n group;  we use  G  Under component-wise 0  into addition^  t o denote t h e i d e n t i t y i n A  34.  For each  g e G , let y  be t h e element o f '  1  V*(h)  =  defined by  h = g  0 Given  A  h e G - {g}  a,B e A , vie d e f i n e elements  a A 0  and  a v 0 of A  setting (a A 0 ) ( g ) = min ( a ( g ) , 6 ( g ) }  g e G  (a v 0 ) ( g ) = max ( a ( g ) , 0 ( g ) }  g e G  (we c o n s i d e r 0 _< l ) .  t o be o r d e r e d i n t h e n a t u r a l way, i . e . ,  F o r a,0 e A  , we w i l l w r i t e . a <_ 0  whenever  a A B = a .  Let for  each  li -  ® (M geG  , cp?) , °  g  a e A , l e t cp =  g  ® <P^„\  . geG ^  a  ® ( f l a , c p ) , and geG . °  a  s  and l e t u  g ;  geG  a  (a f i n i t e p r o d u c t i n w h i c h t h e factor's commute). P r o p o s i t i o n s 1.7 and 1.9  we know t h a t  ( P ) ^ c  a  normal b a s i s f o r M , t h a t  [?H,W,A,o; -* U ]  e r g o d i c C-system, and t h a t  G  isomorphic t o  ® geG  g  the group  G  s ;  From ^  s  a  n  ortho-  i s a f r e e and  G p  i s t h e r e f o r e a hyper•  f i n i t e f a c t o r a c t i n g on a s e p a r a b l e H i l b e r t space; i t f o l l o w s from [3; Prop. 5-5]  °^  = G[7S|,34,A,a -» U ] i s s p a t i a l l y  (G ,cp ® 6) . g  a e  TT  =  that  G  p  moreover,  i s o f type I I I .  As  has s e r v e d m e r e l y ' a s an i n d e x i n g s e t i n t h i s  c o n s t r u c t i o n , G^ I s a c t u a l l y independent o f t h e p a r t i c u l a r choice of G . Let ^ = 1V[ ® <C£  35.  T^p  Lemma 5.10 Proof.  is  a  7H.  That  r e g u l a r MA s u b a l g e b r a o f  i s a MA s u b a l g e b r a o f  P For a l l u n i t a r y operators  Proposition 1 . 5 .  G  G  p  .  i s part of  P U e T(l  and a l l  a e A . . }  ( u ® I)7H  As  ( u ® I ) * = u 7H u * ® CA  p  =  u  p  . G„ = R(U ® I , U„ ® V„ : a e A ,  *fyp  U  i s regular.  a u n i t a r y e % ), !  F o r each  g e G , let i  P  Y.'-(¥)'  and f o r each  F  o  -  ( A ) .i F  .  s  a e A , let I  a = 0  M  TT p.. . U(g)=l. g  otherwise  Y  (a f i n i t e p r o d u c t i n w h i c h t h e f a c t o r s commute). that  -  (5-2) Each  P P a  P  p  = P  a A p + a v  ^aA  f  o  r  a  1  - . 1  P  i s a s e l f - a d j o i n t o p e r a t o r on  Notice  W  a  >*  e  A  •  satisfying  36.  • P <p_ = cp . As ?>t i s MA on Li O Li s e p a r a t i n g f o r 7VL . S. = » ® H  Let  cp O  3  i s b o t h c y c l i c and -  = 7K® 7H. .  and l e t  i s a MA von Neumann a l g e b r a on V!  H  £  ,  Then ? H  (cp ® cp )  .  fl  •  u  p  .  cijptii  i s an  o r t h o n o r m a l b a s i s f o r S ,' and c p Q ® c p Q i s b o t h c y c l i c and s e p a r a t i n g f o r TVt . Lemma 3-11 f  (i)  = { £_ c a<a  P  : c  Q  a s t r o n g l y dense s u b - * - a l g e b r a  e £  and a e A} i s  of  containing  j  ^  =  a%<a  P  {  « ^ P P  I t i s clear that  :  C  ^6  e  '  C  of  a  g  e  J  , for.  -  v  .  =  n  .  d  5  1^ e  A}  and t h a t ^ F o r each  '  ( ¥ ) *  €  containing  l i n e a r space c l o s e d under t h e ^ - o p e r a t o r . F  .  .  i s a s t r o n g l y dense s u b - * - a l g e b r a Proof.  1^  Iz  .  is a g e G ,  -  r  f - f e ) * (  p  o - ^ ) '  g (3.3)  FS - V P U - P )  To show t h a t be g i v e n ;  (P  Y  +  P„)  •  i s c l o s e d under m u l t i p l i c a t i o n ,  from ( 3 - 2 ) and t h e o b s e r v a t i o n .(UAB + avB) A (aA0) = 0  i t i s s u f f i c i e n t t o show t h a t f o r each  that  , y e A ,  l e t a,Be A  37.  P  f o r some  c  =  2  Y  E c P a a a<Y  e C .  If Y = 0  t h i s i s o b v i o u s , and i f  Y e A - {0} ,  Y(g)=l  Y  Y  Y(g)=l = Y  g  •  °  P  IT (g)=l  [^== ^pTl ?!  ^ v  1  Y  °  +  p  g  o  ]  °  >  °  w h i c h i s o f t h e r e q u i r e d form ( i n t h i s c a l c u l a t i o n  we. used  (3.3)). t  T h i s shows t h a t which contains I  i s a sub-*-algebra of  and g e n e r a t e s  ,  By t h e .double commutant  theorem, t h i s p r o v e s ( i ) . The p r o o f o f ( i i ) i s s i m i l a r . . Lemma 3-12 such t h a t  If  a < a ,  S =  c •  a<a  0  :e  cJP ' . T = E  a<a  and  d„ ^ 0  imply  a = 0 ,  (STcp ,cp ) = '(Scp ,cp )(Tcp ,cp ) . Q  Proof.  A simple  are  o  o  o  o  '•' 7 " ""'  o  calculation:  (STcp ,co ) = • =  E c d u> ^ p S. LX Q  5a<a  c  a a d  then  '(P cp , P co ) Q  .7  38.  = co d o  F o r each ga  of  A  a e A  and  g e G  we d e f i n e an element  by s e t t i n g (ga)(h) = a ( g h )  he  - 1  Notice that F o r each  (gh)a = g(ha)  g e G ,  f o r a l l g,h e G  and  v  a e A  v  defines a unique u n i t a r y operator  u  u  g  u  g  =  *rf  u *  =  Dn  '  (i)  M e H  g e G  7  g * 5  Lemma 3.13 . •  U  on  g i s a u n i t a r y representation of '  Q  I f an  , where  (ii)  G  Q  G  34  and t h e map  on  34 ;  %  h  e  G  -  U  &  M U* = M S  i s an i n f i n i t e subgroup o f  i s a f r e e and e r g o d i c  ( i ) F o r such an  >  moreover, a  g e. G  satisfies  [?k,34,G,g -» U ]  Proof.  a e A .  the r e l a t i o n U gcpa = cgp a  g -• U  G.  M  e  .  for a l l G , then  C-system.  and f o r a l l g e G  Mcp = U M U* cp^ = U M <s . o g g o g o rt  Y  A  Q  ,  0 39.  If  MCD = £ c ° . aeA  cp ,  then  a  £ c cp = U £ c cp = E c cp = E . a ^a g „ . a „ . a ^ga aeA aeA aeA aeA &  and thus  c„ = c _i a g a  implies that unless  fora l l a e A .  {ga : g e G }  a = 0 , and" t h e r e f o r e M = c  Q  an Let for  Q  fit  e || Mcp || 6 = — > 0 , 1 +.e  By Lemma J . l l ,  As  q  c  3  t h e r e i s an  cp i s Q  [Wl,tt,G,g -» U ]  such t h a t as cp  M e %.. g• i s separating  Q  E  c  a<a  a  *  P  e J  U  Q  consequently  such t h a t  a  || Scp || > || McpJI  Q  - 6 > .0 , and  -  || (S - M)cp || < 6 = c ( | | Mcp || - 6) < e|| Scp || . o  a  o  has f i n i t e s u p p o r t ,  h e G - {e} w i t h  e C .  we c a n f i n d an Now  - 1  a  d  0  a(h) = d(g h).= 0 .  S S* = . £ c a,8<d f o r some  = 0  and 6 = e( || Mcp l | - 6) . o •  S =  || (S-- M)cp || < '6 . '.Hence  As  a  I f t h e system i s n o t f r e e , t h e r e i s  he f i x e d b u t a r b i t r a r y ;  ,  CP .  Q  we know t h a t  M e %. - {0} and a " g e G - {e} e > 0  a e A - {0}  I .  ( i i ) From t h e p r e c e d i n g i s an e r g o d i c C-system.  Mcp = C  , '  _ 1  i s i n f i n i t e , we must have  Q  s e p a r a t i n g f o r TR ,  As  c -i cp„ e a a » ^  P P  cZ 3  a  P  R  = E a<a a  d a  a  P .  a  A p p l y i n g Lemma 3.12 and (3.-3) ,  -  || ir  "  u  O  so? - u s P cp || g 'o g o o h  2  Y  = || F " II Q s  = (F«"  =  M  Sep  l h  y  q  - S ? cp || O O"  2  S S* c p , c p ) . ( F ^ S S* cp  lh  o  - 2(?j F ~ g  l h  o  +  S S* cp ,cp )  O O = 2(p,- p ) || S c p J I v  2  2  ^0*0' .  On t h e o t h e r hand, • || 11  U o g  Sco - U S o g  cp || < o o — 4  11  || P^ p - P^ p II + —< " o Ug S c^o o Ug M c "o 11  + || Ug M  cp - U g S ?o cpo|| o ^o  11  11  < '|| Scp - Mcp || + || Mcp - Scp || o  o  < 2 e. || ScpJI  Combining .these two c a l c u l a t i o n s * e || ScpJI 2  2  2 e  Q  o  .  yields  > 2(p - p ) || ScpJ| 2  2  :  2  >_ p - p  2  .  2 As  p - p  > 0 , this contradicts  and t h e system Let  [T^LM^G  the a r b i t r a r i n e s s of  g - U ] • i s therefore, free. S  B(p,G) = G[$l,M,G,g - U .] .  41.  Lemma 3.14  fc(p,G)  i s a f a c t o r o f t y p e r e a c t i n g on  a. s e p a r a b l e . H i l b e r t space. G  B(p>G)  i s h y p e r f i n i t e whenever  i s either hyperfinite or abelian. We use P r o p o s i t i o n 1.5--  Proof.  f a c t o r f o l l o w s from Lemma 3 . 1 3 . normal f a i t h f u l (U  8(p,G)-  As  t r a c e on  8(p,G)  And as  M - (Mcp ,cp ) i s a f i n i t e Q  f o r a l l M e TK  +  Therefore  and  i s o f type I I I , 7?L cannot  G  c o n t a i n any m i n i m a l p r o j e c t i o n s , w h i c h i m p l i e s t h a t i s n o t o f t y p e I .•  i sa  satisfying  M U* cp cp ) = (Mcp ,cp )  i sfinite.  That  B(p,G)  B(p,G)  i s a f a c t o r o f type  I I ,l L  If  G  i s a b e l i a n , then  by P r o p o s i t i o n 1.6. say  G = U ' G n=l n  ,  Suppose now t h a t  where G, c Q .  n  F o r each  = a((M ® I ^ ) ( U M e 7H  each  B  n  g  h  G  c« • •  d  f i n i t e subgroup o f G . B  (B(p,G) i s h y p e r f i n i t e , i shyperfinite,  , and each G • n  i sa  n , let  ® V ) : g  g e G  f o r some  n  and  h e G ) ;  i s f i n i t e - d i m e n s i o n a l as a v e c t o r space, and more-  over, ' 6(p,G) = R(B  : n = 1,2,...)  [13;  Using j?.(p,G)  Theorem X I I ] ( o r [6; p. 299])> we c o n c l u d e t h a t :  i s hyperfinite.  Let and  = A x G , the group-theoretic  $  f o r a = (a.,g) , l e t  l i n e a r isometry  y  T(cp  R  ®  a  U  = U  a  ® U  a  .  g  . o f . 34 ® ^ ® 34 ® G  a  34 ® §  with;  f o r a l l ' a,£  A  R  and I t i s s t r a i g h t f o r w a r d t o p r o v e ( c f . P r o p o s i t i o n 1.9) [ ih  § -  U ]  G = G,[ifl. TA S ,a. - -U ] , t h e n  of  G  P  onto  e <j  a = (a,g) 7((M  A - fA?'  3  ® a(p,G)  g e  eA  jY  G  .  that  i s a f r e e and e r g o d i c C-system, and t h a t  if  3  product,  There i s a u n i q u e  onto  ® cp ® g) =.cp ® cp ® ( y , g )  A  direct  .  G  i s an' isomorphism  1  N o t i c e t h a t f o r M,N e  and  ,  ® I£)(U ® V ) ® (N ® ' i ) ( U a  a  G  g  ® V ) ) r  _  =  1  g  (3.4) = ((M ® N) ® I ^ ) ( U  D e f i n i t i o n 3-15 a subalgebra and  -G(G ). Q  G  O  .  :  of  )  G , define  of  q  and' s u b a l g e b r a s  (B(p,G)  O  )  =  B(U  g  $.(G )  = r-?h  3(G )  =r  0  o  i  .  a  as f o l l o w s :  ( P , G , G  %  ® V )  F o r each subgroup. G  ^ 1 ( P J G , G  of  a  G  P  ®  V  p  *  ®  g  :  g  G  e  ? I ( P , G , G  Q  O  )  ) ?  ?i(p,G,G )r-i o  -  1  ..  N o t i c e t h a t these s u b a l g e b r a s a r e a l l p r o p e r .  ( G  Q  )  43.  Lemma 3 . l 6  Let G  7l(p,G,G )  algebra  be a. subgroup o f G .  Q  The sub- .'  i s MA In' &(p,G) . i f and o n l y i f . G  0  Q  ,  satisfies (a) : G ' o  i s a b e l i a n and {g g g : g e G } i s . °o o o o _ 1  v  i n f i n i t e whenever Proof.  g e G - G  Suppose t h a t t h e subgroup  d i t i o n (a).  Then  G  .  Q  satisfies  q  con-  ?l(p,G,G )' i s an a b e l i a n a l g e b r a , and t o  show t h a t . i t i s MA i n &(p,G) , .we m u s t . v e r i f y t h a t 6(P,G) a ( 7 l ( p , G , G ) ) ' c 7l(p,G,G ) o  Let  B e S(p,G) r\ (7l(p,G,G ) ) ' o  given. h * G (3.5)  with  B ~ [M : g e G ] be g  From Lemma 1.2 we have t h a t f o r a l l g e G and 0  •  +*  (3-6) w  .  0  '  Y  (3,7)  B(U • V ) £  H  H  H  B  U*  = M  H  G  <p\ (U, ® V. ) B cp U*. = U, M _ i gh h h' e gh h h gh v  y  (<P* * ^ oe^oo ) o= ( e  E  B  2  M k  k kis.o ^ o M  €  c p  c p  Q  )  =  2  U* n  I' M o " keG K  where t h e e x p r e s s i o n s ( 3 - 5 ) and ( 3 - 6 ) a r e e q u a l . then  M  = U, M  implies'that  U*  M  0  e € .' <fi v  g h e G  for a l l h e G  2 0  '  If g eG  , w h i c h , by Lemma 3 - 1 3 ,  I f g e G - G ., t h e n f o r a l l . • o  , . II MP II 0  = II U  h  M . h  l g h  U* cp M o  = || M _ h  lgh  cp || o  ;  44;  "by ( 3 - 7 ) and c o n d i t i o n and that  ( a ) , t h i s means t h a t  consequently that  = 0 .  M  B e T K p ^ G j  .  J[ (p,G,G )  G  is finite. s  F = (g S g^  ( a ) f a i l s , t h e n , as  a b e l i a n , there i s a  Q  :.g e G 3 0  Q  Let "B =  B e U5(p,G) h e G  1  0  ,J.  any  i s a MA sub-  such t h a t  Q  \  then  Q  I f condition  Q  G  now i m p l i e s  '  7t(p,G,G ) . a b e l i a n i m p l i e s g e G -  Q  3.8  Corollary  C o n v e r s e l y , suppose t h a t a l g e b r a o f B(p,G) .  cp = 0 ,  M  Q  2 heF  U  ® V  n  ; '  n  and, as  '  i B | 7Z(p,G,G ) .  g e F ,  Q  For  and k e G ,  kh e F otherwise - 1  and  . .'• ;  •  <j>* ( U ® V ) B cp U* h  = U  h  K  h  e  <p*_ h k x  T  B <p U -, U* h k . e  I -{0  where we used Lemma 1.2.  =  X  -1, h k e F otherwise ,  "  As  h  - 1  h  k e F  i s equivalent t o  '  •1th" e F 1  all  (h e G  h € G  , k e G)  B commutes w i t h  i . e . , B e (71 ( p , G , G ) ) ' .  U  h  ®  for  This i m p l i e s that  Q  7\ ("£>,&,G ) i s n o t MA i n B(p,G) , w h i c h i s a c o n t r a d i c t i o n . 0  Hence c o n d i t i o n  (a) must h o l d .  Lemma 5.17 with  Let  G  A ~ [M-.': a eG]  be a subgroup o f  Q  i s an element o f  n  G .  An  A e G  ( i ) # (G ) i f  and o n l y i f M  eft®  a  C  M  a e {0} x G o otherwise ,  M„ = 0 a and  (ii)  -cf ( G )  i f and o n l y i f  Q  e  K  M Proof.  o  C  *  aeA  x G,  o  otherwise  = 0  a  Using  ?(G )  ®  m  .  [6; p. 57] a n d - ( J . 4 ) /  =?  [fa® C£ V * & ( p , G - , G ) ] 7  -1  0  =7 R(M ® l£ ® U  ® V  g  g  :  M e 7H , g e G ) 7 Q  € G)  S  and"-?(Gj = ?G^ ® 7tfp,G,Gj7 P = 7 R((M ® I ^ ) ( U  a  Q  - 1  ® V ) ® (U a  aeA  ® V ) :  Me^,  , g e G ) T -1 Q  _ 1  =  a ( ( M ® I ^ I j ) ( U  ® V  M  M  )  a e A , g e G  :  Q  MeTH,  ).  The d e s i r e d c o n c l u s i o n s now f o l l o w from Lemma 3 - 7 .  Lemma 5.18  L e t 71  on t h e H i l b e r t space X .  Let  that •  M =  be an a b e l i a n von Neumann a l g e b r a  K , and l e t x and  E M. N. i e l 1  j  and  1  be a non-zero v e c t o r i n  be two f a m i l i e s i n %  N = S N. N* i e l 1  II  Proof.  M x||  <  2  e x i s t i n 71 i n  1  the s t r o n g t o p o l o g y / a n d suppose t h a t  N <_ I .  £ || M. x|| iel  such  2  Then  .  As lH i s a u n i f o r m l y c l o s e d commutative  B*-  a l g e b r a w i t h i d e n t i t y , t h e Gelfand-Naimark r e p r e s e n t a t i o n theorem ( s e e e.g. [7; p. 876]) g i v e s an i s o m e t r i c *-isomorphism  f ^ o f 71  A  H a u s d o r f f Space.  Let F  onto  I >_ N _> N  p  ,  1 >_ f  P a s s i n g back t o % ,  N„ = E N. N* . 'ieF '  1  1  >_ X I w -I F ieF i f  K  |M | b  some compact  be an a r b i t r a r y f i n i t e s u b s e t o f  I , and s e t . M™ =• £ M. N, and • ieF • 1  C(X), X  2  < -  E i  €  P  >  a  :|.M, | 1•  n  2  d  1  consequently  , and t h e r e f o r e  As  47.  || Mrp x|| •  2  = (M  X,X) 1- 2  (M- M* x,x) = S || M. x| ieF ieF  p  1  . T a k i n g t h e supremum over a l l f i n i t e  1  1  P e l , we a r e done.  In o r d e r t o s i m p l i f y t h e n o t a t i o n , l e t X the i d e n t i t y i n Q  . Lemma 3.19 operator  U e G  given. •  ( c f . [20; Lemma 15] )• U ~ [M  (N ) b  •  b e 3  ;  IF~ o f - $  e > 0 be  of Q  3  containing  o f elements o f ?  ( i ) . || M cp Scp  1  b  . (ii)  Q  i f V.=  o  such t h a t  b  o  (3.9)  || cp* [tJ T  Proof.  .(5.10)  d  U* - V T  for a l l b e £;  o  a  d  =  2  U* cp © c p j l Q  i n [0] x G . M  where -.the sum converges s t r o n g l y . such t h a t  D  V*]  F i x c . and d  %  of ^  , there i s a  S (N, ®IA)(U, ®V. ) , t h e n f o r a l l betf 9 c,d € {0} x G ,  I g = cp* tf U* <f>  $  3  such that:.  - N cp ®cp || < |  D  subset  and an o  family  .'. ' y  Let a unitary  : a e9 ]  Then t h e r e i s a f i n i t e s u b s e t  f o r any f i n i t e subset  (3;8)  a e Q , l e t T„ = U_ ® V  and f o r each  with  denote  a  M*  <  .  By Lemma 1.2,  ,  Hence t h e r e i s a f i n i t e  48.  ( £ a* Y 3  M*  TA  o  a  a  ®  cp  °  0  F i x a f i n i t e subset ¥ =  ,  cp  ®  cp  )  °  0  3  cp  of 3  =  £  a&  * o  containing  II  M  a  4  0  3  ®  cp  cp  II  -  0  <  2  2  5  ^  6  and l e t  S (M^ ® I ) ( U ® V ) be3 b  A g a i n u s i n g Lemma 1.2, w e . f i n d t h a t  * c  *d  0  E ae  t?  ^ hK  [ ^ -1 U T ca  4 *  a  E M -I -] U ae9 c a ^ d " 1  a n )  =  fi  S  ae$ and  u  a  c  U* _ i ] U _ ! [ U M*_! U*] U* _ ca ca a ca a  a  a  ±  M* -, U* a" c •  c  1  r_! c ad  (J*-., c  similarly  (3-12)  <j>* W T W* <p U* = c a ^ c  £ ' M U."M*_i . U* , ^' c -^ad c a  a e  c  -1 where t h e sum i n (3.11) converges s t r o n g l y and 3' = J? A c g d (we u s e t h e c o n v e n t i o n t h a t t h e empty sum i s z e r o ) . o f (3.10) and Lemma 3-18, we o b t a i n || <j>* [U T  d  U* 1 W T  = || E M U a  «^'  a  c  W*] i  d  M*_-i 1  c  ad  t  U* c p OP cp. Q  U cp ® cp c^o  By means  49.  < II  2  ae^-^  :  I Ia|3=?  +  , M  a  U  c  M i c ad  a c C - ad l *  S  5  c  • • aeff-ff  c  ad  U  c  c  °  cp ® cp || + -°  K °%  ®  °  c  °^o'H  9  '  " + ( Z l|M cp ®cp [| )* . a$<? 2  a  o  o  < ( 2 || M * _ i <P « <P II ) * aetf-I? c ad ° o /  +  2  Q  0  J.O  I t f o l l o w s from (3.10) t h a t each  i s i n the u n i t a  ball o f m  .  Hence, by Lemma 3.11 and t h e K a p l a n s k y d e n s i t y  theorem [6; p. 4 6 ] , t h e r e i s a f a m i l y i n the u n i t b a l l of H  where (3.8)  \  l  n  (N^^g^  1 6 1 1 1 6 3 1  *  3  such t h a t  *o ® ^o ~ \  Vq®  ^O'I ±  m  i  n  <f  >  I5n')  i s t h e number o f elements i n 3 .  i s satisfied. • •  °^ e  Letting .  have t h a t ( c f . (5112))  '  f o r  a 1 1  b  '  6  In p a r t i c u l a r ,  V = S ' (N. ® I ) (U. ® V, ) , we. be3 D  15  D  D  >  50.  .  !U*A  T ..¥* - V  = ||. S ae3 1  E  C ^  -  +  <  (M U  M*_!  ^ a ^c  || M  S ae3  V*]- ^  d  c  I' ae3,  ••• .  T  D  a  c  a  U  U  ,11  c  c  c  a  ® cpj|  Q  U* - N  U  d  ad  c  a  a  d  J c. *°o ad. 5 c  - N*.  l a d  =  N*_! . U*)«p ®ep || c  N * _ l „ U*(M c  cp  ad, ." c  (M*.  c  U*  lad  ) U*  - N ) a  ® %°  H  +  • cp || +  %  o  cp o  ® cp || o  S II '(M*_! - N*_ ) cp ® cp || + . ae3' c ad c -'-ad o o !  x  +  £  ,JK  f  i  a  " V  * o * 'o'l  Combining t h e l a s t two i n e q u a l i t i e s b y means' o f t h e (3-9).  t r i a n g l e i n e q u a l i t y gives the estimate Lemma 5.20 subgroup o f G  ( c f . [20; Lemma 1 7 ] ) .  Suppose  G  i sa  satisfying P  ($) :  g i v e n a f i n i t e subset  \  t h e r e a r e i n f i n i t e l y many (i) (ii)  h,k e F  and h g  F  g  Q  and a- g e G ,  e G  k"" = g 1  Q  i f g | N ( G ) , then Q  of, G  g g  Q  such  Q  imply g" | G  h = k  1  Q  that:  Q  .  .  51.  N ( f ( G ) = N(JJ(G )) = J ( N ( G ) ) .  Then  0  Q  Proof.  Q  I t i s easy t o see t h a t  i ( N ( G ) ) c N(?(G )) , o  I(N(G )) c N$(G )) .  Q  o  Q  i C o n v e r s e l y , suppose t h a t we a r e g i v e n a u n i t a r y o p e r a t o r U e 5  s a t i s f y i n g .one o f  (3.13)  U $ (G )U* =  (3.14)  U 3(G )U*  Q  (?(G ) Q  = 1(G ) .  Q  Q  We w i l l be done i f we can show t h a t Let  U ~ [M  : a  °  a  °  U e ^ ( N ( G ) ) .. Q  ] > and f o r each  a,0eA  a e ^  a  , let  p  \  1  where t h e 8 ( a ; a,©)  a r e complex numbers.  Suppose we knew  that (3.15)  8 ( a j a,8) = 0  .whenever  (3.16)  6(a; a 0 ) = 0  whenever  If  }  a'|'A x N(G ) > t h e n fi cp ® cp  separating f o r for a l l  > M  = 0 .  = 0 ;  as  6^0 a | 4 x ( N  cp ® m  is  And i f a e A x N(G ) , t h e n  a_,B e A , M  a  cp ^p a  R  = (P ® P ) M a  = P  G  ® P  p  s yeA  ft  cp ® cp Q  Q  9 ( a ; Y,0) cp ® cp -  Q  )  52. = .( 2 e ( a ; Y , 0 ) P YeA . and t h e r e f o r e U e and  M  e 7*1 ®  a  (N(G ) ) .  cp ) ® cp ,.  a  T  H  Lemma 3.17 now i m p l i e s t h a t  .  Hence i t i s s u f f i c i e n t t o show t h a t  (3.15)  (3-16) h o l d . Fix  e > 0  (cx,g) e £j  an  a f i n i t e subset  of  3  '(^)-beg;  o  and, w i t h  J  elements o f  f  By t h e . f i n i t e n e s s o f 3  3 =  ((ajg)}  and\the d e f i n i t i o n o f a(b; 3 , Y )  (3.17)  a(b; P,Y) P  " N. =  U  3  S B,Y<.a  >  a  s a t i s f y i n g ( 3 . 8 ) and ( 3 - 9 ) •  .tf  t h e r e a r e complex numbers '  and e , we g e t  A p p l y i n g Lemma 3.19 t o U  be g i v e n .  family  and an ( a ^ o ^ ) e A x A , and l e t  and an ® P  A  p  (Lemma J . 1 1 ) ,  $  a e A  such,that  fora l l b e 3 ;  V  Y  w i t h o u t l o s s o f g e n e r a l i t y , we may assume t h a t  ±> 2  a  a  i ." •  From (3.8)^ (3-18)  | > ||  cp ® cp - \ o  Q  cp ® cp || Q  o  v  > |9(b; B , ) - a ( b ; 6 , Y ) | Y  for  all b e 3  and a l l B,Y _< a .  Let  F = ( h e G : (B,h) e 3 a f i n i t e subset o f (B)  to the set  that (3.1-9)  a  g  - 1  G  containing  g .  F  and t h e element  f o r some M i ] , Applying g  has f i n i t e s u p p o r t , we c a n f i n d a a A g g  g" a = 0 1  Q  condition  and u s i n g t h e f a c t g  o  e G o  such t h a t  53.  (3.20)  h,k e g " F  (3.21)  i f g i N(G ) , then  and h g  1  k  Q  g g  Q  = g  - 1  g  Q  imply  Q  | G  _ 1  Q  h = k  .  In order t o s i m p l i f y the n o t a t i o n , l e t h = g g let  d = (0,g g  c = ( 0 , g ) , and l e t  g" ) .  Let V  1  Q  as i n ( 3 - 9 ) o f Lemma 3.19 , l e t 3?' = 3 A d 3 c " S = <{>* f T Notice that  V* <p U  c  v  1  Q  g  be  , and l e t  . •  d  2?'  (a,g) e 3?' , a l s o t h a t  i s n o t empty.  Now  (3.12))  (cf.  2 N U, N*_T ;' a d -L  S =  ae3  d  = . 2 , ~ a,be<5  U* d  AC  N a Udj N? b U*d  ..  b=d ac _1  If  a = (0,k)  relation Hence  and b = ( y , l )  cT~ ac = b and  (0,g g " g  implies that  1  0 = y  a r e elements o f 3 , t h e  (g k) g _ 1  Q  (g  -  1  1  i)' = g 1  ( 3 . 2 0 ) may be a p p l i e d , g i v i n g , k = & .  Q  - 1  ;  k g ) = (y,-t) as  k,£ e F ,  Therefore  and t h e double sum reduces t o a s i n g l e sum.  a = b ,  ' On s u b s t i t u t i n g  ( 3 . 1 7 ) i n t o t h i s sum we g e t S =  2 N, U , NT U* be!?' 10  =  d  ., 2  -  b  a ( b ; 0 , y ) a ( b ; 6,TU ( P ® P ) • Q  0,Y,6-,n<a V  P  6  • V  0  S  54.  £ a ( b ; B , ) a ( b ; 6 ri)(P ®P )(P ®U P U*) : b 6c? B,Y,S,n<a /  Y  3  p  v  6  h  £ a ( b ; P , Y ) a ( b 6 , T U PpPg ® P P b £<? B,Y,6,r\<a /  5  y  From ( 3 . 1 9 ) and t h e assumption t h a t ' y,T) <_a  • follows that that  Y = "H =  •  a 2  ( 8 6 ® Y h r , *o V o P  P  '  P  = .(P P fl  6  cp ,cp )(P o  o  , 1  ®V  ha  2  *° ® ^  cp ,  Y+hTl  . 6  = 6  Q  and y =  otherwise,  therefore (Scp ® cp , I 9 P c  o  a 2 + h a 2  cp ® cp ) Q  Q  £ | c ( b ; S , a2 )'| be3' B<a 0 ;  1  |a((a,g);  a^Og)!  2  To show t h a t ( 3 . 1 6 ) h o l d s , suppose t h a t (a,g)  £ A x N(G ) Q  and t h a t  T = cp* U T  c  = 0 . U* <p U* t  Let \  •  =  cp ) .  Q  0 . and  together  Hence, f o r a l l . 3 ^ , 6 , ^ <. ° \>  1 P  •  <_ a , i t  + ha^ = Y + hri  and  h T 1  Ti  r\  =  imply-  55.  As  U  s a t i s f i e s one o f ( 5 . 1 5 ) , ( 3 . 1 4 ) ,  (5.21)).  T = 0 (Lemma 3-17 and  The i n e q u a l i t y (3-9) now g i v e s  > ||Tcp ® cp - Scp ® o  I  Q  cp H  o  0  I (Scp © cp , cp ® cp ) I Q  Q  Q  b  >. ! a ( ( a , g ) ; a , 0 ) | 2 C o m b i n i n g . t h i s e s t i m a t e w i t h ( 3 « l 8 ) , we g e t |6((a,g); a , 0 ) | < J0((a,g); a^O) - tr((a,g); 0^,0)1 + x  + !c((a,g); _< e  As  e > 0  i  0^,0)1  .  was a r b i t r a r y , we c o n c l u d e t h a t  and t h e r e f o r e t h a t ( 3 - l 6 )  8 ( ( a , g ) ; a.^,0) = 0  holds.  To show t h a t (3.15) h o l d s , suppose t h a t . a' ^ 0 . 2  If then  g I N(G ) , then,as b e f o r e ,  T = 0 ;  and i f g e N ( G ) , Q  T e 7H ® Cs, f o r s i m i l a r r e a s o n s .  I n any c a s e , 0  and t h e r e f o r e ( u s i n g  >  (3.9))  I I T c p Q ® cp Q  -  Scpc  ® cpo  ||  > !(Tcp ® cp - Scp ® cp , cp ® V ^ ^ l c  Q  Q  Q  o  56.  = |(Scp ® cp , I ® P 0  *  a 2 + h a 2  ® <P )I  0  0  |a((a,g); 0 ^ , 0 ^ ) 1  1  As b e f o r e , t h i s i m p l i e s t h a t  = 0 .  9 ( a , g ) ; a^,a^)  T h i s completes t h e p r o o f o f Lemma 3 . 2 0 . Lemma 3-21  There i s an i s o m o r p h i s m  §  o f . CQ  Q  onto  ?1(P>G,G ) s u c h t h a t 0  s ( V j= U ® o g g .g  f o r a l l g e G-  $ (fll(G,GJ)  f o r . a l l subgroups  v  y  to  = 71(p,G,G^)  n  G . Proof. 34 ® G  with  F o r any  of  G  L e t r\ be t h e u n i q u e u n i t a r y o p e r a t o r on ri(cp„ ® g ) = cp  T e C  Q  ^ let  that  $  with  $ (I) e I .  Q  o  ® g' f o r a l l a e & , g e G .  $ (T) = T|(I ® T)TI  .  _ 1  W  i s a normal * - i s o m o r p h i s m o f C-  G  into  Iti s trivial .£(34 ® G)  I f g e G , then f o r a l l . a e A , h e G ,  §oJ V gj m a ® n = TI(I ® Vg„ ) r\  mex ® n  1  lv  = cp ®(gh)  A  g a  = (U  ® V„)(cp ® n ) a  >  $ -(V ) = U ® V . U s i n g [6; p.57], o g g g t h a t f o r any subgroup G of. G , . and t h e r e f o r e  (  Q  we have  57.  *0(7H(G,G )) = $ (R(V 0  Q  : g € G ))  g  Q  = a(5 (V ) : g 6 G ) Q  g  Q  = 7l(p,G,G ) ; Q  i n p a r t i c u l a r , t h i s implies that *o^G^  = o^G,G)) = §  Lemma 3 - 2 2 into  §(m §(G Proof.  [6;  p  .  There i s a ^ - i s o m o r p h i s m  Gp ® B(p,G)  ^  ft(p,G,G)  5  such t h a t f o r any subgroup  o f G <8> p  G  of G ,  Q  ®1»l(G,G )) = THp ®71(P,G,G )  p  0  0  ®Jl(G G )) = G J  Q  ®?l(p,G,G ) .  p  0  The r e s u l t f o l l o w s e a s i l y from Lemma 3 . 2 1 and  pp. 57 and 6 o ] . Lemma 3 . 2 3  F o r any subgroup  Proof.  G <z G , we may c o n s i d e r o ' *  space o f  As  & .  ( c f . Lemma 3 . 2 1 )  I f T e 7H.(GjG  )  G  of  q  G , G o  CQ  t o be a sub-  and g € G - G  $'(T) • e 7l(p,G,G )  , then  and t h e r e f o r e , ' u s i n g  Corollary 3.8, 0 = (cp*. r , ( l ® T) r f  1  <P cp , cp ) e  Q  Q  = ( n ( I ® T) r f cp ® 'e* , cp ® 1  Q  = (l®Tcp ®ce Q  = (T ^ , |)  .  Q  , cp ®g ) >  c  ~ $£((>  -  58.  Now  G  i s i n v a r i a n t under  and  g e G  Q  W(G,G ) , f o r i f T e 7>t(G,G ) 0  , then b y t h e above c a l c u l a t i o n ,  Q  T g" =  S (T g, n ) n = I ' (T e , ( h g " ) ) n = heG heG : L  =  Hence t h e r e s t r i c t i o n  A  S (T ft) ( k g ) keG^ o  T e 7H.(G,G ) o  §'(T) o f a N  an o p e r a t o r on  6  Q  .  v  I t i s easy t o v e r i f y t h a t  normal *-isomprphism o f  A  ?*t(G,G ) o  into  to G o §'  is  is a  6^ o  U s i n g [6j p. 5 7 ] , *'(7K(G,G )) = $ ' ( R ( V  : g.6 G ) )  0  =  =  R ( * ' ( V  CQ  Q  G  )  :  g  e G  Q  )  .  o P r o o f o f Theorem 5-1-  As f)/[  MA i n - G  is  r e s u l t o f S a i t o and Tomiyama [22] i m p l i e s t h a t i s MA i n B(p,G)  .  condition  G  P  ® (B(p,G)  i f and o n l y i f  B u t b y Lemma 3.16,  D e f i n i t i o n 5-15), G  Q  ®7i(p,G,G  )  i s MA i n  (a) holds.  ^-isomorphism o f  of  71 ( p , G , G )  , a  t h i s i s t h e case i f any o n l y i f  P r o o f o f Theorem 5 . 2 .  ?l  fYl  P  G  As  ® &(PJ>G)  J N(7l) 7 ~  ® B ( p , G ) ., f o r  1  A -f AJ  - 1  "onto  G  = N(7  ft?'"" )  i s a normal  ( c f . the text preceding 1  f o r any subalgeb r a  59-  7'N(7L)7"  1  = V R(U : U e G and  ® e(p,G)  p  U?lU*  = ^(T^T'  1  :  U  E  =70)^-  1  ® ^(P^ )  G  G  P  and  and u n i t a r y ,  unitary,  a j l d  U# U* = % )  = R(U : U e 5 W  and  and u n i t a r y , W  Uf  1  =  /  T I T "  1  )  = N ( y 71 T " ) 1  I n p a r t i c u l a r , u s i n g D e f i n i t i o n 3-15 and Lemma 3 . 2 0 ,  = r  N ( ' ? ( G  i  O  ) ) T  =7"1^(N(G0))7 = G  and  P  ®7l(p,G,G )) = G O  P r o o f o f Theorem 3 . 3 p  ®7l(p,G, N(G ))'  ,  Q  similarly. N(G  G  p  ® 71 ( p , G , G ) Q  ?1(PJG,G ) 0  G  ®?l(p,G, N ( G ) ) Q  i s a factor,  i s a f a c t o r i f and o n l y i f  a f a c t o r [6; p. 3 0 ] . preserved.by  As  p  ft(p,G,G ) O  is  As t h e p r o p e r t y o f b e i n g a f a c t o r i s  isomorphisms,  Lemmas 3*21 and 3 . 2 3 i m p l y t h a t '  i s a f a c t o r i f and o n l y i f  {?  &  i s a factor."  6 0 .  B u t . t h e group o p e r a t o r if  G  Q  algebra  i s a f a c t o r i f and o n l y o has t h e i n f i n i t e c o n j u g a t e c l a s s p r o p e r t y .  P r o o f o f Theorem 5.4  ( c f . [5; Lemma l ] ) .  As i n t h e  Q  and o n l y i f 1U(G,G )  i s MA i n <?  If  Q  TUp^G) =  $ (C )  Conversely,  i f Dt(G,G^)  i f then,  i s MA i n  Q  i s MA i n £  O  Q  (a) h o l d s ,  0  7H(G,G )  and so  ,  .  G  ® £  p  7l(p,G,G ) = ® ( ^ ( G , G ) )  by Lemmas 3 - l 6 and 3 . 2 1 , G  G  7 H p ®-7H.(G-,G )' i s MA i n G  p r o o f o f Theorem 3 . 1 ,  0  C?  . .  Q  i s MA i n ( ? , a c a l c u l a t i o n s i m i l a r r  t o t h a t i n t h e p r o o f o f t h e " o n l y i f " p a r t o f Lemma 3 - l 6 that condition  (a) must be  .Proof o f Theorem 5-5  operator  satisfied. F i r s t of a l l , i ti s clear that  G  p  ®7It(G, N ( G ) ) c N ( 7 H  G  p  ®%(G,  Q  N(G )) c N(G Q  To show t h a t t h e . o p p o s i t e in G  ®' C  p  uflf UG By Lemma 3 - 2 2 ,  p  p  Q  p  p  ®?fc(G,G )) 0  ® m(G,G )) Q  .  -  inclusions hold, l e t U  be a u n i t a r y '  satisfying•one of  ®7H(G,G ) u* = K o  ®?H(G,G ) u* = o  $(U)  shows  ® R(G,G )  g  G  O  p  ® W(:G,G ) . O  i s a unitary operator.in  G  ® B(p,G) IT  such t h a t - e i t h e r • *(V)A  p  ® 1 l ( P , G , G ) 5 . ( U ) * = 5(U?l( o  =^  p  ®7>l(G,G ) U*) .  ®ft(p,G,G_)  O  or  $(U) G  p  ® 7 L ( G , G ) § ( U ) * = $(U G P J  o  = G  By Lemma 3--20, U e G  p  $(U) e G  p  p  p  ®7H(G,G )U*) o  ®71(p,G,G ) O  ®7l(p,G, N ( G ) ) , and t h e r e f o r e Q  ® 7U(G,N ( G ) ) . Q  P r o o f o f Theorem 3-6. Theorem J>. J>.  .  S i m i l a r t o the proof o f  .4  . EXAMPLES OF MAXIMAL ABELIAN SUBALGEBRAS  We b e g i n b y s t a t i n g i n f o u r theorems t h e main results of t h i s thesis.  A f t e r a b r i e f discussion of the  c o n s t r u c t i o n s o f t h e p r e v i o u s s e c t i o n , we t u r n t o t h e p r o o f s o f t h e theorems. Theorem 4.1  Each o f t h e t y p e I I I f a c t o r s  G  p  ,  0 < p < | j c o n t a i n s a s e m i - r e g u l a r MA s u b a l g e b r a . Theorem 4.2  F o r each i n t e g e r  m _> 2  and each  p e (0,%),  c o n t a i n s two 2 - s e m i - r e g u l a r MA s u b a l g e b r a s , one o f improper length  m  and one o f p r o p e r l e n g t h  Theorem 4.3  F o r each i n t e g e r  m . m _> 3  and each  p e (0,|-) ,  G c o n t a i n s two 3 - s e m i - r e g u l a r MA s u b a l g e b r a s , P one o f improper l e n g t h m and one o f p r o p e r l e n g t h m . Theorem 4.4  F o r each i n t e g e r  m >_ 2 , t h e h y p e r f i n i t e  11-^ f a c t o r c o n t a i n s (i)  a 2 - s e m i - r e g u l a r MA s u b a l g e b r a o f improper l e n g t h m  (ii)  a 3 - s e m i - r e g u l a r MA s u b a l g e b r a o f improper l e n g t h m + 1 . The f a c t o r s  Pukanszky, [20]. P  £  (®>W  G  p  , p e (0,-|) , were f i r s t studie.d b y  who o b t a i n e d them b y a m e a s u r e - t h e o r e t i c c o n s t r u c t i o n  I n t h i s paper Pukanszky  a l s o c o n s t r u c t s , f o r each "  a ^ each c o u n t a b l y i n f i n i t e group 1 1  G , a type I I I  63.  G(p,G)  factor algebra MA  in  and, f o r each subgroup  0°(p,G,G ) G(p,G)  Theorem 3.1  G(p,G)  of  o  whenever  G  G  of  Q  That  G , a sub-  6>(p,G,G )  satisfies condition  Q  i s n o t d i f f i c u l t t o show.  is  o  (a)  of  I t i s reasonable to  conjecture that N((P  (p,G,G )) = f f ( p , G , o f Theorem 3-2;  under c o n d i t i o n (6)  p r o o f o f t h i s statement G  P  ® &(p,G)  ) )  however, Pukanszky's  i s not v a l i d .  Our  algebra  .  '  G(p,G).  Powers has shown t h a t i f G^  and  Q  i s o b t a i n e d by m o d i f y i n g t h e c o n s t r u c t i o n o f  Pukanszky's  G^  N ( G  Q  are non-isomorphic;  0 < p < q < % , then  unfortunately, his proof '  C*-algebra techniques ( [ l 8 ] ,  depends h e a v i l y on  [19]).  A r a k i and Woods have g i v e n . a p r o o f o f t h i s r e s u l t which.uses o n l y methods, o f von Neumann a l g e b r a s [ 2 ] ; show, t h a t  • - '  (4.1)  G  where  0  P  ® 8 = G  i s the h y p e r f i n i t e  P r o o f o f Theorem 4.1 (3)  i n a d d i t i o n , they"  o f Theorems 3.1  11^  p e ( 0 , | ) ,  f o r ' each  P  factor.  R e c a l l the c o n d i t i o n s  and 3 . 2 ,  respectively.  We  (a)  f i r s t shoitf  that i t w i l l s u f f i c e to construct a countably i n f i n i t e f i n i t e group and and ik  G  w i t h the i n f i n i t e  a n o r m a l subgroup  G  Q  of  G  ®71(PJ J ) G  g  0  i s a MA  hyper-  conjugate c l a s s p r o p e r t y satisfying conditions  F o r t h e n , by Theorems 3 . 1 ,  (6) .  and  subalgebra of  3-2, G  and  3-3>  ® e(p,G)  with  (a)  64.  G  normalizer G  ®  IB(PJG)  P  ® 7l(p,G,G) , a f a c t o r d i s t i n c t from  .  A p p l y i n g t h e isomorphism (4.1) and Lemma  3.14, we a r e done. We now t u r n t o t h e c o n s t r u c t i o n o f such a G  Q  .  Let F  G  and  be a c o u n t a b l y i n f i n i t e f i e l d w h i c h i s t h e  i n c r e a s i n g u n i o n o f a sequence o f f i n i t e s u b f i e l d s ( i n p a r t i c u l a r , we may t a k e f o r t h e F of a f i n i t e f i e l d ) . .  the a l g e b r a i c completion  The s e t  G = {(a,p) : a,8 e F  and  a ^ 0}  becomes a group under t h e o p e r a t i o n (CX,8)(Y,6) = ( a  I t i s easy t o see t h a t finite.  i s c o u n t a b l y i n f i n i t e and hyper-  To v e r i f y t h a t  class property, l e ta all  G  a 6 + 8) .  G  has t h e i n f i n i t e c o n j u g a t e  (a,8) e G <-• {(1,0)}  For"  e,G ,  ( ,S) Y  (Y^)(a,P)(Y^)  _ 1  = (Y a, Y (3 + S ) ( Y  = ( a , -a 6 If  be g i v e n .  a = 1 , then  ' 8 ^ 0 , and so  elements as  6  runs t h r o u g h  3 + 0  - a & + Y S  t h r o u g h i n f i n i t e l y many elements as and i f a ^ l , - a 6 + y P + 5  + Y  y  Q  ,  -Y  6)  _ 1  .  + 6 = Yf3  runs t h r o u g h  runs  F - {0};  runs t h r o u g h i n f i n i t e l y many F .  I t i s easy t o v e r i f y t h a t G  _ 1  = {(1,8) : 8 e F}  _ ' . • .  65.  i s a normal subgroup o f •(a) ., f o r i f  G .  (a,B) e G - G  The subgroup , then  Q  (l,Y)(a,P)(l,Y)"  1  G  has p r o p e r t y  q  a ^ 1 , and  = (a,B  +  Y  so  )(1,-Y)  = ( a , -a Y + P + Y) runs t h r o u g h i n f i n i t e l y many elements as F .  F i n a l l y , we show t h a t  & "',^ 13  n  6 G  G  Y  runs  has p r o p e r t y  Q  through  (B) .  Let  be g i v e n , w i t h , say, g  = (a ,p )  i  i  i = 1,...,n .  ±  Let H = {(1 - c ^ ) " a f i n i t e subset of  F .  B  g  Q  e F>  H  and i f  g  ±  ^  1  Q  - 6-j) : a  If  g  = (1,B )  g"  = g  Q  1  Q  ±  ±  v  J  = (a. a " Hence then  +6".) ( a " - , - a " ^ . ) 'l' j > J.J 1  8  o  'a ' =• 1 , and.thus i  J -  P  i  = 8^  P r o o f o f Theorem 4.2  . + B. )  = -B. + a.B + B. . j i o ^ i  I  - L  1  v  = (1 - a. )~" "(B. - B.) w  1  , r-a.a~^"B . + a.B  1  a. = a . , and so x o B  B.)-  o  = (a.,a.B i iro  f o r some  , then  ( l , B ) = (a ^ )(l^ )(a o  / 1 and 1 < i , j < n} ,  ±  , a contradiction;  G  a. / -I- , l  therefore  jfj  , i.e.,g  ±  = g^ .  F i x a~ p e (0,|)  " . and an i n t e g e r  m >_ 2 . • Suppose t h a t we had a c o u n t a b l y i n f i n i t e group  If  hyperfinite  w i t h t h e i n f i n i t e c o n j u g a t e c l a s s p r o p e r t y and a  6 6 .  subgroup  G ' of  (4.2)  ( i ) G satisfies condition o  (4.3)  (ii) G  G  such t h a t  Q  O)  G  G  Q  ® &(p,G)  p  N (G )  $lp ® Tl(p*G,G )  Then, from S e c t i o n 3>  ®$L(G,G )  =  '  a  n  d  e  does n o t have t h e i n f i n i t e  class property while  ffl  )  a  c  condition  conjugate  does.  i s a 2-semi-regular  o f improper- l e n g t h  m  i s a 2 - s e m i - r e g u l a r MA s u b a l g e b r a o f  of proper l e n g t h  h  o f Theorem 3 . 2  ( i i i ) N(G )  MA s u b a l g e b r a o f  G  0 <_ k <_ m - 1 , s a t i s f i e s  ,  K  •(4.4)  m  Q  N (G )  •  5 K ( o  % N(G ) ^ v  Q  (a) o f Theorem 3.1  m .  As  (B(p,G)  and  and G  ®  @  a r e b o t h hyper-  f i n i t e 11-^ f a c t o r s , two a p p l i c a t i o n s o f (4.1) completes t h e p r o o f o f t h e theorem.  Hence i t s u f f i c e s t o c o n s t r u c t such a  group  G^ .  G 'and subgroup  A g a i n , l e t ' F . be a c o u n t a b l y i n f i n i t e ^ f i e l d w h i c h i s t h e i n c r e a s i n g u n i o n o f a sequence" o f f i n i t e Let  G-.. be t h e group, o f a l l  over  F  matrices  (g. .)  with  (4.5)  §n ^  (4.6)  g  i j L  =1  (4.7) '  &  u  =0  and l e t G matrices'  (m+2) x (m-i-2)  subfields.  Q  0  be t h e subgroup o f  (g^-j)  i  n  G  with  i = 2,... ,m+2 G  i > j , . c o n s i s t i n g o f a l l those  6 7 .  . 12 = 23 g  s  g . =0  j = 4,. . . ,m+2  g  3 < i < J  2j  The group  G  =0  i ( j  i s c l e a r l y countably  Anastasio: has shown t h a t  G  •  i n f i n i t e and h y p e r f i n i t e .  has the i n f i n i t e c o n j u g a t e c l a s s '  p r o p e r t y and t h a t t h e subgroup  G  s a t i s f i e s ( i ) , ( i i ) , and  Q  .(ill) [1]. ./  P r o o f o f Theorem 4.3. Theorem 4.2. p e (0)i)  ing G  Let the f i e l d  and an i n t e g e r  group o f a l l  P  m >_ 3  (m+2) x (m+2)  be as b e f o r e , and l e t a be f i x e d .  matrices  (g - -) n  ( 4 . 5 ) , ( 4 . 6 ) , and ( 4 . 7 ) , and l e t G c o n s i s t i n g o f a l l those m a t r i c e s  • l2 g  . Then :  The p r o o f i s s i m i l a r t o t h a t o f  . G  =  s  g  13  =  g  2j  =  g  ..  23 g  g  =  s  Q  n  Let over  G  be t h e  F  satisfy-  be t h e subgroup o f  (g--?) • i n  G  34  with  •  24 3j  =  0  = o  i s a countably  *  J  =  5,...,m+2 4 < i <j . .  i n f i n i t e h y p e r f i n i t e group w i t h t h e  i n f i n i t e conjugate c l a s s property  ( s e e [ 1 ] ) ; moreover,  6 8 .  • (i)  G  (ii)  G  satisfies condition  Q  p N(G ) ^  Q  5 N ( m  Q  G 0  ( a ) o f Theorem 3-1 ) =  G  >  a  n  d  0 _< k _< m-1 , s a t i s f i e s c o n d i t i o n  e  a  c  h  N k  (?o  )  >  (6) of  Theorem 3-2 (iii)  . N(G )  and N ( G ) 2  do n o t have t h e i n f i n i t e  conjugate c l a s s property w h i l e As'before, t h i s i s s u f f i c i e n t Before proceeding must f i r s t  N^(G )  does.  t o e s t a b l i s h o u r theorem.  t o the p r o o f o f Theorem 4.4, we  prove i  .  Lemma 4.5  Let p  be a p o i n t i n ( 0 , ^ ) , l e t G be a  c o u n t a b l y i n f i n i t e group, and l e t If  G  Q  G ' be a subgroup o f  G .  s a t i s f i e s c o n d i t i o n (B) o f Theorem 3'2, then NC7l(p,G,G )) o  Proof.  That  = 7l(p,G,N(G )) . 0  f| (p, G,N( G )). c N( 71 (p, G,G ).) i s t r i v i a l . Q  Q  For the converse, l e t a u n i t a r y operator ;  be g i v e n .  U?L(p,G,G ) O  Then  G_ ® B(p,G)  I ® U  U  i n B(p,G) , w i t h  U* = 7 l ( p , G , G ) O  i s a u n i t a r y operator i n  such t h a t  ( I ® U)7Up ®7l(p,G,G ) ( I ® U ) * = ^Jtp ® U ( P ^ G , G ) . Q  A c c o r d i n g t o Theorem 3-2, t h i s f o r c e s and t h e r e f o r e  U e7l(p,G,N(G_)) .  O  I ® U- e G  p  ®7l(p,G,N(G  Q  69.  P r o o f o f Theorem 4 . 4 . p  i n (0,1) he f i x e d .  (m+2) x (m+2)  L e t the f i e l d  matrices over  i n t h e p r o o f o f Theorem 4 . 2 .  F  Then  fo(p,G)  q  • By Lemma 4 . 5 and By Lemma 3 - 2 1 ,  m .  Lemma 3 - 2 3 , and ( 4 . 3 ) ,  -•  N(7l ( p , G , G ) ) = 7 l ( p , G , N ( G ) ) Q  O  N (-n(p,G,G )) 2  0  be as  Q  i s a MA sub-  O  has improper l e n g t h  G  G of  i s t h e hyper-  7l(p,G,G )  a l g e b r a o f 0(p,G) (Lemma 3.16 and ( 4 . 2 ) ) . 71(PJG,G )  and a p o i n t  F , t h e group  and i t s subgroup  f i n i t e II-j^ f a c t o r (Lemma 3 . 1 4 ) and  (4.3) ,  m > 2  L e t an i n t e g e r  = 7l(p,G,N (G )) 2  0  =. £  (  N  G  )  SC 2  .  N  !  0  As t h e n o t i o n o f a f a c t o r i s an i n v a r i a n t under isomorphisms, (4.4)  shows t h a t 7t(p,G,G ) <  O  This proves ( i ) . the  i s 2-semi-regular. The p r o o f o f ( i i ) i s s i m i l a r ,  groups and subgroups from t h e p r o o f o f Theorem 4 . 3 b e i n g .  employed.  REFERENCES  [l]  S. ANASTASIO, Maximal a b e l i a n s u b a l g e b r a s  i n hyper-  f i n i t e f a c t o r s , Amer. J . Math., 87 ( 1 9 6 5 ) , pp. [2]  955-971.  H. ARAKI and E. J . WOODS,  A c l a s s i f i c a t i o n of factors,  pre-print. [3]  D. BURES,  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 products,  C o m p o s i t i o Math., 15  (1963),  . pp. 169-191. [4]  . D. BURES, A b e l i a n s u b a l g e b r a s pre-print.  o f von Neumann a l g e b r a s , '' .  [5]  J . DIXMIER, Sous-anneaux a b e l i e n s maximaux dans l e s f a c t e u r s de type f i n i , Ann. o f Math., .59 ( 1 9 5 4 ) , pp. 2 7 9 - 2 8 6 . '  [6]  J.' DIXMIER, L e s 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 , G a u t h i e r - V i l l a r s . , P a r i s , 1957.  [7]  N. DUNFORD and J . T. SCHWARTZ, L i n e a r O p e r a t o r s , I n t e r s c i e n c e P u b l i s h e r s , New York, 1963.  Vol. I I ,  [83 • H. A. DYE, On groups o f measure p r e s e r v i n g t r a n s f o r m a t i o n s I I , Amer. J . Math., 85 ( 1 9 6 3 ) , pp. 551-576. [9]  [10]  p. R. HALMOS, Measure Theory, New Y o r k , 1950. p. R. HALMOS,  D. v a n No s t r a n d , ,.  . I n t r o d u c t i o n t o H i l b e r t space and t h e  theory of s p e c t r a l m u l t i p l i c i t y , Y o r k , 1951. [11]  '  Chelsea, '  New  '  c. R. KERR, A b e l i a n von Neumann a l g e b r a s , M.A. U n i v e r s i t y o f B r i t i s h Columbia, 1965-  Thesis,  71.  .'"[12]  P. J . MURRAY and J . von NEUMANN,  On r i n g s o f o p e r a t o r s ,  Ann. o f Math., 37 (1936), pp. 116-229-  ^ [13]  F. J . MURRAY and J . von NEUMANN, IV,  On r i n g s o f o p e r a t o r s  Ann.of Math., 44 (1943), pp. 716-808.  ' [14] J . von NEUMANN,  Zur A l g e b r a d e r F u n k t i o n a l o p e r a t o r e n  und t h e o r i e d e r normalen Operatoren,  Math. Ann.  102 (1929)^ pp. 370-427. [15]  J . von NEUMANN,  On i n f i n i t e d i r e c t p r o d u c t s ,  Compositio '  Math., 6 (1938),"pp. 1-77. [16]  J.' von NEUMANN, On r i n g s o f o p e r a t o r s I I I , Ann. o f Math., 41 (1940), pp. 94-161.  [17]  J . von NEUMANN, Theory,  On r i n g s o f o p e r a t o r s .  Reduction  Ann. o f Math., 50 (1949), pp. 401-485.  Cl8]  R. T. POWERS, R e p r e s e n t a t i o n s o f . u n i f o r m l y h y p e r f i n i t e a l g e b r a s and t h e i r a s s o c i a t e d von Neumann r i n g s , pre-print. . •  [19]  R. T. POWERS,  Representations of uniformly h y p e r f i n i t e  a l g e b r a s and t h e i r a s s o c i a t e d von Neumann r i n g s ,  Bull.  Amer. Math. S o c , 73- (1967),' PP• 572-575. ;  [20]  L. PUKANSZKY,  Some examples o f f a c t o r s ,  Publicationes  Mathematicae, Debrecen, 4 (1956), pp. 135-156. . •  [21] • L. PUKANSZKY,  ••'-..  • type I I ,  On maximal a b e l i a n s u b r i n g s o f f a c t o r s o f Canadian J . Math., 12 ( i 9 6 0 ) , pp. 289-296.  [22] . T. SAITS and J . TOMIYAMA, Some r e s u l t s on t h e d i r e c t p r o d u c t o f W*-algebras, Tohoku Math. J . , 12 ' (I960), pp. 455-458. . . •" S  '•"[23]  S. .SAKAI, The t h e o r y o f W*-algebras, Y a l e U n i v e r s i t y , 1962.  Lecture Notes, -  72.  [24]  J . SCHWARTZ,  Two f i n i t e , n o n - h y p e r f i n i t e , . non-  isomorphic f a c t o r s ,  Comm. Pure A p p l . Math., 16  (1963), pp. 19-26. [25] ) [26]  J . SCHWARTZ, Non-isomorphism o f a p a i r o f f a c t o r s o f type I I I , Comm.Pure A p p l . Math., 16 (1963), pp. 111-120. I . E. SEGAL,  E q u i v a l e n c e s o f measure spaces,  Amer.  J . Math., 73 (1951), PP. 275-313. [27]  I . E. SEGAL,  Decompositions  of operator algebras I I ,  Memoirs o f Amer. Math. S o c , 9 (1951), pp. [28]  SISTER R. J . TAUER,  Maximal a b e l i a n  1-66.  subalgebras i n  f i n i t e f a c t o r s o f type I I , Trans. Amer. Math. Soc,  114 (1965), pp. 281-308.  [29]  SISTER R. J . TAUER, S e m i - r e g u l a r maximal a b e l i a n suba l g e b r a s i n h y p e r f i n i t e f a c t o r s , B u l l . Amer. Math., S o c , 71 (1965), pp. 606-608.  [30]  SISTER R. J . TAUER,  •  M-semi-regular  hyperfinite factors,  subalgebras i n -  Trans. Amer. Math. S o c , 129  (1967), pp. '530-541.  -  1  

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