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Inner equivalence of thick subalgebras Kerr, Charles R. 1968

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. INNER EQUIVALENCE OF THICK SUBALGEBRAS by CHARLES R. KERR B.A., Washington S t a t e U n i v e r s i t y ,  1962  M.A., U n i v e r s i t y o f B r i t i s h Columbia, 1 9 6 6  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  .  i n the Department ••  of MATHEMATICS  We accept t h i s required  THE  t h e s i s as conforming to' the  standard  UNIVERSITY OF BRITISH COLUMBIA October, 19 68  In  presenting  an  advanced  the  Library  I further for  this  thesis  degree shall  agree  scholarly  in partial  fulfilment  of the requirements f o r  at the University  of British  Columbia,  make that  permission  purposes  by  h i s representatives.  of  this  written  thesis  i t freely  may  be g r a n t e d  of  gain  Mathematics  The University of British V a n c o u v e r 8, Canada  December 24* 1968  by  Columbia  shall  copying  t h e Head  It i s understood  for financial  f o r reference  f o r extensive  permission.  Department  Date  available  that  n o t be a l l o w e d  and  thesis  Department or  that  Study.  of this  o f my  copying  I agree  or  publication  without  my  ABSTRACT I n t h i s t h e s i s we c o n s t r u c t subalgebras  £  of factors  i s maximal a b e l i a n i n  G.  G.  e  some examples o f t h i c k  is thick in  G  if  ( e ' fl G)  We are concerned w i t h t h e i r  e q u i v a l e n c e : g i v e n the t h i c k s u b a l g e b r a s does t h e r e e x i s t a u n i t a r y . U e G  & and nr  such t h a t  Examples o f t h i c k s u b a l g e b r a s  inner  i n ' G,  U e U * = J? ?  w h i c h are not maximal  a b e l i a n have been g i v e n by D i x m i e r and K a d i s o n .  L a t e r Bures  c o n s t r u c t e d numerous examples w h i c h he d i s t i n g u i s h e d by use o f y  certain  invariants. We use B u r e s ' s c o n s t r u c t i o n t o g e t ,  G.  o f t y p e s 11^,- 1 1 ^ , I I I ,  of t h i c k subalgebras l e n t to  £:  when  of  i ^ J  G  uncountable  in certain  families  such t h a t  factors  {&^: i e J }  i s not i n n e r e q u i v a -  (We are a b l e t o add one example to~  those c o n s t r u c t e d by B u r e s ) .  I n each f a m i l y ,  the  be d i s t i n g u i s h e d by means o f B u r e s ' s i n v a r i a n t s , f o r c e d t o show t h e i r n o n - i n n e r - e q u i v a l e n c e  cannot and so we are  by d i r e c t  calculations.  - iiTABLE OP  CONTENTS Page 1  CHAPTER ONE  INTRODUCTION•  CHAPTER TWO  THICK SUBALGEBRAS OP A  CHAPTER THREE CHAPTER FOUR  THICK SUBALGEBRAS OP A  II  -FACTOR  7 16  III-FACTOR  THICK SUBALGEBRAS OP THE- HYPERFINITE I I - j FACTOR ON A SEPARABLE HILBERT SPACE ..... •  APPENDICES  i'  ...  ...  r  I n t e r n a l r e f e r e n c e s appear thus: § 4 " ' and  "(B.2)"  _^ -39 62  BIBLIOGRAPHY .'  of  22  "(4.9)"  f o r "item  nine  f o r " i t e m two o f a p p e n d i x B". A l l  other references are b i b l i o g r a p h i c a l .  - i i i ACKNOWLEDGMENT I a m ' g r e a t l y i n d e b t e d t o P r o f e s s o r D.J.C. B u r e s f o r his  help  i nwriting  assistance  this  thesis.  acknowledge t h e  o f . M r . O l e A. N i e l s e n .  I am a l s o g r a t e f u l financial  I must a l s o  support provided  the U n i v e r s i t y  of British  C o u n c i l o f Canada.  f o r t h e p r o l o n g e d and generous  b y t h e Department  o f Mathematics o f  Columbia, a n d b y the^ N a t i o n a l 'Research >  ')  CHAPTER ONE INTRODUCTION  The purpose o f t h i s t h e s i s i s to give and d i s c u s s some f a m i l i e s o f n o n - i n n e r - e q u i v a l e n t algebras (1.1)  o f a von Neumann a l g e b r a  Definition  G  Let G  t h i c k subalgebra o f  be  ("t  (vNa).  vNa. is  „c [The  / c\c  ordered p a i r s  (MA)  G  in  is a  G") i f  •  i s maximal a b e l i a n  t c G  The.sub-vNa  TSA- i n  • (B.l)  t h i c k sub-von Neumann  ) (B.2),  that i s ,  „cc  ( B . l ) and  (B.2)  r e f e r t o the appendices,  page 39 tf.] Note t h a t , s i n c e a  TSA  £ c p  c c  i s always a b e l i a n . Note a l s o t h a t an  MA  subalgebra i s a  over, i f G . i s a f a c t o r o f type I then a l l ' vNa  e  f o r any s u b a l g e b r a ft,  TSA  are  MA.  Indeed, K,  on a H i l b e r t space  that i s ,  G  TSA.  [Dixmier, 1 9 5 7 , page 1 2 0 ] , must be isomorphic t o a  such t h a t  a'  i s abelian,  '  -  u  B' c ( B')' = a , .  so t h a t  IB' = iB . Since c  G  More-  i s a f a c t o r , so i s  IR :  so  8 = d8r(M)-  that  invariant S  is  is  ( B . l ) under  TSA  ^-(K),  in  that  MA,  or  non-MA  (Thus  G  the r e l a t i v e  the isomorphism  i f and o n l y  <P  comrnutant i s  of -G  (c?e)° = ^ ( e ) 0  i f  £  is  MA.  is  .  i n non-type  I f a c t o r s we c o n s t r u c t -  i f G  ®, MA  in  ;  TSA  ,  which  ,  t h e "von Neumann d o u b l e comrnutant t h e o r e m "  hold  onto  i s , i f and o n l y i f  But are  And, s i n c e  does n o t  i s not''a t y p e I f a c t o r ) .  We  construct  (1.2)  p  (1.3)  fora l l  y±  'i s  TSA.  (1.4)  for  in  such  that  G  =377, a m a x i m a l  i ,  substantial ' in  (p,^ : ietf}  families  subalgebra  abelian  (MASSA, a p p e n d i x D)  G. £. and p  i /  are not inner  equi-  valent.  (1.5) . D e f i n i t i o n e and ^ of ' M  :  Let  are equivalent  f> and in  B-  be sub-vNa i f there  [aeA(ft) ] ' ( A.3) . f o r w h i c h  equivalent  ::  i f a. i s i n n e r  ,[a G(8) €  of a  vNa I R .  i s an.automorphism  a ( g ) = 3?.' e and 3 . (A.2) ].  a  are -inner  I f ft and tf a r e e q u i v a l e n t as above, t h e n b y a(ft°) = 3 ° . then  a  Hence, i f ft and tf a r e a b e l i a n w i t h  induces  p  an automorphism  ft  (B.l),  =rs ,  0  c  of. ft - = r s : C  0  P = (a|ft ) e A(*,ft ) C  (A.5) . s u c h  p(ft) = tf. M o r e o v e r ,  that  PgP  C  e tf° ' f o r g e  _ 1  P g P e e°  f o r h e tf° , •  _ 1  where ft° =  {aeA(ft°) : aE=E  (1.6)  .  where  p e A( 8, ft ). •'  f ^ V  1  holds  i s thus  'for a l l  existence of a necessary  P e A(ft )  G(5?^£ ) C  (A.6)..  p_° -  from space G  Of?  tf  !  c  the  P  Thus t o v e r i f y  so, t o get the  abelian algebra  and so  that  (1.6)  c o n d i t i o n f o r the equivalence o f  examine t h e a c t i o n o f t h e 077 ~group And  such  0  ft and tf a r e i n n e r e q u i v a l e n t , t h e n to .  (A.8),  Eeft^}  :  t h e a b e l i a n sub-vNa ft and tf w i t h  belong  ;  = tf° ,  0  The  V "  ft°  and a f u l l  [ASA §10]. of  (1.4),  &(g 077) }  : ietf},  (1.6)  If must  we must  (A.7).  we b e g i n w i t h a n  (F.7) 077 - group  G  [obtained  a g r o u p o f m e a s u r a b l e t r a n s f o r m a t i o n s on some measure •  (B.4,5)].  The c o n s t r u c t i o n o f  and a n i s o m o r p h i s m  (1.7)  <P 077  cp  is  o f 077  MASS A  (H)  into  in  G  G.  g i v e s us a such  that  vNa  4. G ( G , c p c 2 # ) = cpGcp"  (1.9)  A(c,cpc37 ) = c N(G)cp"  fixed  Now  i f  thick  of  the type  {H. : i e U } to  o f the  (G.7,8)  G  (A.,12).  A(<^7) vNa  (J.13).  G  -groups  which  i t turns out t h a t the  = cp( °)  • . . ' ! . • ' '  H i  G  S . °  with  this  p o i n t we  =. .cp>  can discuss the inner equivalence •  £. and. £ .  the  same b e h a v i o r w i t h  uniform m u l t i p l i c i t y ciency  type  morphic  show t h a t  Note  (K.10),  thickness  We  to  in  i s a family of  I n e a c h f a m i l y w h i c h we  to  G  of  (A.8,11)  in  At  control  angles"  algebras  >i  are  the normalized  will  at right  ,  1  P  G  group  "act  1  N(G)--is  where The  (1.8)  (.row  ( L , when  (K.4).  ),  G  is a  II]/'-  Thus t h e s e  summarize  (K.7)  invariants  t h e examples  and a u t o -  will  i n the f o l l o w i n g  two c o l u m n s we  of the factor  tt&xraly  does n o t change w i t h  G ( G , < ^ 7 )•  .,factors  G  In the t h i r d  arise  >  vary  not serve •  from  of  table.  G .= G (  G . The d e f i c i e n c y  w h i c h d e p e n d s o n l y on t h e a c t i o n  columns,.isomorphic  factor),. defi-  are not inner equivalent.  that i n the f i r s t  DT  have  to the following .invariants:  'automorphic n o r m a l i t y  the  change t h e t y p e  regard  c o n s t r u c t , a l l members  type o n ^  and f o u r t h  two a p p a r e n t l y  y  -  different not  groups  affect = H/ , 3  G(G,  the d e f i c i e n c y but  This variation type  or the u n i f o r m m u l t i p l i c i t y  i t does change t h e b e h a v i o r  hull  (K.l)  ("Auto H u l l "  row).  gives  a pair  of non-equivalent  G{G,<3&)  in  o f the  Hence i n t h i s TSA  i n the  of  automorphic  case  same  does  each  factor.  H^  Type of G  ^ ( R (2.1)  G ==  0,^77 )  lations  of  R  2  p  lu.  =( v  T  areirrational-slope  DT  • Auto Hull Inner Equivalence  (3.6)'  Ill  Auto  F  (3.6)  (2.8-12)  Normal  a X i d  e  (  C  T  S  )  0  (^.3,22) I  (4.3,22) (4.24)  2  Auto  •Auto N o r m a l  (4.26)  (3.10)  Thick  (4.27)  Uniform M u l t i p l i c i t y 2  (u+rj a n d -v a r e l i n e a r l y i n d e p e n d e n t a l l r _ w i t h r a t i o n a l components, t h e n  nu nv equivalent e  ,  =  (3.6)  (2.7)  If for  3s  e  [0,1]  arei r r a t i o n a l s i n  III  ^ 0  (2.6) •  (2.5)  *2u u (3.6) =  ! (2.2,3)  ^00  s,t  vectors.  0  Lebesgue  (4.12) R a t i o n a l (4.2,3,12) Rational translations flips. modulo 1  2  ).? U = F u . .1 l u u  '(2.2,3) t  in R  on a  (4.12,13,30)  space  £^(0,1)  t r a n s - (3.1) Rational transl a t i o n s and m a g n i f i c a -  Rational  u,v  TSA  and h y p e r f i n i t e  ) Lebesgue . ;  tions  The  11^  separable  on  G(  (3.5)  Ill  "oo  •  ^nv^ 1,2).  8 1 1 ( 1  (n=  a  r  e n  o  t  i  n  n  e  r  "  (4.25)  (4 . 29 , 30,32) P inner (s-t)  n s  and e  n t  are-  e q u i v a l e n t i f and o n l y i f i sr a t i o n a l ( n = 3,4).  7.  CHAPTER- 2 1 1 ^ - FACTOR  THICK SUBALGEBRAS OF A  Let the j(  o  £QQ(  =  X  ( R  2  R  be e u c l i d e a n  2  2-space,  •• ^ l ^ i p l i c a t i o n  r  ,Xg)  (  B , i |  <277  be  a l g e b r a a c t i n g on  where  ')i  and l e t  i s Lebesgue p l a n e  measure.  p y e R  Any v e c t o r formation  T  R  of  i n d u c e s a one-to-one p o i n t -trans-  A  2  :  •'  C  '.  . T x = x+y This sets.  T  y  x e  R '  2  p r e s e r v e s Lebesgue measurable s e t s and  Hence i t induces  an automorphism  \^ - n u l l  of c 2 ^ ( B . 5 ) :  T  2 where  ". i s m u l t i p l i c a t i o n .by Let  JJQQC'R  be t h e e % ? - g r oup  G-^  G^ = { T  g e  R  2: r e R ,  r  has r a t i o n a l  T h i s c o u n t a b l e 0/7 - group a c t s f r e e l y  (A.9)  components} and e r g o d i c a l l y  onc37. Let  be t h e a l g e b r a o f  i s o m o r p h i s m o f Q?J onto ,.  ( 2 . 1 ) Since  cpj^, a 'G^  (H), with  MASS A  cp t h e  i n G^- J  acts, e r g o d i c a l l y so a l s o does  8.  G(G [for  [ G ]  1 3  .  [where  7r  x  infinite  •  E  '. ( J . 1 3 ) , .  by  normal f a i t h f u l  MASSA  of  G  is a  1  contraction  factor.  (Cl)  then  as  a trace  as  in  (B.4)]  s e m i - f i n i t e normal- f a i t h f u l  i t i s easy to of  > E  n  Mapping t h i s has  so,  is  trace  ;  projections  '  and  a  on-the  (J.3)  G-L Now  the  i s the  . i s thought  2  •  1  1  down.to t h e  properly factor  = cp[G ]cp"  (P.6) ]  see  1  If of  cp^)  n  +  type  Thus  X E 2  into  n  < +  G-j_... shows t h a t 1957,  [Dixmier,  Gj  • c3ff P •',  that  ..  1  fE }  e x h i b i t a sequence'  such  sequence  continuous  co  the  factor 3].  proposition  i s a factor of  type  11^  G^  .  .  o (2.2) irrational  the  u  =  group o f  L e  (B.ll).  • Let  slope. F  .  ^  f  now  Let.  ^u  :  k e R  )  u  (T  U  be be  )  any  vector  the  "'  ^lu ' <v  R  with  $77-group  anxLlet  >•  a l l translations' along  ^lu  cyclic  of  ^  e  u.  the \respective :  fixed  algebras  (2.3) is  TSA  in  the v e c t o r  :  r  cover  Given a vector  r  easily  that  with rational  (r-u) - .. ..  r  II ul  #  perpendicular the plane  (b)  >  to  u-- i s a l w a y s non-zero..- Now  u  with a family  of width  the corresponding  n = \  E  -{S  : neuu}  we  can  of s t r i p s which are u  |ip ||  projections  ^  e  >  .  n we  have fE  and  n  for a l l T 1  But  l u  components,  2  ( a ) \ d i s j o i n t and p a r a l l e l t o  For  shows  component^  p = of  The c r i t e r i o n . ( G . 7 )  1 c  P °  u  E  =  E  T  =  (G.2  and E . 2 )  TSA  in  1  < (I - E  p n —  t h i s means t h a t  G-  -n  E  = -I  n  n  r. n  0  v  _  =  for a l l  T  • J  J  T,. I P,,' on  E'(T ,F ) r  )  n  ^  u  E ( r  e  v  I  (G.l)  or  e )  ' Hence, by  '  (G.7),,  ^  l u  is  10. (2.4)  lu e  l u  S i n c e we  c  is also  F  J  G-^  in Now  (J.l4)  of  i  'lU °  TSA  (2.5) the  F  o b v i o u s l y have  £-^  u  arise  has  deficiency  from  u.  parallel  and w h i c h have w i d t h  at  ||u|| . (2.6)  in  (K.10):  c o v e r i n g .the p l a n e w i t h  strips' which are p e r p e n d i c u l a r - to most  1^  type  §15  ASA  induces  The  deficiency  as f o l l o w s :  an a u t o m o r p h i s m  v  type  A rotation  v(F )v~" " = u  u  .is c a l c u l a t e d  o f axes i n the  such  fe } ® { t r a n s l a t i o n on  where t h e t e n s o r p r o d u c t  tf^  plane  °? OT7  1  of--  that y-axis)  o f automorphism groups  (J.17).  ,  i s defined i n  . N e x t an ( a l g e b r a i c ) i s o m o r p h i s m  p  serves to  map  onto  (the way  tensor product that  algebras) i n such  a  .'  1  j°vF v~jO~ u  (where  of-the m u l t i p l i c a t i o n  K  1  =  fe} ® K  i s an e r g o d i c a l l y a c t i n g  group on  S^i®,!]),  f  o  r  . 1 1 .  which the f i x e d  algebra  ( J . l 8 ) ..  is  ~ -oo(H) ® < £ p  (A.11),  Hence  pv(P °°)v-y u  Now  A(.£  for by  [0,1])  has e r g o d i c  another theorem o f  group o f type  This  (2.7)  P °°  ..  =  (H.6).  11^  ( ^  l u  u  See  auto ^  (J.12)-and s o ,  [6,1])  III.  Thus  i sa #  I u  I I I i n G^. and  I?  l u  are not (inner or  i n G-^.  for  s  {e} $> A ( £  has type  u  We' show t h a t  e  auto  [by  iSj  ft^  u  The a u t o m o r p h i c h u l l  (A.10)  :  subgroups o f type  shows t h a t  outer) equivalent  (see  A(^[0,1]'T  (J.19),  ASA  I I I , that  d e f i c i e n c y type  normal.  ®  t h e g r o u p o f t r a n s l a t i o n s modulo 1 - -  instance,  has  fe]  =  i  = C  { ( e  l u  l  .(K.l)  )  u  and  0  n A ( G  are automorphically  of  p  ^ ) l  p.  lu  :  0  °)  ) ° ° n (A. 1 2 )  N([  G  ]))°)  l  f o r N( [G-jJ) ].  And i n t h e same way,  = c p ( f F ° ^ N( [G^ ]) ]°) 0  u  .Now f o r a n y r e a l number  k  and a n y  T  r  i n G-^  12.  -r T T-  T.  T  =  Ku r K u so  that.  T  K  1  C P  U  U  :  u  that  Thus  (auto  (  (  where  °  and', t f ^  N  ° a"  )°°° =  and, s i m i l a r l y ,  u  we t a k e  (auto  normal  1  =  such  up t h eq u e s t i o n  such a  P =  y /  u  = '^lu*  o f inner  vectors.  (1.6)  The  only i f  3s°  3  •  -1  Now  tf-j_ )  that  p\.= cp acp ,e [G-jJ,  1  ( T J ^  i na .  be i r r a t i o n a l - s l o p e  l v  .^ ( F ^ ^ B -  x  T u  c a nbe i n n e r e q u i v a l e n t  v  i s a e cp[G-^]cp~^  T c G  NdG,])} ^ ( 0  °  L e t u and v  i s , letting;  sides,  a r ea u t o m o r p h i c a l l y  lu  (2.9);  )  U  Finally  •a  that  T  o f both  u  and 3 ^  equivalence,  there  we h a v e  £]_ ) = ' f t ^  (2.8)  TSA  1  (• )°'  (r )°3  and so  C N([G 1).  U  u  taking  (F\.8)'  1  (T ) a (T )°°,  whence,  so  cN([G ])  eN(G )  U  (T ) Since  -r  = F °° v  p  •  h a s .the f o r m  s(teT) t E " t  -  ,  ( F . l ) and a l s o ,  by  (P.4),  .  13-  PW " 3  =-"E("s,t€T). ( s ^ t " )  1  ( t)(^\n\y  1  tE  ]  or ST. P ku where  R  = "r(re&) -  1  T  A  , P " ku+r -r 3  i sa collection  o f rational-component  Suppose now t h a t ( u + p ) and  ( 2 . 1 0 )  v  the f o l l o w i n g  are l i n e a r l y  rational-component'vectors Then, f o r e a c h strips  S  r e  parallel  2  X  [  holds:  independent f o r a l l  .  t h e whole p l a n e may be c o v e r e d to  v  and.so narrow  n -(ku r).^  S  vectors.  = '°  T  +  with  that  v  that i s ,  ( 2 . 1 1 )  ;  Qo T *S  k l ] +  JQJ  'ku+r^S-  =  0  ,  where •• Q  = M  Q  Xs  S Hence we have  •  °  :  T  =  e [(F ) ° ]  1 I U  ,  R  1 F__  on  E'f^e- ,^)  no m a t t e r what  1  ' I  ( G . l ) and"'so, b y  = E'(p  T k u  equivalent  i f  ( 2 . 1 0 )  in  Q ^ .  ( 0 . 2 ) ,  p-i,F °°) v  p e [ G ^ ] we u s e .  Thus we s e e t h a t c a n n o t be i n n e r  P  v  . ;  holds,  rf^-and  tf  lv  14.  E v i d e n t l y the  same s o r t o f argument  that  (2.10)  also interferes with  lence  of  and  e  l u  : ieS"}  i  of  to  possible inner  show  equiva-  .  (2.12) [e  the  serves  Thus, t o get TSA  with  above, take  f o r index  ional-slope  vectors  the  uncountable f a m i l i e s  (1.2,3,4),  set  3"  we  would,  an u n c o u n t a b l e  such t h a t  for'  u,v  e. 3"  i n both  family of condition  cases  irrat(2.10)  holds. For that  instance,  this  a condition stronger  (2.10)'  (u+r)  and  than-  (v+t)  are  rational-component vectors  Next, is  (2.10)  linearly  r and  done b y is  if. there  slope are  m.  For  rational  first  noting  >  independent  for a l l  t.  f o r e a c h i r r a t i o n a l number  a vector with  m « m'  c o u l d be  ;  irrationals  m,  u(m)  m,m/  component v e c t o r s  i+m^  = write  r.r'  for  *  which  u(m)  + r  and  u(m/)  + r'  are  linearly  dependent,  or  equivalently  m+r-p slope[u(m)+r] =  It set  i s easy to of  see  1  +  that  irrationals.  =  r  «  1  +  r  m'+r' / =  i s an  slope[u(rn' )+r' ] .  equivalence  relation  on  the  15-  Nov; t h e r e must be u n c o u n t a b l y  many e q u i v a l e n c e  c l a s s e s , s i n c e each e q u i v a l e n c e c l a s s i s o n l y c o u n t a b l y infinite. satisfy  Moreover, i f m « m'  u(m)  then  u(m)  and  u(m')  (2.10)'.  Hence f o r of  fails,  we c o u l d take t h e u n c o u n t a b l e  o b t a i n e d by c h o o s i n g one  m  family  f r o m each c l a s s .  16. CHAPTER THICK  SUBALGEBRAS OP A TYPE  We  continue''with  Any point  A  This  A.  Hence,  for  positive  transformation  (3.1)  x =  t  ^  a  t  a  t " ^  i  s  t  ^  t  since  t  r  note  t  (3.4)  > 0  i f  ^  r  induces  a  one-to-one  Itself:  e R  2  sets  and  an automorphism  r  1  B  =  r  t  \  a^.  0  - n u l l  set  o f 37} :  ±  Hence  1  = k \ (M) 2  automorphism  transformation  a^_ e N ( G - ^ )  -then  cN([G^])  an example  are not i n general  a e N([G J),  the  ,  This'provides  a  then  that  we h a v e  e G  r  e G^,  by the point  ( «5)  groups  X (aM) 2  as i n Chapter Two.  i s r a t i o n a l , , we c a n show i f  i s rational.  of f u l l  that  t  measurable  so that  [(3*3) lizers  t  u c e c  a T a .-  (3.2)  G-j^  2  = T  r  onto  induces  Indeed, i n <  A oT_ oA^.-l  i t  i f  L  r  2  Lebesgue  2  e N([G- ]). T  R  FACTOR  (R ,\ ).'  Now a^  of  number  x  (B.5)/  a l l . g e  real  III  and  D77  t x  preserves  by  THREE  (P.8).  to show-that  f u l l .  norma  To s e e t h i s , -  1  for is  a l l  M e£7^,  where  a t r a c e as i n  (BA)].  x (ts)  > 0  a  And f o r a l l  t  2  M £(??7+.  (with the o r i g i n  Finally,  2  5  since'any ,  so  (3-5) c a l l y ) b y ' G^  '  2  X  5  to  N([G ])  L e t now  G  L  H  [N([G^ ] ) ] .  cannot  (algebrai-  group  Let  G  L  be t h e a l g e b r a o f  2  with  1  2  _1  2  i s the isomorphism Since  (H)  <P<^7) = cp[G ]cp~ •  2  preserve  a*  be t h e g r o u p g e n e r a t e d  2  A ( G , <%#?) = cpN([G ])cp  2  But  t>0 and r a t i o n a l ] c N([G- J)  2  G  the automorphism  '  P  fa :  G(G ,  hence  s  plane  i s not f u l l . ]  1  and.the  t  where , cp  maps a n y h a l f  "  0  XUHP  ( F . l ) and b e l o n g s (3A),  S c R ,.. so t h a t  + a ,M  c  satisfy  t > 0/..  on t h e b o u n d a r y ) orrto i t s e l f  a* = *a M exists  2  '  2  a l l  X  2  a l l Lebesgue measurable s e t s  2  [here  2  X (a M) = t x ( M ) for  i s a constant  " -  = t x (s)  2  for  k  7  G^  onto  i s e r g o d i c , so a l s o  i s a factor the trace  o f OTJ  ,  X , 2  the is  ( J . 1 3 ) . ...Moreover, while.the  a^  MASSA  cp DT] i n (  G(G ,$3fl), 2  and  [ G ^ ] - members  do n o t .  Hence, b y  18.  (J.l6),  a  2  factor  (3.6)  As i n C h a p t e r  i  s  a  o f type I I I .  (T  slope vector. . Define For  )  u  r  such that  (5.7)  J 3 ' ( a  T  T  r  , F  u  A  and . F t > 0  a l l rational  vectors  Two we l e t  ^ . T ^ e, R  )  u  u  he an i r r a t i o n a l (2.2).  as i n  and r a t i o n a l - c o m p o n e n t we h a v e  0  =  o P To p r o v e  this  sup{E } = I  find  (a E )( t  i  T r  (2.3),  as i n  parallel: to  u  and o f p o s i t i v e  not  meet  ing  t o these 'strips  i t s own  t h e c a s e where  plane  T  r  that  t = 1  and  r and u  that  r ^  - image.  0.  a r e never.....  c a n he c o v e r e d w i t h s t r i p s  t h i n n e s s so s m a l l t h a t  exhaust  such  •  we u s e t h e f a c t  the entire  )  or  E.) =0  Consider f i r s t Just  a f a m i l y " {E.} c ( F i , •/ ( a ^ i ^ E ^ E . ^ = 0  and for.-'all  i  (5.8)  we must  parallel  each s t r i p o f 077  The p r o j e c t i o n s  I , - b e l o n g t o . F °,  will  correspond-  and f u l f i l l  (3.8).  As f o r t h e c a s e and  T  origin  r  t / 1,  consider the action  when t h e y a r e r e s t r i c t e d and p e r p e n d i c u l a r t o . u .  one) o f t h i s l i n e ;  l e n g t h whose  t o the l i n e We show t h a t  Indeed, g i v e n a p o i n t  x  from  every point (but of positive  its T  on t h i s  t  through the  c a n be c o v e r e d b y an. i n t e r v a l  A^ - image i s d i s j o i n t  of • A  r  line,  - image.  there are  t h r e e p o s s i b i l i t i e s : t x = x + r , t x > x+r, and t x < x + r .  In  19. the  first  This  case,  x  i s the aforementioned e x c e p t i o n a l  i s m e r e l y a s e t o f measure z e r o .  choose  e > 0  so  point.  In the second case  we  that  s u p [ x + r - , x + r + ] = i n f [ t ( x - ) , t ( x + ) J. , s  that  e  i s , so t h a t  e  x+r+g = t x - t g ">  e  Thus  _  _ tx-(xtr) t+1  c a s e as w e l l ) .an i n t e r v a l [ x - e , x + ] c a n E  so  The  desired  e  r  '  0  X fA [x- , x + ] t  •  .  Q  e > 0  1  or •  y  (and i n the t h i r d  be f o u n d w i t h  e  e  E ^ z^ffl^  that  n  T [x-. ,x+_ ]} p  e  e  a r i s e when we  = 0  , ,  -',  take the c y l i n d e r s o f .  p these  intervals in: R  (3-9)  ' 0 = E'(a T ,F ) t  whenever  a  tj  ^  T r  Now all  and p a r a l l e l  G  0  e  r  u  E(a  and  r'  Thus we  only  u  to note that  which are f i n i t e  "  since  a^ e N ( G ^ ) , T„a. T a a T  products- l i k e  a.  / 3  where  has r a t i o n a l c o m p o n e n t s .  t' > 0  Hence  0 = E ' ( g , P ) .= E ( g , e ) u  for a l l  g e Gg. *2u  is  thick  in  And s o , b y  (G.7)  a  * 0  =  G, . 2  Just  as i n  have  ,P ) •  c a n be b r o u g h t down t o t h e f o r m rational  t V  u.  •  i t remains  - members  >  to  (2.4)  we  also  have  is  20  e  as  a  TSA  2u  «p  =  in  G .  e q u a l l y w e l l here while  ^  a  t o show t h a t  p„  2u  (2.5)  and  (2.6)  serve  has d e f i c i e n c y t y p e  1^ ,  d e f i c i e n c y type I I I .  s  (3.10)  A g a i n as i n  irrational-slope equivalence  The arguments o f  of  vectors. £  2  We  and'ft  u  2 v  ( 2 . 8 ) . we l e t u  and  i n v e s t i g a t e the p o s s i b l e or  ^  2  and ^  u  2  y  in  V  be inner  Gg.  To e a s e t h e n o t a t i o n we w i l l w r i t e ,  (t,r) =  Then we have  (3.11)  a r t  some f i g u r i n g t r i c k s  (t^r)"  1  =  ( f ^ - t r )  (t r)(tV ) ;  3  Wow  where so  that  (tt'/-+r') t'  2  I -i \ -1  a  =  l e t a e [ G ] . ' We c h e c k w h e t h e r o r n o t  a(l,u)a The  r  TI oo e F ^  has t h e f o r m a =  "?,. ( t . , r . ) E . "  t^ > 0  i s r a t i o n a l and  r^  h a s r a t i o n a l components,  2.1. (3.12)  where,  atl^ujaf  t. . > 0 . Now  .isrational we  adapt  0-=E>[(.t j,r 1  for  a l l  =  1  i , J 'if  1 ; J  and  (3.7)  v  ' (2.10)  holds.  so  a(l,u)af^ £ Hence i f  cannot  F v  °°  o  (2.10)  : ies } -  taking for such that  satisfying  i f  be done as i n  u , v e 3" (2.12).  a e [Gg] TSA  3  we  t;ake.  and  tf  2y  c^.  Two we  obtain uncountable  (1.2,3,4)  then  (G.6),  ]  h o l d s , the  IT 'an u n c o u n t a b l e  situation:  Hence b y  m a t t e r what  be i n n e r e q u i v a l e n t ' i n As i n C h a p t e r  {£.  n  0  components.  E ' t G t ^ r ^ + t j U ) ^ ^ ]  +t u),F ]. = 3  has r a t i o n a l  to this  . 0 = E'[a(l,u)of ^ F ^  and  r i.  i n both cases  families in  G  0  by  family of irrational-slope vectors (2.10)  holds.  Again this  can, •  22.  CHAPTER POUR THE F L I P THICK SUBALGEBRAS OP A  In these multiplication  Off  examples  I I - FACTOR 1  will  a l g e b r a , where -X  i s Lebesgue  We b e g i n b y c o n s i d e r i n g ' t h e measure p r e s e r v i n g p o i n t 0  < r < 1 " split [0,1]  and  then  r e f l e c t each p a r t  measure.  [0,1]:  of  i n 'two p a r t s ,  S' x r.  -i  for.  ;  £+1  S y o  a  by  o-  Figure  g i v e s us t h e t r a n s f o r m a t i o n  S ,  defined  f o r 0 < r _< 1  '  (4.2)  S x =  0  j  r-x (r+l) - x x '  i f 0 < x < r i f r < x < 1 ,otherwise-  6  < 1 .  Note t h a t . ° \ r  -  through i t s midpoint.  (4.1)  e*-3)  ([0,1],\)  = [0,r] u [ r , l ],  x  for  £  f o l l o w i n g one-to-one  transformation  the i n t e r v a l  0  This  be t h e  S^  p  i s the i d e n t i t y  --"g.8 - i - M „ g  S r  •-  map, and so  o -\ t  23.  for in  g e £[0,1] (B.7).  d e f i n e s an automorphism  a  o f ^7/  r  as  Then i t i s easy t o see t h a t E ( a , e ) = '-0 r  for  0 < r _< 1  ( E . 2.v).  L e t now, f o r f e M f (m) A  be t h e F o u r i e r  , of  f  = .SgtO,!]  and. i n t e g e r  coefficient  f (m) = f f ( x j e ' o A  Q  1  2 7 r i m x  dx  !  r e l a t i v e to>the complete o r t h o n o r m a l b a s i s 27rimx  r  fe Then f r o m  ., an i n t e g e r a  : m  (4.2) we have (f.S ) (n) = f A  :  f  f(l-x)e  1  _ 2 7 r i m x  dx . .  o  f(y)e- ™( -y)dy  1  2  1  o = f (-m) A  or (4.5)  (f S )A(m) = fA(- ) 0  1  m  and f u r t h e r m o r e (fcS ) (m) = A  r  f  f(r-x)e" ™dx + J f 2  o  1  '' r  l-x)e- ™dx 2  ( r +  m,  = f  f(y)e-  2 7 r i m  ( r  d y + l^f( z ) e "  y )  2 i r ± m { r + ±  r ~—' [ f f(y)e dy o  1 2-jrimz + I f( z ) e dz r  iJL  e  -  e  -27rimr  2 7 r i m r  3  2 7 r i m y  f A (  _  2 7 r i m z  ^  m )  that i s , (4.6)  (f S ) (m) = eA  0  f (-m)  2 i r i r a r  A  r  o  But t h i s reminds us t h a t (4.7)  (foT " ) (m) 1  A  r  = e-  (4.5,6,7)  so' t h a t p u t t i n g  f (m) ,  2 7 r l m r  A  together  gives  ( f °S ) (m) = e--27rimrfj ( - m ) A  2 7 r i m r ;  ;  A  r  -27rimr/(f°S ) (m) A  1  = ( f =>S "T - ) (m) , 1  1  and  A  r  also (f»S ) (m) =  e  A  r  =  2 7 r i  (-  m ) r  f (-m) A  (f°T ) (-m) A  r  = (f»T oS ) (m) I A  r  1  so t h a t almost everywhere (4.8)  '  S  r  =  S  1  o  1 T  r  =  T  r  o  S  i  -  and h e n c e  •  (4.9)  a  .  r  = r c .= or r  1  1  1 r  Prom t h i s we. have some f i g u r i n g  (4.10)  ;  a T R  T  s  C T  = aiT_ 'T .= r  S  r  a T _ . =:  s  1  S  R  T  '  a  T  T  C T  T  f f  r l l - s a  a  CT  ~ r-s  T  T g ^ T g "  =  1  T  a  T  s  0  1  T _  T _  r  s  l - s - (r+s) T  a T a r  s  -1 -  r  a a T a T r  T  (4.12) the  s  1  r -s l T  a  1  C  T  r  l -r T  =  Now we c o n s t r u c t  T  -s  a  11-^ f a c t o r .  L e t G-^  -group  G-, = ( T l e t G^  •(a By  °'r+2s  T  -1  and  r  also  (4.11)  he  a _g  - s r l " ) - . . s + r l . - . r+s.  Pr s and  tricks  be t h e  r  : r  : r  i sa rational  OTJ-group' g e n e r a t e d b y t h e s e t i sa-rational  ( 4 . 9 , 1 0 ) . • we- c a n l o o k u p o n :  number 1  number] . G^  .  as t h e group  . generated by  26.  G  3  fa l,  u  that  1  i s , since  f o r n = 3,4,  Thus  (A.9)  .n = 3,4  .  G ( G  The  G  that  o f (2.1)]  on 0/7  .  he t h e v N a o f (H) :  n  --<#  n  n  , ^ )  =  ^ [G ] n  (J.13)  are factors  i n  and f i n i t e  n  M  n  a n  "  > .  1  o f c o n t i n u o u s [ b y arguments t y p e : the- G  Now we c a n ; n i c e l y  crib  n  a r e II-]/-  a proof that  h y p e r f i n i t e -from v o n Neumann: t o s a t i s f y  4.6.1  777 o f RO.IV  o n page  Collected  nm  :  m  6  cu  )  subalgebras o f G .  (4.14) ;  (page  Works) we must e x h i b i t  fv  v  acts' e r g o d i c a l l y  ;  i s MASSA  (4.13)  of  l  l e t G  G  that o Off  are  group  [ ( 4 . 4 ) and (E.3)]  \ so  the countable  and f r e e l y For  a-j_ e N ( G ^ ) ,  f o r which n  K ' c y , ,_ x  -nm  "n(m-fl)  factors. the G  the d e f i n i t i o n  290 i n volume  a sequence  like  I I Io f fhe~  27. (4.15) '.  V nm  (4.16)  -G = ^ U . K ]  v  has f i n i t e  n  m  For  vector-space dimension  '  n m  each p o s i t i v e i n t e g e r  m  L(m) = l e a s t common m u l t i p l e and  t h e n l e t 077^  projections  he t h e sub-vNa.  InQTT^  P(m,k)  ITBT  f l , 2,...,m}  of  o f 377  generated by the  correspond t o the i n t e r v a l s  l < k < L ( » ) .  1  Furthermore,- s i n c e c a n be a p p r o x i m a t e d  which  define  any measurable  "from o u t s i d e "  [0,1]  subset o f  arbitrarily  closely i n  measure b y a c o u n t a b l e u n i o n o f r a t i o n a l - e n d p o i n t - i n t e r v a l s , we have  Q77 ] ^ ' = a-uo r um 077 'm K  L  '  J  Now f o r p o s i t i v e i n t e g e r  G^ 5  This  G-^  obviously  m  G  3m  Note a l s o t h a t q,  hence  c  G  /  i f T  r  has f i n i t e  3(m+1)  r  l e t G-^  be t h e  - group  the lowest-terms : denominator o f r is <m  = m  m,  ' e G^ , m  order  '  (0 <. r < l )  G  then  3  =  u  and a l s o  m 3m' G  r = ^/j^m)  f  o  r  s  o  m  e  integer -  28. , P(m/K.) = P[m,(k+q)modL(m)] , r  so  that  Let M  • ^5 c  ( H f 4 ) be t h e g r o u p o f u n i t a r i e s o f  m  defined by  (ll f) = f ° ( T r  _ 1 r  )  everywhere, f o r f e £g[0,l]  \-almost  T ( M ) = U MU * R  r  r  and t h e c o r r e s p o n d e n c e with  U  r  <-> T  above,  that, i f 5  r P(m,k) = r  =  z j f f  Then  i s an i s o m o r p h i s m o f (^ ^  R  T e G~ with r 3ra.  m  .r = V / \ L( m) T  as  U P(m,k)tyr  P[m,(k+q)modL(m)]  To g e t a c o r r e s p o n d i n g r  M  e ^m"'  R  then  (4.17)  o  T  G3m Note a l s o  f  and  G^  and ^ k > >  m  <  w  m  e  n  °te  that  'r ?m> e  G  (Vf) ^ . ( s ^ ) =.fS 1  for  f e £2[0,1],  such  that  r  ,  defines a u n i t a r y operator  V  r  (H.l)  29. The correspondence -''cr group  G  4  = G  m  V  U a G  3 m  x  r  sets up an isomorphism of the ^  and the group  5 r Q  <J V ^ m "  = ^  Note that since a P(m k).= V P ( m k ) V * = P[m,L(m)-(k-l) ] , 1  3  1  J  1  OjpZ^ c c 2 %  i t i s also true that  so that  G^ <2^ < 2 ^ c  m  More-  over, G  4m  c  G  4(m+l)  Now for  nm  ,u  m= 1 , 2 , 3 , . . . ,  ..  n = 3,4,  v  n  n  the  n  '  G  4 ~ m 4m U  G  algebras  - n V / nm  yJ  w  *  have the properties ' ( 4 . 1 4 , 1 5 , 1 6 )  Obviously nm  K  n(m+1)  c  K  As for the vector-space  c  n '  a  dimension of  ^  n m  s  w  e  note that  the algebra generated (only .algebraically) by the  n  v  /  m  /  c^  set  n^nm >  u  ;  consists of f i n i t e l i n e a r combinations of products  [$ n^)u (^ )  (4.18)  with  n  T  r  e ^' G  n  nm  .  r  ,  And so the dimension of  c_^^  m  is  finite.  n m  j,  30. Hence, s i n c e t h e weak o p e r a t o r  topology  is a locally  convex  topology,  is  a vector  space w i t h  f i n i t e dimension.  To show f i n a l l y t h a t c  n  G  note f i r s t  Now  we  let  U  e  G  e %{G> ,'V pfy n  S =  ^ c  nm(a)'  h  i s  t  1  )  E  cp^ ^?'in  e ^[G^^ , 1  / / a  :  n  then  ,  there  i s an i n t e g e r  -is -induced b y a  a  G  c  m(a)  such  that  This  W ^ ^ .  t h e sum •  ^  a  W )  $  n (  W  a )  E  the strong operator  induces  g.  on  ° t  *  g  of  cp^^.  a  a e ^/f  each a  ( t p ^ " fE } c  and  R  = exists-in  •  (D.l)  induce  r  *E(ae^)  G  t  means t h a t  and  .  p l a y on t h e r e g u l a r i t y  For a  >  ]  that  ' where  mV  R [ U  =  'This  a topology,  belongs  to  ^ {o^^^pT/)  <$<3<f. n  W  obviously belongs  to  B .  Moreover,  since  31. U and W it U  induce  the'• same a u t o m o r p h i s m  must be t h a t  U = MW  f o r some  g  on t h e  M e fy i ^ f ? ^ •  MASSA ' H  e  n  c  e  M8 c 8 .  e  n  n  This proves finite  (4.20)  We  Hilber.t  s p a c e . •. •,  •• N o t e f i r s t [Dixmier, z e M  1957,  = ^[U^  n  show t h a t  that  page 5 ] ,  (H.8).  n  G  ]  and so  i s hyper-  3,4.  for n =  separable  that  G  c a n be r e p r e s e n t e d  n  on a  . .  since  has a s e p a r a t i n g  t h e ' vNa '  vector  has a s e p a r a t i n g  vector  Now  E = pr[Q z] e G ' , n  so  that  t h e map  Moreover, A  = 0,  of  c  n  A  onto  show t h a t that A e G  there  n  But t h i s  A -* Ag  (G )g. n  means  i s an i s o m o r p h i s m -  E  E  (4.20)  e > 0  and  |y-Az| < /-* e  >3  m  S  i n some  e  5  space.  .  Then t h e r e  K 3£ nm'  such  i f we  To do t h i s ,  and t h e n ,  -  nm  |Az-Sz| < /  be a c c o m p l i s h e d  are given.  = R.[U K J  must be a n  will  i s a separable  such that  n  Thus t h e map  purpose  rng  A  i f AE = 0 .  onto  (G ) -  y e rng E n  i s a homomorphism o f  i f and o n l y  A = 0.  . hence  The  A -» Ag  = 0  E  n  n m  suppose  i s an  since  c  that  K  n ( ? n + 1 )  ,  .  32.  (4.19)j  Now b y  (4.18)  products in"  S  S  T  such  |y-Tz| where  T  Tz  o f the  the c o e f f i c i e n t s  (complex),  so o b t a i n i n g an  . That i s , '  e  €  f o r some  s e t o f such  T  linear  m.  combination  of the  \  i s obviously countable; linear  subset  a r e d e n s e , as we showed above,  rng E  of  the f a m i l y  r n g E.  Since  i s a separable  space. (4.21)  II-j - f a c t o r  it  <  combination  By n u d g i n g  < /y  |Tz-Szj  i s obviously a countable  Tz  Hilbert  that  (4.18)  of The  are  'm = m'.  is'a rational-coefficient  products  the  with  linear  we c a n make them r a t i o n a l  operator  of  is. a f i n i t e  This proves  that  on a s e p a r a b l e H i l b e r t  isomorphic  [Dixmier,  i s convenient  isa space.  hyperfinite  Hence  G^  1 9 5 7 , page 2 9 1 , Theorem 2 ] .  and  G^  -However,  t o m a i n t a i n a d i s t i n c t i o n b e t w e e n them f o r  a while. (4.22)  e the  fixed  morphism  a n  c s  ^  zero  show t h a t rational  define  &  group g e n e r a t e d - b y  t h e auto-  '  n  is.an  :  n  a l g e b r a . o f /the c y c l i c c , 0  s,  * &?,  B  Now i f - s can  r e a l number  -p[(o )°] c  -  n s  F o r each  • -..  i r r a t i o n a l number,  £ * Is t h i c k ns r,  in  G • n  0 < s < 1, • we  Indeed,-we h a v e , '• }  f o r non-  33.  E( e ) E( c ,e)  =  r  and  0  =  v  0  ;;.  • E(a ,r ) £  r  = E ( a  s  = E(a _ ,e) = 0  e )  V  s  E(a ,a ) = E(a a ,e) = E( s  r  r  s  T r  • •  r  •  _ ,e) * 0 s  that  E(  for  (4.4)  also  (4.23)  so  by  a l l  tells  0 s  , g ) = E(g,e) = 0  g e G  us  ~  n  that  e  n s  (4.24) p  has  fe}..  Hence t h e  is  The  TSA  thickness  criterion  for  3,4.  in  test  d e f i c i e n c y type  ilo  '  (J.l4) I  : Let  0  n .=  shows e a s i l y t h a t Q  .  <=c/^7  P  3  w  while  Q-^  Ll [ s , ( | +-|)]  corresponds  to  ,  '  ;  [ f , s ] U [(|"+ - i ) , l ] as  i n the  figure,  Q  Q  2^2  0  so  that  we  have  „  '  s  the•  correspond  set C0 |]  (G.8)  .* l  "  to  TSA the  34.  along with have  a  = e.  2 g  t h e type  Ig.  (4.25) in  the  Thus  (a )°°  and  s  has d e f i c i e n c y  (L.5),  By  11^ - f a c t o r  TSA in  Hence t h e g r o u p s  (£  n s  )°  type I g .  has u n i f o r m m u l t i p l i c i t y  2  a. n  Hence f o r n = 3 , 4  each i r r a t i o n a l  of deficiency'type  I  yields  a  and u n i f o r m m u l t i p l i c i t y  0  11^ - f a c t o r  the h y p e r f i n i t e  s  on a s e p a r a b l e  We go on t o show t h a t  and  2  space.  a  r  e  r  e  a  H y  different. • . (4.26)To p r o v e t h i s , any r a t i o n a l  . so t h a t  CJ  3s  let s  .(K.7)  i s a u t o m o r p h i c a l l y normal  be an i r r a t i o n a l  in  [0,1].  in C  Then f o r  r,  s r s"  CT  T  a  1  s 3 s~' " G  )  CJ  1  =  =  G  s r  a  T  3  a  C  i  T  l  s  - r  =T  d  h  e  n  c  e  (a ) c N(G ) c N([G ]) s  and  5  3  finally  (a ) c ( a ) ° ° s  Take  (•)°  s  of this  last  (a )°3-[(a )° s  §  N(.[G ]) c ( a ) ° °  n  3  in  A(P77)  n  W([G ])] 3(a ) 5  •'. .  and see t h a t 0  0  .. -  s  s  0 0  °  = (a )° . s  35.  (K.7),  Hence b y  i s a u t o m o r p h i c a l l y normal i n G^«. On the o t h e r hand,  (4.27)  ( K . 4 ) i r i -G^. I n the p r o o f o f t h i s , the n e c e s s a r y  thick  c a l c u l a t i o n of. auto (&>,„) is  an  i s automorphically  i s s i m p l i f i e d b y the f a c t t h a t  (J.l4)  Sg-group.  and so b y  t h i s l a t t e r , f u l l o ^ - g r o u p has'a v e r y s i m p l e s t r u c t u r e :  if  a e [(a )],  where  c  (J.15)  Now  then  s  (4.28)  ( a )  a = "s: , a. E " j j- j j 2  a  n  = a  . l E  2  E  +  so that-  and a  =  1  =  = e.• Hence  0  *s l + 2 ' E  E  and Eg b e l o n g t o (cr )°. g  Consequently ( F . 2 )  Now (4.28)  i f  with  a e (a )°° ~ [ e l , then  Eg < I .  «a a  _ 1  r  And b y  (aja^" )  ^ 2 ^ ^ .  ]  a  l  r  C T  r 2 a  .=  1  1  - 1  =  aa  CTg  r  °s r a  =  s  e  =  T  has the form  ( F . 4 ) we have,  i n w h i c h we have the cases a _a a  a  c  a  s-r  2  s  _  r  ( a  K  E  k  for rational  ) ( ^ ) *,  r,  36,  a  so  2  r  a  a  l "  1  =  e a  r°s  =  r-s  T  >  that  ( ^ a  )  1  = "a _ 2s  +  If  E(aa a  r'  _ 1  r  and  T  E (ao E ) + r _  T  r  .  1  (even  E ^ a a ^ )  s  + a  ,a ;)  = E( a , a , ) E ( a  , r  = E(a ,T  r  _ 1  r /  )  r  E ^ a ^ )  r  E (aa E )" 2  r  ( F - 5 ) and  < E  )E (aa E )  = 0  2  C T j 5  2  •  2  then by  E )  r  r  i f r = r').  Hence  G^ - member a t most o n aa^af  ^ [G^].  1  (o )°° s  so  p,^  i  g  (4.23).  < I  a  2  1  can agree  < I.  And t h u s  ( E . l ) with a  i f &. e ( a ) ° ° ~ f e " b s  T h i s means  n  ^ %  s  fe]  Now we  cannot  _  r  T( r / 2 )  hull  •  •1 s (r/2)'" T  under  auto-  the inner equivalence o f  s. *  i s any r a t i o n a l ,  C T  Is i n v a r i a n t  be ( o u t e r ) e q u i v a l e n t .  investigate  forirrational  If  r  aa a f  the automorphic  (4.29) p  2  2  automorphically thick.  s  s^c,  morphisms,  E  r /  N([G,J) =  n  Since  the.  s  i s any r a t i o n a l ,  r  r  and  1  also  E(ao a  then  r  CT  s+2(r/2)  ...  then  ~  T  ( /2) r  s+r  CT  €  G  3  c  G  4  a  n  d  37.  by  (4.11),  so t h a t  T  (r/2),  s  ( a  f o r t h e c y c l i c , groups  ) T  (r/2)  ( s+r).  =  CT  ;  and T  The  o (r/2) (' s)°?. ~ ^ s + r ^ ' C  a  a  e  t h i c k subalgebras  equivalent  i f r (4.30)  irrational.  c ^ e G^,  (4.3D  .  inner  On t h e o t h e r  b o t h be  r  v  hand, l e t ' s  and  t  An automorphism, c e [G-^] h a s t h e f o r m ( F . l )  c = "y. with  a n d £„/ ., .\ - a r e a l w a y s ns n^ s-i-r; i s a rational.  c .E. "  and s o  e ^ c -  (F.4)  =  1  c ^ c . ^  (c E )(ca E.)^  1  k  k  s  or s+r where  ft  .....  r  i s a family of rational  numbers.  E t c ^ c " , ^ ) = T. ' ^ ( a  ,a )P  1  r  by  (F-5).  This last  if  ( t - s ) i s not r a t i o n a l .  :  ca c Q  _ 1  i  s + r  i s obviously  ft^  = S  r  zero  r  E[  f o ra l l  V  (  f  i  +  t  l  )  ,e]P  r  c e [G~],  Hence i f ( t - s ) i s n o t r a t i o n a l ,  [( a . ) ] = ( a . )'  Thus  t  Hence  00  for  a l l  c e [G^].  and  only  i f (t-s) i s a ratibnal  g  and  p.^ number.  inner  equivalent " i f  38.  The same goes f o r  (4.32)  we a g a i n have c . 'e G-, 1  in  3  ca c "  c. e o , G i 1 P  or  s  so t h a t the.  r  a  s  = V-sJo  0  r sjo  T  a  =  r s p a  T  T  =  a  T  =  a  - 1  c^c"  that  ( r and jo  t  can agree  irrationals  i  with  o\j.  .  -  n = 3,4^  i f and o n l y if... ( t - s )  and is  set  3*" o f  w h i c h - d o not - d i f f e r among themselves (1.2,3,4)  i n the"• h y p e r f i n i t e f a c t o r o f type  a separable H i l b e r t  space.  are  p  rational.  Thus:,;, t a k i n g an u n c o u n t a b l e  TSA  on a  t  amounts, we can get two o f the p r o m i s e d families of  (E.l)  then  (o )'°°  Hence f o r b o t h cases  (4.33)  rational):  (r-!-s)-p  c e [G-^ ] .  inner equivalent  are  is,  i [(c )] =  1  w h i c h appear  do not d i f f e r by a r a t i o n a l ,  none o f t h e s e automorphisms  for a l l  ( c . a c. ) 1 s K  either  r-s+p  r+s p  non-zero p r o j e c t i o n ,  c e [G^]  ^p+(r-s).  =  s and t  If  (4.10).  if  where t h i s time  (4.31),  . can be one o f the f o l l o w i n g  (4.31)  a a o  from  as i n  1  n = 4^:  by" r a t i o n a l sorts...of v 11^  on  39. APPENDICES The f o l l o w i n g appendices c o n s i s t l a r g e l y o f a summary o f ASA. A:  Notation  B:'.  R e l a t i v e Commutants,  MA . A l g e b r a s ,  £•'Multiplication  ' Algebras C:  C o n t r a c t i o n s and S t r o n g  Piniteness  D:  Substantial  E:  Agreement o f Automorphisms  P:  C u t t i n g and P a s t i n g  G-:  S t r o n g O r t h o g o n a l i t y and T h i c k n e s s C r i t e r i a  H:  The  J:  Group Type  K:  Automorphic H u l l s and D e f i c i e n c y Type  L."  Uniform M u l t i p l i c i t y i n  Subalgebras  ASA . C o n s t r u e t i b l e A l g e b r a s  11^ -  Factors  A:  Notation G  Let  2^ (G)  (A.l)  be a von Neumann a l g e b r a i s the s e t o f u n i t a r i e s  such t h a t  et  ;  i s induced by some  (A.3)  . A e n i n n e r automorphisms o f G .  are c a l l e d  ( G ) i s t h e group o f a l l automorphisms o f  A  I f now  (A.H.(A.5)  i s any sub-vNa o f  U A(G,C9?7)  A(a,D77)  if  '  q.  G , then.  fuea(c) :  =  vj?7m=j77)  i s t h e group o f a l l automorphisms o f  which are the r e s t r i c t i o n s t o a £  of G  a  U e E^(G) :  ' aA =. UAU* a  then  of••• G .  G ( G ) i s t h e group o f a l l automorphisms  (A.2)  Such  (vNa),  a  n  d  o  n  ly  i  f  *j77 a  o f automorphisms o f • G  = ($\07?),  where  —•  3 e A(G)  and  $371=371. (A.6)  &(Cx,377)  i s t h e s e t o f automorphisms  [0,077) ~  a r e i n d u c e d by  members: M e <377  . aM = UMLT* f o r some (A.7)  (A.8)  U - e 2 ^ ( a,3?) • :  .: • An  G-group  If G  a  i s any subgroup o f A ( G ) .  i s any  G _ g r o u p , then  o f o^/  7  whic  41.  G° =  is  a sub-vNa  (A.9)  of  G  =  c  i s any  on  G  i f  subalgebra  G° =(£' •  of  G  If  then  G,  : gE = E f o r a l l E e  A(G).  of  Thus we  G  and  The  (3.3)  (F.7),  will  £}  i s a b e l i a n , - £°  often-speak  G-group  the  G  is a  full  we  =  (seA(c)  G  g i v e an example  N(G)  may  n o t be  in  : sGs  - 1  =  A(G)  MA  algebra  i s the  G-group  G]  showing  full..  R e l a t i v e Commutants,  of the f i x e d  ('0°)°.  =  0 0  normalizer of  N(G)  B:  G]  G-group.  (A.12)  In  £  {geA(a)  a subgroup  ( A . 11.) G°  for all- g e  acts ergodically  e°  (P.7)  : gA = A  q.  .And. i f  (A.10)  is  (AeG  t h a t , even i f  G  is full'  1  Algebras,; £  Multiplication  Algebras. (B.l)  If . G  t h e n we  have  - a  1  3  sub - vNa  G° = <C •  C  is a  vNa  the r e l a t i v e =  of  Q  and  tf  commutant  i s any subalgebra-, o f of  tf  in  "G,  G :  tf'  n  Q.  Moreover,  Note the  that (,-)°  .G  is a factor  i f and  - operation i s preserved  only i f by_  vNa-  42. isomorphisms: of  G C  E  8.  onto  • (CP£) . e £,  Let  On  a and  Then,  B  i f  be  vNa  £ c G  we  («PE)T  <P  an  isomorphism  o b v i o u s l y have  i f .T 6 ("Pe) 0  t h e o t h e r hand,  T(cpE) =  with  in  cp(^ ) c c  then  for.all  ;  . cp[(cp" T ) E ] = <P[E(cp" T)] , 1  so  that  C  P" T  1  e £°  1  and  <?(£ ) = ( ^ e ) c  (B.2) abelian  tf  T e ^(ft ),  •  0  i s maximal a b e l i a n  subalgebra • which  subalgebra . of  (B.5)  If  [Dixmier,  1957,  (B.4)  G.  tf  is  5  hence  0  in  MA  in  i n ^ M ) ,  G  as a H i l b e r t Let  measurable , where.  •£  This  only  i s an i n any  i f tf = tf°.  A(tf) cr G($( i i ) , tf)  then  (X,£,x)  Let K  Let  2  on  be X  i s a Banach  the s e t of a l l complex-valued  which  a r e bounded  a l g e b r a under  X-almost  every-  t h e norm  x  cp  6  ^ For  =  ^ ( X ^ x ) .  each  cp  e  a-  £ ( X , £ , x)  be  Q  be  llcpll^ = i n f [ a > 0 : XfxeX: |cp( .)|>a} = 0 ] , where  abelian  space.  (X,£,x)  functions  tf  253].  page  a - f i n i t e , measure space..  - considered  i f  i f and  • j ! ^ Multiplication Algebras.  totally  G  i s not:, p r o p e r l y c o n t a i n e d  is  MA  (MA)  ^  define  (M f ) ( x ) = c p ( ) f ( x ) x  t h e map  M^  f e J-f  on  j{  :  4 3 .  for  X-almost a l l x e X.  o p e r a t o r on  i s a.. MA cally  y,  Each  i s a bounded  linear  and t h e s e t  Q)  sub-vNa o f t h e vNap^'M ). Moreover, J?/^ i s i s o m e t r i -  *-isomorphic  to. ' £  under t h e c o r r e s p o n d e n c e  iMqJI _= •lollop, The measure f a i t h f u l trace  X  The t r a c e  X  \  on  x(M )„ =. J cp  %  'X  cp <^-?> M^ :  =.(V*.;"  gives r i s e t o a s e m i - f i n i t e normal : ' •  cpdx  ; C p  e  .  i s f i n i t e i f t h e measure space  4)  (X,2,X)  i s totally  finite. (B.5)  Automorphisms  of  z  Algebras  A l l o f o u r examples  arise  as f o l l o w s . Continuing w i t h the s i t u a t i o n o f a group o f one-to-one which a l s o preserve for a l l g e G ;  X - n u l l sets: i f S e  i f x(S) = 0 ,  which i s e q u i v a l e n t t o  (B.6)  then  be  \  a  then  x(gS) = 0  gS e £  f o r a l l g € G,  \ = X°g- i s . a measure on O And so t h e e q u a t i o n  ( u f ) =-(r:/|^-J * g - i , f f  letG  E-preserving point transformations of X  In t h i s case t h e f u n c t i o n 2  (B.4),  where  g e G, f e }i  The map  g —^ U  • defines  Q)  defines  a u n i t a r y operator  on  a unitary representation  of  G  i n d u c e d by t h e s e  .U  on  V  Moreover, f o r a l l (B.7 1 J)  U M U - *  ^  cD°g  [U :g  obviously.  0  The group o f automorphisms  act ergod'ically  x(S  then  '  f = f °g  \(S) = 0  X - a.e.  Lemma  G  for a l l  0  or  a c t s on. X  3.3.2,  to t h i s I s :  i f and o n l y i f  page  g e G ,  x(X~S) = 0 .  A condition equivalent  [RO I I I , C:  i f and o n l y i f  will  S e £,  gS) =  A  and o n l y i f  (B.9)  on ^JfJ  (A.9)  as ' f o l l o w s : f o r a l l  if  Thus  e G] c £ ^ > ! ) , a 7 ) -  g  (B.8)  . ',  cp=g  e £  -1  and g e G,  = M . - 1 ,  g to g  where  -P e £  C  199,  f  If-  f e £ ( X , £, \ ) , 2  i s constant  X-a.e.  v o l . I l l o f the C o l l e c t e d W o r k s ] . '  C o n t r a c t i o n s , and S t r o n g F i n i t e n e s s .  (Cl)  A  vNa  a-.' i s s t r o n g l y f i n i t e  [ASA §6] over  its  ^/  sub-vNa  Of  i f there' i s a n o r m a l f a i t h f u l c o n t r a c t i o n o f  down t o  ,  l i n e a r map  tt : ^  &  that i s , there e x i s t s a normal f a i t h f u l p o s i t i v e  TTN = N  Q7  such t h a t f o r TT(MBN)  =  M,N e  M(TTB)N  37  and  B e R,  4 . 5  D:  Substantial  (D.l) in  In  G  Subalgebras  ASA  vNa  •G  these  i n G.  regular  [ D i x m i e r , 1954]: G = ^ ( G,c%0 - .  i sstrongly f i n i t e  conditions  a s i n ( C . 1)  over  the contraction  unique.  of  G  involve  abelian  s u b s t a n t i a l sub-  a l g e b r a s w h i c h a r e c o n s e q u e n t l y maximal a b e l i a n : Agreement  o f Automorphisms,  (E.l)  I fc^/  and t  a g r e e o n E e Q?f,  such  (E.2)  7  i s any a b e l i a n  vNa  and  s , t e A(c2fr), • t h e n  define agree  }  on  P)  • agree  E(s,t)  (iii)  rE(s,t) = E ( s r  (v)  =  on  . (ii)  (iv)  2  ,y-  : st  = sup{P  .s,t  zffl  i f s F = t P for. a l l F  Then (i)  s  •  F o r s , t e MQff)  3  MASSA.  Freeness.  [ASA §4]  t h a t • F _< E .  E(s t)  (E.3)  i s  down'-.to  * • A l l o f o u r examples  E:  i s substantial  G  i f is  Under  57 a sub-vNa cj?7 o f a  S(t  _ 1  An^^-group'  .'  :  s,e) - 1  , tr" ) 1  = supfPe^  H  _  E(s, t)  E(s,t)E(t,r) < E(s,r) I-E(s,t)  •  acts  i f r e .•  k{Q/f)  '*  . '  : ( s P ) ( t ? ) = 0)  freely  on  i f E(h,e) = 0  46. for  h e H ~  (E.4)  {e}  R e v e r t now  non-A-null sets such  S e g  notation  and  (B.4,5).  of  a l l . g e G ~{e]  If for a l l  there  exists  F e  that  ,  F c S t h e n the acts  on Q77'.  i s an Cutting  (F.l)  \F >' 0 ,  group o f automorphisms  freely  g -» F:  to the  Moreover,  and  Pasting  0/7  he  • Suppose we  (not  i n d u c e d by in this  an  In the  c'ase t h e  abelian  vNa  and  IT  (B.4)  of  representation  let  have  r E = x ; s ( E )  ( %  the  Automorphisms C  "  n e c e s s a r i l y a l l non-zero) such  Then t h e map  = 0 ,  gF]  k(Dff).  .  : s ej) cOTf  s  ~s  n  isomorphism..  Let  [E  \[F  s  ^s "£  c  = I .  that  • ~  r  s'  v  s E " a  • defined  s E ")(M) = £ S  u l t r a - s t r o n g operator  g  [ASA  by  s(E M)  M e  s  topology  (F.2)'  ( ' E g B E g T ^ ^ s "  (F.3)  For  1  i s an  (.sEg)''  t € A(c2?7)  t "°s r s E s ' = ."Hos ( t s' ) E_" s v  §4]  •  automorphism  . /  ^ of  £  47. "r  J  (note that not  s E " t = *r s S  S  6  t  (st) ( t ^ E j " ' s v  c a n commute w i t h a l l t h e "r  commute w i t h  s €  ,  h u t may  still  s E ''-.) S  (P.4)  x  S  '  I f we have a = "v. a. E . " '  -i and  B €  x  1  h(37?),  then  . apq  = *  - 1  (a^)(aB^E.)"  (a.Ba^ ) 1  2  i  k  '  •j  -  r and  further, i f  .  8 = "v  b' F "  then  (P.5)  (P.6)  .  In the notation o f  (F.3)  E(t,  s  *z  j$  If  s  c  s E * ) ' = J] E s  K(D?7)  s  a  n  s E " , f E } cJfiP) s  [-e/ ]. ?  be a n a b e l i a n g r o u p w h i l e  I f we l e t , f o r a e A(<3/7), E(a,e/)  •  1  then  If jq/^ i s an.c7^7-group, t h e n so i s c  (E.2),  E(s,t)  [ ^ ] = {aeA(«%0 :. a = %  (F'3)y  and  = sup(E(a,s)  : s e ^ / }  s  As p o i n t e d  ~ out i n  may n o t b e .  48. then  [J] = f a e k{j77) : E{a,tf) = 1} Note a l s o  that  (A.8)  •  =*f° ' I f of  (P.7)  :  " .  [</] = [[«/]]  i s an c ^ 7 - g r o u p  be c a l l e d  a f u l l 377  (P.8)  I f -e^  and  then  will  -group.  i s ancT^-group  a e N(V)  and  (A. 1 2 ) ,  ' then  for e = ^(se^)  we  so t h a t  [<>/]  a[^ef  and  ]a  c  1  _  a e N( [ff ] ) . N(/) Strong  (G.l)  "?XsesJ) ( a s a  =  1  •a[^]a  "a  e  nave aca"  G:  s E "  1  )  (aE )"  [jj/]. ' S i m i l a r l y =  c  [jj/]  so  that  Thus  c N( [ / ] ) .  O r t h o g o n a l i t y and T h i c k n e s s Let  <077  If  a e  i f  ^[j»f]a.  a  [/]  "be an a b e l i a n  A{3?7)  and  i s strongly orthogonal to  [ASA 511]  - 1  P  s.£777^  a  n  d  H  Criteria.  vNa.  E\ on  there  i s an o ^ - g r o u p , P"  or-  " _l_H a  i s a mutually  t h e n we s a y :  on  P,"  orthogonal  49. c(H°)  {E }  family  ±  P < and  z  ±  E  such  P  that  ±  fora l li , E  a(E  ±  Define  (G.2)  P ±  0  )  also  E"'(a,H) = I - s u p f P e ^ Then  on  P}  (F.6) E(a,H) < E'(a,H)  and  : a ^ H  '  •  '  ;  also E'(a,H) = E ' ( a j H ] ) =  (F.6)  and  (G.3)  E'^H  0 0  )  (A.11).  And i n t h e e v e n t  [or an  - group  that  H  i s a group  of f i n i t e  order  ( J . l 4 ) ] , and  E ( a , e ) = E( a, H) for  some  a e k(QTJ),  then  also  E ( a , e ) =.E(a H) = E'(a,H) J  (G.4). some  Note a l s o 3 ^c/t Y  >  that  i f ^ / c A ( o ^ , and  then  ,. a E  a  H  on  E P Q  6 J__H  on  P  for  50. on  P,  then . P J _ H  Hence i f P __[_ H  on  P  . '. '  •  Conversely,  P  e  i f .Y J_H  on  E^P  for a l l  cd •  (G.5) .  P <  (G.6)  -E'(  I  fora l l P e  we h a v e  Y,H)  And i f  P < I - E'(P,H)  P €  ^  L e t J77  be  .  t h e n a g a i n we have. P < I - E'( Y,H) . (G.7) in  [ASA § 1 2 ] .  Thickness C r i t e r i o n  G.  Let  H  anQ77~group  be  MASSA  f o r which  E'(a,H) = E ( a , e ) for  a e  all"  (A. 6 ) ,  .and (G.8) in  a s e t such that  then  H°  Suppose  group  &(g,D77) >  [S-] =  i s t h i c k -',(1.1)  Thickness C r i t e r i o n  G-  S^-  S ,  in  [ASA § 1 2 ] .  G  L e t 0?7  i s an 077~group o f f i n i t e  H  (J.l4),  and  (F.6) (H°) be  c  =  Off.  MASSA  o r d e r , o r i s an  and  ' :-  E(a,h) < E(a,e) for  all. h e H  then  H:  H°  and a l l  i s thick  in  a e S,  G  with  The C o n s t r u c t i b l e A l g e b r a s  (H.l)  :  Any a b e l i a n  vNa  a s e t .such t h a t (H°)  of  c  • [S] =  G(G,'C/^),  = 017-  ASA- ..  •  -  c a n be r e p r e s e n t e d on some  51. X  H i l b e r t space If  G  G c  in  Q  i s any f u l l  -group  is  (F.7)j  MA on  !H  t h e n we have  G^>! ),^7) o  So we can l e t ^ ^  QT/  such a f a s h i o n t h a t  he the u n i t a r y M ),«3^)  =  0  Then  [ASA s 9 L  group  :  U  there.is  induces, some ' geG]  a H i l b e r t space  isomorphism  cp  [cp(l) = I ] . "of Q?7 i n t o  isomorphism  .f  of  ^  X),  (f^-{ )i), $0^)  into  an  and a group  such t h a t  ( § ! 7 Z ( < ^ ) ) = [*\U{dV)) (H.2)  The a l g e b r a .  G = a U(</ ) u $277} Jf  has the f o l l o w i n g p r o p e r t i e s : _ (•H.3)  . cp ^ i s  MASSA  (D.l)  in  G.  ( ^ ) = Uw,Oih  (H.4)  ?  (H.5)  G(G Po%) •= W  (H.6) '  h^G^dft) = cpN(G)cp  c  J  "  1  _1  [see (A.12) ]  And moreover (H.7)  The n o r m a l f a i t h f u l c o n t r a c t i o n  cpc^7 i s g i v e n by  7r o f  G down t o  C  :  TTA = cp( ff*A^),  <jfr  where  i s an  This  [Dixmier.,  }i  of  subspace G : i f  1957,  page 5,  Lemma  If  x  onto a c l o s e d  Q  ^> \\  x  subspace o f  .A  Q  1963,  pages  The  {0}  for  d e f i n i t i o n 3]-,.  C o m p a r i s o n w i t h , von  [Bures,  =  [ ^1l ]  vector  G'  is cyclic for  Q  is a separating  is a separating  (H.9)  A e n  isometry  for  separating  ^x  •  e c^7/ f o r a l l A e G].  [ ^*A^  (H.8)  52.  A  e G,  Hence we  vector  and  hence i s then  A =  0  have  f o r Q/7•  then  s  G«  for  Neumann's C o n s t r u c t i b l e A l g e b r a s  •  171-2]  collection  [0%'& ,G , g~*Ug ] Q  Q  is said  t o be  of  in  a  C-system i f  [as  in  (1)  <)TI  is  (2)  G^  is a  (3)  g -» Up.  (bA 5),  normal basis  {g:  equation  0  is a representation  G  o f a measure L e t • •G  in^(M )  group  where  }  formations  MA  . be  Q  i s t a k e n as  a group o f p o i n t  geG }.  space w i t h  Represent-  G  Q  in  a complete 11. (o&(G )) Q  '  v g  h  = ^h  Q  trans-  space]. a Hilbert  Q  G  T  h e  G  Q  ortho:  t  n  e  for  g e G-  defines  a. u n i t a r y  V  on  G  for a l l g e G  .  Q  Define  G  o  \ZG  =  ^  C Q  €  U  W  f  S  ) Now  the  an ^ This  C-system i s s a i d  u ^=  fo}  g  i s equivalent  g * o G  e  G  o^  '  ]  • t o be f r e e i f  ~/fe}  v  t o the requirement t h a t  i n d u c e d b y t h e • U £>V acts g g G i s isomorphic to t h i s  freely'  the  (E.4).  •g(M<&l) = ( U ® V ) ( M ® l ) ' ( U f c V ) * =  for We  case  g  g  (U MU *)®I  g  g  g  t h e n have  is  (H.5)'  G ( G  ^ O  O  MASSA  assertion  proved G  (H.6),  o f some f r e e  G Q  Q  feature  o f the  ASA  G  may  the  Hence i n t h e s p e c i a l ).{  that  n o t be t h e " f u l l  ASA  §9  control; o f the dimension o f the H i l b e r t  show t h a t  construction i s (H.5)  be  closure"  of  C-system.  On t h e o t h e r hand,  operates.  s9  proof of which r e q u i r e s  f o r t h e c a s e where Q  in  ='[G ]  Nov;, t h e added  to  in this  Q  07)<£  the  And  g e G , M e.3l7.  (H.3)'  the  (Off&C) - g r o u p  $>,<£) - g r o u p :  Q  g  .,. •  case  c a n be t a k e n t o be  construction space i n which  (4.20) separable.  we  loses G  must us.e  (H.8)  54.  J:  Group Type Let  [ASA §8] 0?7  b e  a  abelian  n  vNa  and l e t - G  be any f u l l  07? - group. (J.l)  A trace  x  on 0?7  i s called a  G-trace•if I t i s  G-invariant:  T  (gM) =  If "T  for  M ejrf'  , g e G .  G - t r a c e , .then we w i l l  Analogous e x p r e s s i o n s  will.appear:  f o r s e m i - f i n i t e n e s s ; • FNFT,  SFNFT,  write  SFNT,  G-FNFT,  G-SFNFT  faithfulness.  (J.2)  ' The group  there  define  isa  "G  on Q?7  (J.5)  Theorem  G-FNT  t  such t h a t  If 077  G,  i t extends  G , then  v  v  ,  3  oaOTy.  then  .(v'°^)  isa  and i t I s f i n i t e o r semi-  i s . That i s , G( C. 0?7)  G - SFNT  ( w i t h c o n t r a c t i o n TT),  3  f i n i t e a c c o r d i n g l y as Conversely:  in G  G( G, D77) - t r a c e o n ^ j  f i n i t e a c c o r d i n g l y as  G  A n a l o g o u s l y we  type."  i s MASSA  ;  i s a normal  on  TM > 0.  and o f type I I I i f t h e r e a r e no  n o r m a l t r a c e on  SFNFT  i s o f f i n i t e type i f f o r . a l l M > 0 i n  i s o f p r o p e r l y . i n f i n i t e type i f t h e r e a r e no  G-FNT  v  G  i s of semi-finite G  and  (M) .  i s a f i n i t e : normal  i s G-FNT."  G-SFNT  Ol?  x  T  G  i s f i n i t e o r semi-  i s .  I f Off i s MA . i n . G and .p.. i s an ()x\J7?) - i s an  SFNFT on  J77.  can a l s o be c l a s s i f i e d w i t h r e g a r d  to discreteness:  (JA)  07?  E e  V  is  (gP)(E-P); - 0 for a l l (J.5)  g e  .  and a l l  G  Theorem. S i n c e  ;  eJff  P  (J.6)  Theorem  abelian, of  then•  If G  e  such t h a t  <  F  0/7-group,  -  E,  E o.Qff^  is  . :\  '-  r: .-  - ;  g e G such t h a t  for a l l  F  is a full  ..'G  G - a b e l i a n i f and o n l y i f E < E(g,e)  if  G-abelian  gE = E . -  Q77  is  MASSA  is abelian,  in  G ,  and  E  is  i . e . ' E ' i s an a b e l i a n  G ( G , ^ ) -  projection  Cx i n the u s u a l sense [ D i x m i e r , 1 9 5 7 , page 1 2 3 , d e f i n i t i o n 3,] • The  (J.7)  <077 ~g P  G  rou  n o n - z e r o p r o j e c t i o n o f 077  i s s a i d t o be o f type I i f e v e r y  majorizes  a non-zero  G-abelian  projection. (J.8) ' if  The p r o j e c t i o n s  E , P e 077  A  R  G-equivalent,  E  E ~ F (G),  there i s a family (P  s  : s e  G } ' c ^  P  such t h a t s (J.9) G(  P= E S  s  • Theorem  h.,D??) -  equivalent  If  • 3?7  i  S  s  MASSA  equivalent projections in  in  s  S  (p ) = s  P  G and . E , F  o f c?7'/3  then  E  %  are and  F  G i n the u s u a l sense [ D i x m i e r , 1 9 5 7 , page 2 2 5 ,  are  56, Definition 1 ] . (J.10) if  G  i sof  type  there i s a family  projections  (E  I ,  where  n  i s a c a r d i n a l number,  : i e n ] C-Qtff  i  such that  of  G-equivalent  G-abelian  '  7 ( i e n ) E. = I . 1  (J.ll)  The p r o j e c t i o n  i z e s no n o n - z e r o type i f  (J.12)  I  i sa  E e Q?7  is  G-continuous  •G-abelian projections. G-continuous  G  I s o f continuous  projection.  Summary  Type o f t h e f u l l Qjy - Group G  i f i t major-  ,  •  Definition  I .  N o n - z e r o Off - members m a j o r i z e n o n zero G - abelian projections,  I  I i s t h e sum o f n G - e q u i v a l e n t abelian projections.  P  G-  n Ii^  "co  III  'G N  has c o n t i n u o u s  finite  typ.e,  G has c o n t i n u o u s ' p r o p e r l y I n f i n i t e f i n i t e type. '. . •  T h e r e a r e no n o n - t r i v i a l .. G-SPNT  on  semi-  OV,  57. (J.13)  The M a i n Theorem If  Off  i  of  a n o' = (G,C1?7). G  a  Oft-group (J.14)  n  t h e n we h a v e  o  h a s t h e same  v N a - t y p e as' t h e t y p e o f t h e f u l l  Gt(Q. 0ft). 3  Type - 1^  Some Let  S  i n a,  MASSA  s  ASA $ 8 .  Groups  ( 2 _< n _< oo )  QT7 be an a b e l i a n  v N a . A n cZ/7-gTO\xo  G  i s an  - group i f (i)  G  i scyclic  o f order  G - (g)  n :  g  n  = e P  (ii)-  There I s a f a m i l y g  (  F  i  )  ' = (i+l)mod n F  ( F ^ : i e n } cQT? •••  1  e  such  that  n  and  2 ien  i=  F  In t h i s [since is  I  n  G a G°°  i f type  I  case  [ASA §13],  (:A.ll). ]  and a l s o  [or, i f G°  the F  are G°°-  i  G°°- abelian.  i s thick,  G°  I t a l s o t u r n s out. t h a t  any  (J.16)  S  - group  A Criterion  [Dixmier,  :  1957,  G.  for  page 13^,  Hence  G°°  r  [G] = G ° ° for  equivalent  has d e f i c i e n c y  "\  (K.10)].  (J.15)  .  1  Type  - TTT Groups  proposition  4]-  .  type  53. 077  If  (J77.  Is  MASSA  G , i f  in  X  i s an  SFNT  on  and i f  G {a€G(c,a#) : X(aM) = X(M), M €C^7 } +  x  is  an e r g o d i c a l l y a c t i n g p r o p e r  both  G(Gy2?)  (J.17)  have  l e t 377^  Then  i s a unique  [Dixmier,  of  automorphism:  then  3  §13-].  be an a b e l i a n  1957, page 60,  i  G(G, L%7),  type I I I . .  [ASA § 1 2 ,  T y p e - I I I Groups  i = 1,2  For  a ^ € A(377^). there  G  and t h e f a c t o r  Some N o n - E r g o d i c  sub-group  vNa  and l e t  proposition  ( a ^ ® a ) e A (<2^  2]  ^077^)  2  such  that  (a  for  ® a ) (M  1  2  M.  denote  If  G gG l(  If  ±  [077^ %377^)  the  (J.18)  G  x  G  fg-^gg  =  2  ® M ) = ( a ^ j ^ a Mg) 2  i s a n (077^ -  : g  2  I fo^ ^ 7  type. - I I I  {e}  ® G.  (J.20.)  2  i  s  count a b l y  07?2_ ~  sub-group o f  G , 2  i s of  I f 077  ±  6 G , ±  ® (T  ( {e} ® G ) ° =  (<J.19)  then  G- & G L  2  will,  group  group,  I = 1,2}  2  and. i f  then  •  decomposable,  t h e n any f u l l  . -  a c t i n g 077^-group,  i s an.ergodically  2  group,  G^  If  G  2  i s a' f u l l  i s an e r g o d i c  [CTTT^ ®  t y p e - II-,  ~ g*"oup  containing  type I I I .  i s c o u n t a b l y decomposable  and i f off/  is i s o m o r p h i c  59. to  ^oo^ ; ]  the  (J/^© K:  0  has  (L  Automorphic Let  (K.l)  ~ algebra,  1  type  Hulls ft.  and D e f i c i e n c y  be a t h i c k  .£.  =  h u l l  (K.2)  Also, so  ft.c  i f  that  s  since  MA.  (K.5)  of  ft  f o r a l l  to see  8,  i n  E  ft  c  n  said  that. I s ,  a u t o ft =  g  c  e  IR.  <  such  that  P.)  f o r a l l E e t  by  e A(ft )  ft,  then  s(ft ) C  (K.7)  e°  5  The  -  •  0,  t o be automorphically  (K.2),  i f  and only  thick  i f  auto  i f  ,  A(s,e ') = [ e ]  TSA  e  i s  said  C  c  c  n  ft ,  :  e q u i v a l e n t l y . '"  (K-.6)  =  A(a ft )  0  i s  vNa  that  auto e = [ £ ° n A d ? , ? , ) ] ft  the  seA(fl)  i s thick,  TSA  of  ,  f o r a l l  sE = E  e  c  group  [ASA § 1 0 ]  i s  an automorphism  (s|e ) ft  The  is  or  =  -  ft°  • and  Induces  (K.3) (KA)  easy  a u t o ft c  s €..A(KO  .t Hence,  i t , i s  = E  Type  sub-algebra  sB=B  {Bei??!:  SE  Now  (o#? $)D77r)  the  III.  The automorphic  auto  then  t o be automorphically  normal  i f  ft  •  66. (K.8)  auto  (K.9)  Since  £; = .-£'  for  (ag)  a 6 A(e)  we  have  ' = a(e )  auto(ae)  c  c  a u t o m o r p h i c n o r m a l i t y and a u t o m o r p h i c under e q u i v a l e n c e (K.10) of  e  If x  and  holds. tf  o £ .  (1.5),  then  i s the type  (J.12)  . with there  £° = r r ^ C  and i f  i s ' f3 e A( B £ )  For  £  P e  w h i c h we  in  C  3  (J.12)  £ and  such  that  shows t h a t  £  11^ - F a c t o r s  following i s a sketch of  C(P)  be.'an a b e l i a n sub-vNa (£°)  P  we have t h e  u  ASA  §1. •  o f the  11-^ - f a c t o r  £-support" o f  G.  P ,  = i n f (Ee£ : E > P] P  have  (I)  E e  C(EP) = EC(P)-  C(suptP) =-sup{C(P):  (ii)  (L.2)  TSA. £  have t h e same d e f i c i e n c y t y p e .  Let  for  thickness are invariant  Consultation of the t a b l e  Multiplicity  (L.l)  of the  £ and tf a r e . TSA  The  for  - group  are e q u i v a l e n t  (1.6)  L:  c  e)  (1.5).  The d e f i c i e n c y t y p e  the f u l l '  = a(auto  Pej£>]  £ ;  tPc:(£).' C  L e t now  P  :  dim(•)  denote  the normalized  dimension  function  61. P E e e  G , and t h e n f o r n o n - z e r o  of  p{^^i  define  :P (e ) .0<P<E]  r(E)  =  m(E)  = ; i n f ' { r ( F ) : Fe£ ,0<F<E}  S  U  c  P  e  5  P  Alx^ays we have  mE _< r E .  The f u n c t i o n r\  ing,  while  m  I s d e c r e a s i n g , and i f • ^  r  i s increas-  P c  e  i s an o r t h o g o n a l  family (L.5)  r ( s u p @ ) = sup[rE  : E e SL }  m ( s u p & ) .= i n f [mE : E e 0. (L.4)  A non-zero  projection  x  i f r E = mE = x..  x  i f I  (L.5)  E'e £  'As a n example o f t h i s -  11^factor  such  that  G°  G, Is  } has u n i f o r m  The s u b - a l g e b r a  has u n i f o r m m u l t i p l i c i t y  the  \  and l e t  TSA  in  £  has u n i f o r m  multiplicity  x.  [ASA § 1 3 ] ,  G a k(Qff)  G»- Then  multiplicity  G°  l e t Q?7  be a n  be  MASSA  S -group n  has u n i f o r m  in  (J.l4)  multiplicity  n. (L.6) ion  F i n a l l y note dim(-)  i s invariant  form m u l t i p l i c i t y , (1.5).  that,  since the normalized  dimension  under a l l automorphisms o f  i fi texists,  i s invariant  under  G,  functt h e uni-  equivalence  .62.  BIBLIOGRAPHY D.  Bures [ASA]  A b e l i a n .Subalgebras  [1963]  Certain Factors  Products 169  J.  -  Tensor  1963).  Dixmier  '  .  Sous-anneaux  279-286,  abeliens  max-imaux dans l e s  facteurs  ( A n n a l s o f M a t h e m a t i c s , v o l . 5 9 , n o . 2,  de type f i n i . pp.  C o n s t r u c t e d as I n f i n i t e  pre-print.  ( C o m p o s i t i o M a t h e m a t i c a , v o l . 1 5 , F a s c . 2, p p . •  191,  [195^]  o f von Neumann A l g e b r a s :  1954)..  1  [1957]  Les a l g e b r e s d ' o p e r a t e u r s dans 1 e s p a c e h i l b e r t i e n  (Paris,  1957).  1  R. V... K a d i s on Normalcy i n o p e r a t o r a l g e b r a s . 29,  J.  pp. 459  - '464,  (Duke M a t h . J o u r n a l , v o l .  1962).  von Neumann [RO I I I ] vol.  41,  On R i n g s o f O p e r a t o r s pp.  94-161,  III  (Annals of Mathematics,  1940).  [RO I V ] ' On R i n g s o f O p e r a t o r s  IV  ( w i t h F . J . Murray) - v  (Annals of Mathematics, v o l . 44, pp. 4 l 8 - 808, 1943).  

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