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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 State University, 1962 M.A., University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY . in the Department •• of MATHEMATICS We accept t h i s thesis as conforming to' the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 19 68 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mathematics The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e December 24* 1968 ABSTRACT In t h i s t h e s i s we cons t ruc t some examples o f t h i c k subalgebras £ o f f a c t o r s G. e i s t h i c k i n G i f (e' fl G) i s maximal a b e l i a n i n G. We are concerned w i t h t h e i r inne r e q u i v a l e n c e : g iven the t h i c k subalgebras & and nr i n ' G, does there e x i s t a u n i t a r y . U e G such tha t U e U * = J? ? Examples o f t h i c k subalgebras which are not maximal a b e l i a n have been g iven by Dixmie r and K a d i s o n . L a t e r Bures cons t ruc ted numerous examplesy which he d i s t i n g u i s h e d by use o f c e r t a i n i n v a r i a n t s . We use B u r e s ' s c o n s t r u c t i o n to ge t , i n c e r t a i n f a c t o r s G. o f types 11^,- 11^, I I I , uncountable f a m i l i e s {&^: i e J } o f t h i c k subalgebras o f G such tha t i s not i nne r equ iva -l e n t to £ : when i ^ J (We are ab le to add one example to~ those cons t ruc t ed by B u r e s ) . In each f a m i l y , the cannot 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 , and so we are fo rced to show t h e i r non - inne r - equ iva l ence by d i r e c t c a l c u l a t i o n s . - i i -TABLE OP CONTENTS Page CHAPTER ONE INTRODUCTION• 1 CHAPTER TWO THICK SUBALGEBRAS OP A I I -FACTOR 7 CHAPTER THREE THICK SUBALGEBRAS OP A III-FACTOR 16 CHAPTER FOUR THICK SUBALGEBRAS OP THE- HYPERFINITE II-j-FACTOR ON A SEPARABLE HILBERT SPACE ..... 2 2 • i' - _^  APPENDICES ... r . . . -39 BIBLIOGRAPHY .' 6 2 I n t e r n a l r e f e r e n c e s appear thus: " ( 4 . 9 ) " f o r "item nine o f § 4 " ' and " ( B . 2 ) " f o r "item two of appendix B". A l l othe r r e f e r e n c e s are b i b l i o g r a p h i c a l . - i i i -ACKNOWLEDGMENT I am'greatly indebted t o P r o f e s s o r D.J.C. Bures f o r h i s help i n w r i t i n g t h i s t h e s i s . I must a l s o acknowledge the a s s i s t a n c e of.Mr. Ole A. N i e l s e n . I am a l s o g r a t e f u l f o r the prolonged and generous f i n a n c i a l support p r o v i d e d by the Department o f Mathematics o f the U n i v e r s i t y o f B r i t i s h Columbia, and by the^ N a t i o n a l 'Research C o u n c i l o f Canada. > ') CHAPTER ONE INTRODUCTION The purpose of this thesis i s to give and discuss some f a m i l i e s of non-inner-equivalent thick sub-von Neumann algebras of a von Neumann algebra (vNa). (1 .1 ) D e f i n i t i o n Let G be vNa. The.sub-vNa t c G i s a thick subalgebra of G ("t i s TSA- i n G") i f • • ) (B.l) i s maximal abelian (MA) i n G (B . 2 ) , that i s , „c / c\c „cc [The ordered pairs (B.l) and (B . 2 ) r e f e r to the appendices, page 39 tf.] Note that, since £ c p c c f o r any subalgebra ft, a TSA i s always abelian. Note also that an MA subalgebra i s a TSA. More-over, i f G . i s a factor of type I [Dixmier, 1957, page 120 ] , then a l l ' TSA are MA. Indeed, G must be isomorphic to a vNa e on a Hi l b e r t space K, such that a ' i s abelian, that i s , ' -u B' c ( B')' = a , . so that IB' = iB c. Since G i s a factor, so i s IR : so t h a t 8 = d8r(M)- And, s i n c e the r e l a t i v e comrnutant i s i n v a r i a n t ( B . l ) under the isomorphism <P of - G onto ®, S i s TSA i n G i f and o n l y i f (c?e)° = ^(e 0 ) i s MA i n ^ - ( K ) , that i s , i f and o n l y i f i s MA, or £ i s MA. . ; But i n non-type I f a c t o r s we c o n s t r u c t - TSA which are non-MA , , (Thus the "von Neumann double comrnutant theorem" does not h o l d i f G i s not''a type I f a c t o r ) . We c o n s t r u c t f a m i l i e s (p,^ : ietf} such t h a t (1.2) py± ' i s TSA. i n G (1.3) f o r a l l i , =377, a maximal a b e l i a n s u b s t a n t i a l subalgebra (MASSA, appendix D) ' i n G. (1.4) f o r i / £. and p are not inne r equi-v a l e n t . (1.5) . D e f i n i t i o n L e t f> and be sub-vNa of a vNa IR. e and ^ are e q u i v a l e n t i n B- i f there i s an.automorphism a of ' M: [aeA(ft) ] ' (::A.3) . f o r which a(g) = 3?.' e and 3 are - i n n e r e q u i v a l e n t i f a. i s i n n e r ,[a €G(8) . (A.2) ]. I f ft and tf are e q u i v a l e n t as above, then by ( B . l ) , a(ft°) = 3°. Hence, i f ft and tf are a b e l i a n w i t h ft0 = rs c, then a induces an automorphism p of. ftC- = rs 0: P = (a|ft C) e A(*,ft C) (A.5) . such that p(ft) = tf. Moreover, PgP _ 1 e tf° ' f o r g e ft° V " P _ 1gP e e° f o r h e tf° , • ; where ft° = {aeA(ft°) : aE=E 'for a l l Eeft^} ( A . 8 ) , and so (1.6) . f ^ V 1 = tf° , where p e A( 8, ft0). • :' The e x i s t e n c e o f P e A ( f t 0 ) such that (1.6) holds i s thus a n e c e s s a r y c o n d i t i o n f o r the equi v a l e n c e o f the a b e l i a n sub-vNa ft and tf w i t h p_°! - tfc [ASA §10]. I f ft and tf are i n n e r e q u i v a l e n t , then the P of (1.6) must bel o n g to . G ( 5 ? ^ £ C ) (A.6).. Thus to v e r i f y (1.4), we must examine the a c t i o n o f the 077 ~group &(g}077) (A.7). And so, to get the : ietf}, we b e g i n w i t h an a b e l i a n a l g e b r a Of? and a f u l l (F.7) 077 - group G [o b t a i n e d from a group of measurable t r a n s f o r m a t i o n s on some measure • space (B.4,5)]. The c o n s t r u c t i o n o f (H) give s us a vNa G and an isomorphism cp o f 077 i n t o G such t h a t (1.7) <P 077 i s MASS A i n G. 4. ( 1 . 8 ) G ( G , c p c 2 # ) = cpGcp"1 ( 1 . 9 ) A(c,cpc37 ) = c PN(G)cp" 1 , where N ( G ) - - i s the nor m a l i z e d o f G i n A(<^7) (A . , 12) . The group G w i l l c o n t r o l the type o f the vNa G ( J . 1 3 ) . Now i f {H. : ieU} i s a f a m i l y o f -groups which "act at r i g h t a n g l e s " t o G ( G . 7 , 8 ) i t turns out t h a t the f i x e d a l g e b r a s (A.8 , 1 1 ) > i = c p ( H i ° ) • . . ' ! . • ' ' are t h i c k i n G w i t h S . ° =. .cp> At t h i s p o i n t we can d i s c u s s the i n n e r e q u i v a l e n c e • of £. and. £ . In each f a m i l y which we c o n s t r u c t , a l l members have the same beh a v i o r w i t h r e g a r d t o the f o l l o w i n g . i n v a r i a n t s : u n i f o r m m u l t i p l i c i t y (L, when G i s a II]/'- f a c t o r ) , . d e f i -c i e n c y type (K . 1 0 ) , 'automorphic n o r m a l i t y (K . 7 ) and auto-morphic t h i c k n e s s ( K . 4 ) . Thus these i n v a r i a n t s w i l l not serve • to show t h a t the are not i n n e r e q u i v a l e n t . We summarize the examples i n the f o l l o w i n g t a b l e . Note t h a t i n the f i r s t two columns we tt&xraly v a r y G . = G ( to change the type o f the f a c t o r G . The d e f i c i e n c y type (.row DT ), which depends o n l y on the a c t i o n o f o n ^ y-does not change w i t h G ( G , < ^ 7 )• In the t h i r d and f o u r t h columns,.isomorphic .,factors G a r i s e from two a p p a r e n t l y > d i f f e r e n t groups G(G, T h i s v a r i a t i o n i n G{G,<3&) does not a f f e c t the d e f i c i e n c y type or the u n i f o r m m u l t i p l i c i t y o f = H / 3 , but i t does change the b e h a v i o r o f the automorphic h u l l ( K . l ) ("Auto H u l l " row). Hence i n t h i s case each H^ g i v e s a p a i r of n o n - e q u i v a l e n t TSA i n the same f a c t o r . Type of G • " o o I l l (3.5) 11^  and h y p e r f i n i t e on a separable space (4.12,13,30) on ^ ( R ) Lebesgue . ; £^(0,1) Lebesgue G == G ( 0,^ 77 ) (2.1) R a t i o n a l t r a n s -l a t i o n s o f R 2 (3.1) R a t i o n a l t r a n s -l a t i o n s and ma g n i f i c a -2 t i o n s i n R (4.12) R a t i o n a l t r a n s l a t i o n s modulo 1 (4.2,3,12) R a t i o n a l f l i p s . u,v are i r r a t i o n a l - s l o p e v e c t o r s . s , t are i r r a t i o n a l s i n [0,1] The TSA p = ( T ).? U = F 0 l u . v u . .1 l u u '(2.2,3) ! (2.2,3) (3.6)' *2u = F u (3.6) , e3s = ( C T S ) 0 (^ .3,22) (4.3,22) t DT ^ 0 0 • (2.5) I l l (2.6) • ^ 0 (3.6) I I I (3.6) I 2 (4.24) Auto H u l l Auto Normal (2.7) •Auto Normal (4.26) Auto Thick (4.27) Inner Equiva-l e n c e (2.8-12) (3.10) I f (u+rj and -v are l i n e a r l y independent f o r a l l r _ w i t h r a t i o n a l components, then e n u a X i d e n v 8 1 1 ( 1 ^ nv^ a r e n o t i n n e r " eq u i v a l e n t ( n = 1,2). Uniform M u l t i p l i c i t y 2 (4.25) (4 . 29 , 30,32) P n s and e n t a r e -i n n e r e q u i v a l e n t i f and o n l y i f ( s - t ) i s r a t i o n a l (n = 3,4). 7. CHAPTER- 2 THICK SUBALGEBRAS OF A 1 1 ^ - FACTOR Let R 2 be euc l i d e a n 2-space, and l e t <277 be the £ Q Q ( r •  ^ l ^ i p l i c a t i o n a lgebra a c t i n g on j( o = X 2 ( R ,Xg) ( B , i | ' ) i where i s Lebesgue plane measure. p Any vector y Ae R induces a one-to-one p o i n t -trans-formation T of R 2 : C • ' '. . T x = x+y x e R ' 2 This T y preserves Lebesgue measurable sets and \^ - n u l l s e t s . Hence i t induces an automorphism T of c 2 ^ ( B . 5 ) : 2 where ". i s m u l t i p l i c a t i o n .by g e JJQQC'R Let G-^  be the e % ? - g r oup 2-G^ = { T R : r e R , r has r a t i o n a l components} This countable 0/7 - group acts f r e e l y and e r g o d i c a l l y (A.9) o n c 3 7 . Let be the algebra of (H), w i t h cp the isomorphism of Q?J onto c p j ^ , a MASS A i n G^- J ,. ( 2 . 1 ) Since 'G^  acts, e r g o d i c a l l y so a l s o does 8 . . G ( G 1 3 c p ^ ) = cp[G 1]cp" 1 • [ f o r [ G 1 ] see ( P . 6 ) ] and so, by '. ( J . 1 3 ) , . G 1 i s a f a c t o r . I f 7r i s the normal f a i t h f u l c o n t r a c t i o n ( C l ) o f . down.to the MASSA then [where x 2 . i s thought of as a t r a c e as i n ( B . 4 ) ] i s a p r o p e r l y i n f i n i t e s e m i - f i n i t e normal- f a i t h f u l t r a c e on-the f a c t o r G-L ( J . 3 ) • ; P • Now i t i s easy to e x h i b i t a sequence' fE } c3ff •', the p r o j e c t i o n s o f such t h a t ' E n > E n + 1 . . X 2 E n < + co Mapping t h i s sequence i n t o G-j_... shows that the f a c t o r G^ has continuous type [Dixmier, 1957, p r o p o s i t i o n 3 ] . . Thus Gj i s a f a c t o r o f type 11^ . o (2.2) • L e t now u be any v e c t o r o f R w i t h i r r a t i o n a l s l o p e . Let. ( T U ) be the c y c l i c $77-group a n x L l e t F u = f ^ u : k e R ) >• the group o f a l l t r a n s l a t i o n s ' a l o n g u. L e ^ ^ l u ^ l u ^ e t h e : \ r e s p e c t i v e f i x e d a l g ebras ( B . l l ) . " ' ' <v ( 2 . 3 ) The c r i t e r i o n . ( G . 7 ) e a s i l y shows t h a t # l u i s TSA i n : G i v e n a v e c t o r r w i t h r a t i o n a l components, the v e c t o r component^ ( r - u ) II ul p = r - .. ..2 u> o f r p e r p e n d i c u l a r t o u-- i s always non-zero..- Now we can c o v e r t h e p l a n e w i t h a f a m i l y -{S : neuu} o f s t r i p s w h i c h are (a) \ d i s j o i n t and p a r a l l e l t o u (b) o f w i d t h |ip || F o r t h e c o r r e s p o n d i n g p r o j e c t i o n s E n = \ e ^ > . n we have fE 1 c P ° v E = -I n u _ -n n and f o r a l l n T E = T E < ( I - E ) 1r. n p n — ^ nJ But t h i s means t h a t T , . I P,,' on I ( G . l ) o r 0 = E ' ( T r , F u ) = E ( v e ) ( G . 2 and E . 2 ) f o r a l l T r e ' Hence, by ( G . 7 ) , , ^ l u i s TSA i n G1- • J ' 1 0 . ( 2 . 4 ) Since we o b v i o u s l y have l u c F ' l U ° J e l u i s a l s o TSA i n G-^ ( 2 . 5 ) Now £-^u has d e f i c i e n c y type 1 ^ ( K . 1 0 ) : the F i of ( J . l 4 ) a r i s e from c o v e r i n g .the plane w i t h p a r a l l e l s t r i p s ' which are p e r p e n d i c u l a r - to u. and which have width at most ||u|| . ( 2 . 6 ) The d e f i c i e n c y type of-- tf^u .is c a l c u l a t e d i n ASA § 1 5 as f o l l o w s : A r o t a t i o n of axes i n the plane induces an automorphism v °? OT7 such t h a t v(F u)v~" 1" = fe }®{translation on y - a x i s ) , where the t e n s o r product o f automorphism groups i s d e f i n e d i n ( J . 1 7 ) . . Next an ( a l g e b r a i c ) isomorphism p serves to map onto (the t e n s o r product o f - t h e m u l t i p l i c a t i o n a l g e b r a s ) i n such a way t h a t .' 1 j ° v F u v ~ j O ~ 1 = fe} ® K (where 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 a l g e b r a i s ( J . l 8 ) .. ~p-oo(H) ® <£ Hence (A. 1 1 ) , p v ( P u ° ° ) v - y i = fe] ® A ( ^ [ 0 , 1 ] ' T : Now A(.£ [ 0 , 1 ] ) has e r g o d i c subgroups o f type 1 1 ^ ( J . 1 2 ) - -f o r i n s t a n c e , the group o f t r a n s l a t i o n s modulo 1 - - and so, by another theorem of ASA ( J . 1 9 ) , {e} $> A(£ [ 6 , 1 ] ) i s a group o f type I I I , t h a t iSj P u ° ° has type I I I . Thus # I u has d e f i c i e n c y type I I I i n G^. Th i s shows t h a t and I ? l u are not ( i n n e r or outer) e q u i v a l e n t i n G-^ . ( 2 . 7 ) We' show t h a t ft^u and are a u t o m o r p h i c a l l y normal. The automorphic h u l l .(K.l) o f p. l u : auto e l u = { ( e l u ) 0 n A ( G p ^ ) l 0 (see (A.10) f o r C l u ° ) .. = s ( ^ u ) ° ° n N ( [ G l ] ) ) ° ) [by ( H . 6 ) . See (A. 12) f o r N( [G-jJ) ]. And i n the same way, auto ^ = cp(fF u° 0^ N( [G^ ]) ]°) .Now f o r any r e a l number k and any T r i n G-^  12. - r T. T T- = T Ku r Ku - r so t h a t . T K U e N ( G 1 ) c N ( [ G 1 ] ) (F\.8)' and s o ( T U ) C P U C N ( [ G 1 1 ) . S i n c e ( T U ) a ( T u ) ° ° , we h a v e whence, t a k i n g (• )°' o f b o t h s i d e s , : ( r u ) ° 3 ( ( T U ) ° ° N N d G , ] ) } 0 ^ ( T u ) ° ° ° = ( T J ^ s o t h a t ( a u t o £]_u) = ' f t ^ u a n d , s i m i l a r l y , ( a u t o tf-j_u) = ' ^ l u * Thus a n d 3 ^ a r e a u t o m o r p h i c a l l y n o r m a l i n a . (2.8) F i n a l l y we t a k e up t h e q u e s t i o n o f i n n e r e q u i v a l e n c e , L e t u and v be i r r a t i o n a l - s l o p e v e c t o r s . The TSA and' , t f ^ v c a n be i n n e r e q u i v a l e n t (1.6) o n l y i f t h e r e i s a e cp[G-^]cp~^ s u c h t h a t • a ° a " 1 = 3s° l u l v 3 t h a t i s , l e t t i n g ; p\.= cp - 1acp ,e [G-jJ, • (2.9); . ^ ( F ^ ^ B - 1 = F v ° ° • Now s u c h a p h a s .the f o r m - . P = y / s ( t e T ) t E t " , wh e r e T c G x ( F . l ) and a l s o , b y (P.4), 1 3 -or P W 3 " 1 =-"E("s,t€T). ( s ^ t " 1 ) (tEt)(^\n\y] ST. P 1 = "r(re&) T , P " ku - A ku+r -r 3 where R i s a c o l l e c t i o n o f rational-component v e c t o r s . Suppose now t h a t the f o l l o w i n g h o l d s : ( 2 . 1 0 ) (u+ p ) and v are l i n e a r l y independent f o r a l l rat i o n a l - c o m p o n e n t ' v e c t o r s . Then, f o r each r e the whole plane may be covered w i t h s t r i p s S p a r a l l e l to v and.so narrow t h a t X 2 [ S n T - ( k u + r ) . ^ = '° v t h a t i s , ( 2 . 1 1 ) ; Qo T k l ] + J Q J = 0 , *S 'ku+r^S-where •• Q Q = M e [ ( F ) ° ] P S Xs v ' Hence we have T 1 I U , R 1 F__ on I ( G . l ) and"'so, by ( 0 . 2 ) , • ° : = E ' f ^ e - 1 , ^ ) = E ' ( p T k u p - i , F v ° ° ) no matter what p e [G^] we use. . ;Thus we see t h a t i f ( 2 . 1 0 ) h o l d s , rf^-and tflv cannot be i n n e r e q u i v a l e n t i n Q ^ . 14. E v i d e n t l y the same s o r t of argument serves to show t h a t (2.10) a l s o i n t e r f e r e s w i t h the p o s s i b l e i n n e r equiva-lence of e l u and . (2.12) Thus, to get the uncountable f a m i l i e s [ e i : ieS"} o f TSA w i t h (1.2,3,4), we would, i n both cases above, take f o r index s e t 3" an uncountable f a m i l y of i r r a t -i o n a l - s l o p e v e c t o r s such t h a t for' u,v e. 3" c o n d i t i o n (2.10) h o l d s . For i n s t a n c e , t h i s ; c o u l d be done by f i r s t n o t i n g t h a t a c o n d i t i o n s t r o n g e r t h a n - (2.10) i s > (2.10)' (u+r) and (v+t) are l i n e a r l y independent f o r a l l rational-component v e c t o r s r and t . Next, f o r each i r r a t i o n a l number m, u(m) = i+m^ i s a v e c t o r w i t h slope m. For i r r a t i o n a l s m,m/ w r i t e m « m' i f . t h e r e are r a t i o n a l component v e c t o r s r . r ' f o r * which u(m) + r and u(m/) + r ' are l i n e a r l y dependent, or e q u i v a l e n t l y m+r-p m'+r' slope[u(m)+r] = 1 + r = 1 + r / = slope[u(rn' )+r' ] . I t i s easy to see t h a t « i s an e q u i v a l e n c e r e l a t i o n on the se t of i r r a t i o n a l s . 15-Nov; there must be uncountably many equivalence c l a s s e s , since each equivalence c l a s s i s only countably i n f i n i t e . Moreover, i f m « m' f a i l s , then u(m) and u(m') s a t i s f y ( 2 . 1 0 ) ' . Hence f o r we could take the uncountable f a m i l y of u(m) obtained by choosing one m from each c l a s s . 16. C H A P T E R T H R E E T H I C K S U B A L G E B R A S O P A T Y P E I I I F A C T O R We c o n t i n u e ' ' w i t h D77 a n d G-j^  a s i n C h a p t e r T w o . A n y p o s i t i v e r e a l n u m b e r t i n d u c e s a o n e - t o - o n e 2 p o i n t t r a n s f o r m a t i o n o f R o n t o I t s e l f : (3.1) A t x = t x x e R 2 T h i s A . p r e s e r v e s L e b e s g u e m e a s u r a b l e s e t s a n d \ 0 - n u l l s e t H e n c e , b y ( B . 5 ) / i t i n d u c e s a n a u t o m o r p h i s m a ^ . o f 37} : f o r a l l . g e ^ ( R 2 , \ 2 ) . ' N o w i f t > 0 i s r a t i o n a l , , w e c a n s h o w t h a t a ^ e N ( [ G - L ] ) . I n d e e d , i f r r e G ^ , t h e n t h e a u t o m o r p h i s m a t T r a t " ^ i s i n < ^ u c e c ^ ( B«5) b y t h e p o i n t t r a n s f o r m a t i o n A t o T _ r o A ^ . - l = T s o t h a t w e h a v e (3.2) a t T r a t . - 1 = r t r e G± , s i n c e t i s r a t i o n a l . H e n c e a^_ e N ( G - ^ ) c N ( [ G ^ ] ) ( P . 8 ) . [ ( 3 * 3 ) T h i s ' p r o v i d e s a n e x a m p l e t o s h o w - t h a t n o r m a l i z e r s o f f u l l g r o u p s a r e n o t i n g e n e r a l f u l l . T o - s e e t h i s , n o t e t h a t i f a e N ( [ G 1 J ) , - t h e n (3.4) X2(aM) = k a\ 2(M) 1 7 " -f o r a l l M e£7^, where k a > 0 i s a constant [here X 2 i s a t r a c e as i n (BA)]. And f o r a l l t > 0/.. x 2 ( t s ) = t 2 x 2 ( s ) 2 f o r a l l Lebesgue measurable s e t s S c R ,.. so t h a t X 2(a tM) = t 2 x 2 ( M ) ' f o r a l l M £(??7+. F i n a l l y , s i n c e ' a n y , maps any h a l f plane ( w i t h the o r i g i n on the boundary) orrto i t s e l f s the automorphism a* = *acM + a0,M " 5 X U H P ' 2 5 X L H P ' e x i s t s ( F . l ) and belongs t o [N([G^ ] ) ] . But a* cannot s a t i s f y (3A), so N([G 1]) i s not f u l l . ] ( 3 - 5 ) Let now G 2 be the group generated ( a l g e b r a i -c a l l y ) by ' G^ and.the group f a t : t>0 and r a t i o n a l ] c N([G- LJ) Let G 2 be the a l g e b r a o f (H) w i t h G ( G 2 , <P<^ 7) = cp[G2]cp~1 • A(G 2, <%#?) = cpN([G2])cp_1 , where , cp i s the isomorphism o f OTJ onto the MASSA cp DT] i n ( Since G^ i s e r g o d i c , so a l s o i s G(G2,$3fl), and hence G 2 i s a f a c t o r ( J . 1 3 ) . ...Moreover, [ G ^ ] - members p r e s e r v e the t r a c e X 2, w h i l e . t h e a^ do not. Hence, by 18. ( J . l 6 ) , a 2 i s a f a c t o r of type I I I . (3 .6 ) As i n Chapter Two we l e t u he an i r r a t i o n a l -slope v e c t o r . . D efine ( T u ) and . F u as i n ( 2 . 2 ) . For a l l r a t i o n a l t > 0 and rational-component v e c t o r s r such t h a t A ^ . T R ^  e, we have (5.7) J 3 ' ( a T T r , F u ) = 0 o P To prove t h i s we must f i n d a f a m i l y " {E.} c (F ) such t h a t sup{E i} = I and for.-'all i , •/ ( a ^ i ^ E ^ E . ^ = 0 or (5.8) ( a t E i ) ( T r E . ) = 0 • Consider f i r s t the case where t = 1 and r ^ 0 . J u s t as i n ( 2 . 3 ) , we use the f a c t t h a t r and u are never..... p a r a l l e l : the e n t i r e plane can he covered w i t h s t r i p s p a r a l l e l t o u and o f p o s i t i v e t h i n n e s s so s m a l l t h a t each s t r i p w i l l not meet i t s own T r - image. The p r o j e c t i o n s o f 077 correspond-i n g to these ' s t r i p s exhaust I, -belong to . F °, and f u l f i l l ( 3 . 8 ) . As f o r the case t / 1, c o n s i d e r the a c t i o n o f • A t and T r when they are r e s t r i c t e d t o the l i n e through the o r i g i n and p e r p e n d i c u l a r t o . u. We show t h a t every p o i n t (but one) of t h i s ; l i n e can be covered by an. i n t e r v a l o f p o s i t i v e l e n g t h whose A^ - image i s d i s j o i n t from i t s T r - image. Indeed, g i v e n a p o i n t x on t h i s l i n e , there are three p o s s i b i l i t i e s : t x = x+r, tx > x+r, and tx < x+r. In 19. the f i r s t case, x i s the aforementioned e x c e p t i o n a l p o i n t . T h i s i s merely a set of measure zero. In the second case we choose e > 0 so that sup[x+r- s,x+r+ e] = i n f [ t ( x - e ) , t ( x + e ) J. , t h a t i s , so t h a t x+r+g = t x - t g or "> • • _ t x - ( x t r ) Q . r e _ t+1 y 0 ' Thus (and i n the t h i r d case as w e l l ) .an i n t e r v a l [x-e,x+ E] can be found w i t h e > 0 so that X 1 f A t [x- e,x+ e ] n T p[x-. e,x+_ e ]} = 0 , , -', The d e s i r e d E^ z^ffl^ a r i s e when we take the c y l i n d e r s of . p these i n t e r v a l s i n : R and p a r a l l e l to u. Thus we have (3-9) ' 0 = E ' ( a t T r , F u ) > E ( a t V , P u ) whenever a t j T r ^ e • • " Now i t remains o n l y to note t h a t s i n c e a^ e N(G^), a l l G 0 - members which are f i n i t e products- l i k e T„a. T a a T can be brought down to the form a. / 3 where t ' > 0 i s r a t i o n a l and r ' has r a t i o n a l components. Hence 0 = E'(g,P u) .= E(g,e) f o r a l l g e Gg. And so, by (G.7)a *2u = * 0 i s t h i c k i n G,2. J u s t as i n (2.4) we a l s o have 20 e2u = «p as a TSA i n G . The arguments of (2.5) and (2.6) serve e q u a l l y w e l l here to show t h a t p„ 2 u has d e f i c i e n c y type 1 ^ , wh i l e ^ a s d e f i c i e n c y type I I I . (3.10) Again as i n ( 2 . 8 ) . we l e t u and V be i r r a t i o n a l - s l o p e v e c t o r s . We i n v e s t i g a t e the p o s s i b l e i n n e r e q u i v a l e n c e o f £ 2 u a n d ' f t 2 v or ^ 2 u and ^ 2 y i n Gg. To ease the n o t a t i o n we w i l l w r i t e , ( t , r ) = a t r r Then we have some f i g u r i n g t r i c k s (3.11) ( t ^ r ) " 1 = ( f ^ - t r ) ( t 3 r ) ( t V ; ) = ( t t ' / - + r ' ) t ' Wow l e t a e [ G 2 ] . ' We check whether or not I -i \ -1 TI oo a ( l , u ) a e F ^  The a has the form a = "?,. ( t . , r . ) E. " where t ^ > 0 i s r a t i o n a l and r ^ has r a t i o n a l components, so t h a t 2.1. (3.12) a t l ^ u j a f 1 = where, t . . > 0 . i s r a t i o n a l and r i . has r a t i o n a l components. . Now we adapt (3.7) to t h i s s i t u a t i o n : 0 - = E > [ ( . t 1 j , r 1 ; J + t 3 u ) , F v ] . = E ' t G t ^ r ^ + t j U ) ^ ^ ] f o r a l l i , J ' i f ' (2.10) h o l d s . Hence by ( G . 6 ) , . 0 = E ' [ a ( l , u ) o f ^ F ^ 0 ] and so a ( l , u ) a f ^ £ F v ° ° n o matter what a e [Gg] we t;ake. Hence i f (2.10) h o l d s , the TSA 3 and tf2y cannot be i n n e r e q u i v a l e n t ' i n c^. As i n Chapter Two we o b t a i n uncountable f a m i l i e s {£. : ies -} s a t i s f y i n g (1.2,3,4) i n both cases i n G0 by t a k i n g f o r IT 'an uncountable f a m i l y of i r r a t i o n a l - s l o p e v e c t o r s such t h a t i f u,v e 3" then (2.10) h o l d s . Again t h i s can, • be done as i n (2.12). 2 2 . CHAPTER POUR THE FLIP THICK SUBALGEBRAS OP A I I 1 - FACTOR In these examples Off w i l l be the £ ( [ 0 , 1 ] , \ ) -m u l t i p l i c a t i o n a l g e b r a , where -X i s Lebesgue measure. We b e g i n by c o n s i d e r i n g ' t h e f o l l o w i n g one-to-one measure p r e s e r v i n g p o i n t t r a n s f o r m a t i o n o f [ 0 , 1 ] : f o r . 0 < r < 1 " s p l i t the i n t e r v a l i n 'two p a r t s , ; [ 0 , 1 ] = [ 0 , r ] u [ r , l ] , and then r e f l e c t each p a r t through i t s midpoint. 0 x S' x r. S y £ + 1 -i o a o-( 4 . 1 ) F i g u r e T h i s g i v e s us the t r a n s f o r m a t i o n S , d e f i n e d f o r 0 < r _< 1 by ' r-x i f 0 < x < r ( 4 . 2 ) S x = j ( r + l ) - x 6 i f r < x < 1 x ' ,otherwise-f o r 0 < 1 . p Note that S^ i s the i d e n t i t y map, and so e*-3) . ° r \ --"g.8 - i - M g „ S r • - ot-\ 2 3 . f o r g e £ [ 0 , 1 ] defines an automorphism ar of ^7/ as i n (B.7). Then i t i s easy to see that E(a r,e) = '-0 f o r 0 < r _< 1 (E. 2.v). Let now, f o r f e MQ = .SgtO,!] and. i n t e g e r m, f A(m) be the F o u r i e r c o e f f i c i e n t f A(m) = f 1 f ( x j e - 2 7 r i m x d x ! , ' o of f r e l a t i v e to>the complete orthonormal b a s i s r 27rimx . , fe : m an i n t e g e ra Then from (4.2) we have ( f . S ) A ( n ) = f1 f ( l - x ) e _ 2 7 r i m x d x . . : o f1 f(y)e- 2™( 1-y)dy o = f A(-m) or (4.5) ( f 0 S 1 ) A ( m ) = f A ( - m ) and furthermore ( f c S r ) A ( m ) = f f ( r - x ) e " 2 ™ d x + J 1 f ( r + l - x ) e - 2 ™ d x o '' r = f f ( y ) e - 2 7 r i m ( r - y ) d y + l^f( z )e" 2 i r ± m { r + ± -r 1 iJL ~—' 3 2-jrimz e - 2 7 r i m r [ f f ( y ) e 2 7 r i m y d y + I f ( z ) e 2 7 r i m z d z o r e - 2 7 r i m r f A ( _ m ) ^ that i s , (4 .6) ( f 0 S r ) A ( m ) = e - 2 i r i r a r f A ( - m ) o But t h i s reminds us that ( 4 . 7 ) ( f o T r " 1 ) A ( m ) = e - 2 7 r l m r f A ( m ) , so' that p u t t i n g ( 4 . 5 , 6 , 7 ) together gives ( f °S r) A(m) = e - 2 7 r i m r f A ( - m ) ;- 27r ;j -27rimr/ (f°S 1) A(m) = ( f =>S1"Tr-1)A(m) , and a l s o ( f » S r ) A ( m ) = e 2 7 r i ( - m ) r f A ( - m ) = (f°T r) A(-m) = ( f » T r o S 1 ) A ( m ) I so that almost everywhere ( 4 . 8 ) ' S r = S 1 o T r 1 = T r o S i and hence • . (4 .9 ) a r = r r c 1 . = o1rr 1 Prom t h i s we. have some f i g u r i n g t r i c k s (4 .10) ; a R T S = aiT_ r'T s.= a 1T S_ R. =: a r _g T s C T r - T s T r C T l " ) - . . T s + r f f l . - .CTr+s. Pras ' T r a l a l T - s ~ T r - s and a l s o (4 .11) T g ^ T g " 1 = T s 0 1 T _ r T _ s a l T - s T- (r+s) °'r+2s -1 -1 a r T s a r - a r a 1 T s a 1 T r T r T - s a l C T l T - r = T - s (4 .12) Now we c o n s t r u c t a 11-^ f a c t o r . L et G-^  he the -group G-, = ( T : r i s a r a t i o n a l number 1 and l e t G^ be the OTJ-group' generated by the s e t • ( a r : r i s a - r a t i o n a l number] . . . By ( 4 .9 ,10 ) . • we-: can look upon G^ as the group generated by 26. G3 u f a 1 l , t h a t i s , s i n c e a-j_ e N(G^), Thus f o r n = 3,4, the countable group G acts' e r g o d i c a l l y (A.9) and f r e e l y [(4.4) and (E.3)] on 0/7 . For .n = 3,4 l e t G n he the vNa of (H) : \l ;--<# so t h a t o Off i s MASSA i n a n> . . G ( G n , ^ ) = ^ n [ G n ] M n " 1 The G n are f a c t o r s (J .13) of continuous [by arguments l i k e t h a t of (2 .1)] and f i n i t e type: the- G n are II-]/- f a c t o r s . (4.13) Now we c a n ; n i c e l y c r i b a p r o o f t h a t the G are h y p e r f i n i t e -from von Neumann: t o s a t i s f y the d e f i n i t i o n 4.6.1 on page 777 of RO.IV (page 290 i n volume I I I o f fhe~ C o l l e c t e d Works) we must e x h i b i t a sequence f vnm : m 6 cu) o f subalgebras o f G f o r which . n (4.14) K ' c y , , _ x v ; -nm "n(m-fl) 27. (4.15) V has f i n i t e v e c t o r - s p a c e dimension v ' . nm (4 .16) -Gn = ^U m.K n m] ' F o r each p o s i t i v e i n t e g e r m d e f i n e L(m) = l e a s t common m u l t i p l e o f f l , 2,...,m} and then l e t 077^ he the sub-vNa. of 377 generated by the p r o j e c t i o n s P(m,k) InQTT^ which correspond to the i n t e r v a l s I T B T 1 l < k < L ( » ) . Furthermore,- s i n c e any measurable subset o f [ 0 , 1 ] can be approximated "from o u t s i d e " a r b i t r a r i l y c l o s e l y i n measure by a countable union of r a t i o n a l - endpoint - i n t e r v a l s , we have Q77 = a r u 077 ] ^ ' K-u L m 'mJ ' o Now f o r p o s i t i v e i n t e g e r m, l e t G-^ be the - group the lowest-terms G^ = : denominator o f r 5 m / r i s <m Th i s G- m^ o b v i o u s l y has f i n i t e order (0 <. r < l ) and a l s o G 3m c G3(m+1) ' ' G 3 = um G 3m' Note a l s o t h a t i f T r e G^ m, then r = ^/j^m) f o r s o m e i n t e g e r q, hence -28. ,rP(m/K.) = P[m,(k+q)modL(m)] , so t h a t Let • c ^ 5 m (Hf4) be the group of u n i t a r i e s o f M d e f i n e d by ( l l r f ) = f ° ( T r _ 1 ) \-almost everywhere, f o r f e £g [0,l] and T R e ^m"' Then T R ( M ) = UrMUr* M z j f f and the correspondence U r <-> T R i s an isomorphism of (^  ^ m w i t h G-3m Note a l s o t h a t , i f T e G~ w i t h .r = V T/ \ as 5 r 3ra. L( m) above, then (4 .17) r rP(m,k) = U rP(m,k)ty-= P[m,(k+q)modL(m)] To get a co r r e s p o n d i n g G^ m and <^>km> w e n ° t e t h a t f o r 'r e G?m> ( V f ) ^ . ( s ^ 1 ) = . f S r , f o r f e £ 2 [ 0 , 1 ] , d e f i n e s a u n i t a r y o p e r a t o r V r ( H . l ) such t h a t 29. The correspondence -''cr V r sets up an isomorphism of the group G 4 m = G 3 m U a x G 5 r Q and the group ^ = ^ <J V ^ m " Note that since a 1P(m 3k).= V 1 P(m J k)V 1 * = P[m,L(m)-(k-l) ] , i t is also true that OjpZ^ c c 2 % so that G^m<2^ c<2^- More-over, G4m c G4(m+l) .. ' G4 ~ UmG4m Now for n = 3,4, the algebras nm , uv n n -nV/ nmy J * n w m= 1 ,2 ,3 , . . . , have the properties ' (4 .14 ,15 ,16) Obviously Knm c Kn(m+1) c an ' As for the vector-space dimension of ^ n m s w e note that c ^ n m j , the algebra generated (only .algebraically) by the set n v /m / u n ^ n m ; > consists of f in i te l inear combinations of products ( 4 . 1 8 ) [$nn^)un(^r) , with T r eG^' n m . And so the dimension of c _ ^ ^ m is f in i t e . 30. Hence, s i n c e the weak op e r a t o r topology i s a l o c a l l y convex topology, i s a v e c t o r 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 G n = R [ U m V ] > . • note f i r s t t h a t Now we p l a y on the r e g u l a r i t y ( D . l ) o f c p ^ c ^ ? ' i n G n : l e t U e %{G>n,'Vrpfy induce g e ^[G^^1, then ' S = * E ( a e ^ ) ( t p ^ " 1 ) E a / / , where ^ c G R and fE a} c cp^^. For each a e ^/f there i s an i n t e g e r m(a) such t h a t T h i s a e Gnm(a)' t h a t i s * a -is -induced by a W ^ ^ . means that the sum • = ^ a W ) $ n ( W a ) E a e x i s t s - i n the s t r o n g o p e r a t o r topology, belongs to ^ {o^^^pT/) and induces g . on <$<3<f. ° n t 'This W o b v i o u s l y belongs to B . Moreover, s i n c e 31 . i t must be th a t U = MW f o r some M e fy i ^ f ? ^ • H e n c e U and W induce the'• same automorphism g on the MASSA ' U e M8 c 8 . n n Th i s proves t h a t G n = ^ [ U ^ ] and so i s hyper-f i n i t e f o r n = 3 , 4 . (4 .20) We show t h a t G n can be re p r e s e n t e d on a separable Hilber.t space. •. •, . . •• Note f i r s t t h a t s i n c e has a s e p a r a t i n g v e c t o r [Dixmier, 1957, page 5 ] , the' vNa ' has a s e p a r a t i n g v e c t o r z e M n (H.8). Now E = p r [ Q n z ] e G n' , so t h a t the map A -» Ag i s a homomorphism o f onto ( G n ) g . Moreover, A E = 0 i f and o n l y i f AE = 0 . But t h i s means A = 0, . hence A = 0 . Thus the map A -* Ag i s an isomorphism-of c n onto ( G n ) E -The purpose (4 .20) w i l l be accomplished i f we show t h a t rng E i s a separable space. To do t h i s , suppose th a t y e rng E and e > 0 are g i v e n . Then there i s an A e G n such t h a t |y-Az| < e/-*>3 and then, s i n c e A n = R.[U mK n mJ - K n m c K n ( ? n + 1 ) , . ther e must be an S i n some 3£ such that nm' |Az-Sz| < e / 5 . 3 2 . Now by ( 4 . 1 9 )j S is. a f i n i t e l i n e a r combination o f the products (4 .18 ) w i t h 'm = m'. By nudging the c o e f f i c i e n t s i n " S we can make them r a t i o n a l (complex), so o b t a i n i n g an ope r a t o r T such t h a t |Tz-Szj < e/y . That i s , ' |y-Tz| < € where T i s ' a r a t i o n a l - c o e f f i c i e n t l i n e a r combination of the products o f (4 .18) f o r some m. \ The s e t of such T i s o b v i o u s l y c o u n t a b l e ; the f a m i l y o f Tz i s o b v i o u s l y a countable l i n e a r subset o f rng E. S i n c e the Tz are dense, as we showed above, rng E i s a sepa r a b l e H i l b e r t space. (4 .21 ) T h i s proves t h a t i s a h y p e r f i n i t e II-j - f a c t o r on a separable H i l b e r t space. Hence G^ and G^ are isomorphic [Dixmier, 1957, page 291, Theorem 2 ] . -However, i t i s convenient to m a i n t a i n a d i s t i n c t i o n between them f o r a w h i l e . (4 .22 ) For each r e a l number s, d e f i n e & : e n s - -p[(o B)°] c * n & ? , the f i x e d a l g e b r a . o f /the c y c l i c group generated-by the auto-morphism c , 0 n a s c ^ n ' • -.. Now i f - s i s . a n i r r a t i o n a l number, 0 < s < 1, • we can show t h a t £ * Is t h i c k i n G • Indeed,-we have, f o r non-ns n ' • } zero r a t i o n a l r , 33. E( v e ) = 0 E( cr,e) = 0 ;;. by (4.4) and a l s o (4.23) • E ( a £ , r r ) = E ( a s V e ) = E ( a s _ r , e ) = 0 • • • E ( a s , a r ) = E ( a r a s , e ) = E ( T r _ s , e ) * 0 so t h a t E ( 0 s , g ) = E(g,e) = 0 ' f o r a l l g e G n ~ fe}.. Hence the t h i c k n e s s c r i t e r i o n (G.8) t e l l s us t h a t e n s i s TSA i n f o r n .= 3,4. (4.24) The t e s t ( J . l 4 ) shows e a s i l y t h a t the• TSA p has d e f i c i e n c y type I0 : Let Q <=c/^7P correspond to the ilo . w set C 0 3 | ] Ll [ s , ( | +-|)] , ' " w h i l e Q-^  corresponds to ; [ f , s ] U [(|"+ - i ) , l ] as i n the f i g u r e , Q Q 2 ^ 2 0 „ ' s .* l so t h a t we have 34. a l o n g w i t h a g 2 = e. Hence the groups ( a s ) ° ° and ( £ n s ) ° have the type Ig. Thus has d e f i c i e n c y type I g . (4 .25 ) By ( L . 5 ) , has u n i f o r m m u l t i p l i c i t y 2 i n the 11^ - f a c t o r a.n-Hence f o r n = 3 ,4 each i r r a t i o n a l s y i e l d s a TSA of d e f i c i e n c y ' t y p e I 0 and u n i f o r m m u l t i p l i c i t y 2 i n the h y p e r f i n i t e 11^ - f a c t o r on a sepa r a b l e space. We go on to show t h a t and a r e r e a H y d i f f e r e n t . • . (4 .26 ) - i s a u t o m o r p h i c a l l y normal .(K .7) i n C 3s ) To prove t h i s , l e t s be an i r r a t i o n a l i n [ 0 , 1 ] . Then f o r any r a t i o n a l r , . CTsTras"1 = a s T r C T s = T - r so t h a t C Js G 3 C Js~' 1" = G 3 a i l d h e n c e ( a s ) c N(G 5) c N ( [ G 3 ] ) and f i n a l l y ( a s ) c ( a s ) ° ° n N(.[G 3]) c (a s ) ° ° .. - •'. . Take (•)° of t h i s l a s t i n A(P77) and see th a t ( a s ) ° 3 - [ ( a § ) ° 0 n W ( [ G 5 ] ) ] 0 3 ( a s ) 0 0 ° = ( a s ) ° . 3 5 . Hence by ( K . 7 ) , i s automorphically normal i n G^ «. (4.27) On the other hand, i s automorphically t h i c k (K . 4 ) i r i -G^ . In the proof of t h i s , the necessary c a l c u l a t i o n of. auto (&>,„) i s s i m p l i f i e d by the f a c t that ( a c) i s an Sg-group. ( J . l 4 ) and so by ( J . 1 5 ) Now t h i s l a t t e r , f u l l o ^ - g r o u p has'a very simple s t r u c t u r e : i f a e [ ( a s ) ] , then (4 .28 ) a = "s: 2 , a. E " j j - j j where a n = a and a 0 = e.• Hence . E l + E 2 = 1 = * s E l + E 2 ' so that- and Eg belong to (crg)°. Consequently (F . 2 ) Now i f a e (a c)°° ~ [ e l , then a has the form (4 .28 ) w i t h Eg < I. And by (F . 4 ) we have, f o r r a t i o n a l r , « a r a _ 1 ^ 2 ^ ^ . ( a j a ^ " 1 ) ( a K E k ) ( ^ ) *, i n which we have the cases a ] _ a r a 1 . = CTgaras = a 2 s _ r a l C T r a 2 - 1 = °s ar e = T s - r 36, a 2 a r a l " 1 = e a r ° s = T r - s > so t h a t ( ^ a 1 ) = "a2s_T E 1 ( a o r E 1 ) + r s _ r E ^ a ^ ) + T r . s E ^ a a ^ ) + a r E 2 ( a a r E 2 ) " • I f r ' i s any r a t i o n a l , then by (F - 5 ) and ( 4 . 2 3 ) . E ( a a r a _ 1 , a r ; ) = E( a r , a r , ) E 2 ( a C T j 5 E 2 ) < E g < I and a l s o E ( a o r a _ 1 , r r / ) = E ( a r , T r / ) E 2 ( a a r E 2 ) = 0 (even i f r = r'). Hence aa a f 1 can agree ( E . l ) w i t h a G^ - member at most on E 2 < I. And thus i f &. e ( a s ) ° ° ~fe"b then a a ^ a f 1 ^ [G^]. T h i s means (os)°°n N([G,J) = fe] _ • and so p,^ i s a u t o m o r p h i c a l l y t h i c k . S ince the automorphic h u l l Is i n v a r i a n t under auto-morphisms, s^c, a n ^ % s cannot be ( o u t e r ) e q u i v a l e n t . (4 .29 ) Now we i n v e s t i g a t e the i n n e r e q u i v a l e n c e o f the. p f o r i r r a t i o n a l s. * ... I f r i s any r a t i o n a l , then T ( r / 2 ) € G 3 c G 4 a n d •1 T ( r / 2 ) C T s T ( r / 2 ) ' " CTs+2(r/2) ~ CTs+r 37. by ( 4 . 1 1 ) , so th a t f o r the c y c l i c , groups T ( r / 2 ) , ( a s ) T ( r / 2 ) = ( C Ts+r). ; and o T(r / 2) C(' as)°?. ~ ^ a s + r ^ ' The t h i c k subalgebras e and £„/r.,v.\ -are always i n n e r ns n^ s-i-r; e q u i v a l e n t i f r i s a r a t i o n a l . ( 4 . 3 0 ) On the other hand, l e t ' s and t both be i r r a t i o n a l . An automorphism, c e [G-^] has the form ( F . l ) c = "y. c .E. " w i t h c^ e G^, and so ( F . 4 ) ( 4 . 3 D . e ^ c - 1 = c ^ c . ^ 1 ( c k E k ) ( c a s E . ) ^ or s+r r where ft i s a f a m i l y o f r a t i o n a l numbers. Hence ..... E t c ^ c " 1 , ^ ) = T.r ' ^ ( a s + r , a t ) P r = S r E[ V ( f i + t l ) , e ] P r by ( F - 5 ) . T h i s : l a s t i s o b v i o u s l y zero f o r a l l c e [G~], i f ( t - s ) i s not r a t i o n a l . Hence i f ( t - s ) i s not r a t i o n a l , c a Q c _ 1 i [( a . ) ] = ( a . )' 00 f o r a l l c e [G^]. Thus ft^g and p.^ in n e r e q u i v a l e n t " i f and o n l y i f ( t - s ) i s a r a t i b n a l number. 38. (4 .32 ) The same goes f o r n = 4^ : i f c e [G^] we aga in have ca c " 1 as i n ( 4 . 3 1 ) , where t h i s time e i t h e r c . 'e G-, or c . e o , G - s so tha t the. ( c . a c. - 1 ) which appear 1 3 i 1 P 1 s K i n (4 .31 ) . can be one o f the f o l l o w i n g ( r and jo are r a t i o n a l ) : a r a s o 0 = V - s Jo = ^p+(r-s). a r a s j o = T r-s+p T r a s T p = a r + s T p = a(r-!-s)-p from ( 4 . 1 0 ) . I f s and t do not d i f f e r by a r a t i o n a l , then none of these automorphisms can agree ( E . l ) w i t h o\j. on a non-zero p r o j e c t i o n , tha t i s , c ^ c " 1 i [ ( c t ) ] = (o t)'°° . -f o r a l l c e [G-^ ] . -Hence f o r both cases n = 3,4^ and p are inne r e q u i v a l e n t i f and o n l y if... ( t - s ) i s r a t i o n a l . (4 .33 ) Thus:,;, t a k i n g an uncountable set 3*" o f i r r a t i o n a l s i which-do not - d i f f e r among themselves by" r a t i o n a l amounts, we can get two of the promised ( 1 . 2 , 3 , 4 ) sorts. . .of v f a m i l i e s o f TSA i n the"• h y p e r f i n i t e f a c t o r o f type 11^ on a separable H i l b e r t space. 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 : N o t a t i o n B: ' . R e l a t i v e Commutants, MA .Algebras , £ • ' M u l t i p l i c a t i o n ' Algebras C: Con t r ac t ions and S t rong P i n i t e n e s s D: S u b s t a n t i a l Subalgebras E : Agreement of Automorphisms P : C u t t i n g and P a s t i n g G-: S t rong O r t h o g o n a l i t y and Thickness C r i t e r i a H : The ASA . C o n s t r u e t i b l e Algebras 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 11^ - Fac to r s A: Not a t i o n Let G be a von Neumann algebra (vNa), then ( A . l ) 2 ^ ( G ) i s the set of u n i t a r i e s of••• G . (A.2) G ( G ) i s the group of a l l automorphisms a of G such that ; et i s induced by some U e E ^ ( G ) : ' aA =. UAU* . A e n Such a are c a l l e d inner automorphisms of G . (A.3) A ( G ) i s the group of a l l automorphisms of q. I f now i s any sub-vNa of G, then. ( A . H - U = f u e a ( c ) : vj?7m=j77) .(A.5) A(G,C9?7) i s the group of a l l automorphisms of which are the r e s t r i c t i o n s to *j77 of automorphisms of • G —• a £ A(a,D77) i f' a n d o n l y i f a = ($\07?), where 3 e A ( G ) and $371=371. (A.6) &(Cx,377) i s the s e t of automorphisms a of o^/7 which are induced by [0,077) ~ members: . aM = UMLT* M e <377 f o r some U - e 2 ^ : ( a,3?) • (A.7) .: • An G-group i s any subgroup of A ( G ) . (A.8) I f G i s any G _group, then 4 1 . G° = (AeG : gA = A f o r a l l - g e G] i s a sub-vNa of q. (A.9) G a c t s e r g o d i c a l l y on G i f G° =(£' • ( A . 1 0 ) .And. i f £ i s any s u b a l g e b r a of G, then e° = {geA ( a ) : gE = E f o r a l l E e £} i s a subgroup of A ( G ) . I f G i s a b e l i a n , - £° i s a f u l l (P.7) G -group. ( A . 11.) Thus we w i l l often-speak of the f i x e d a l g e b r a G° c G and the G-group G 0 0 = ( ' 0 ° ) ° . (A. 1 2 ) The n o r m a l i z e r of G i n A ( G ) i s the G -group N(G) = (seA(c) : s G s - 1 = G] In ( 3 .3 ) we g i v e an example showing t h a t , even i f G i s f u l l ' ( F . 7 ) , N(G) may not be f u l l . . 1 B: R e l a t i v e Commutants, MA Algebras,; £ M u l t i p l i c a t i o n A l g e b r a s . 1 ( B . l ) I f . G i s a vNa and tf i s any subalgebra-, o f "G, then we have the r e l a t i v e commutant o f tf i n G : - - 3 C = Q n tf' a sub - vNa o f Q. Note t h a t .G i s a f a c t o r i f and o n l y i f G° = <C • Moreover, the (,-)° - o p e r a t i o n i s p r e s e r v e d by_ vNa-42. isomorphisms: Let a and B be vNa w i t h <P an isomorphism of G onto 8. Then, i f £ c G we o b v i o u s l y have cp(^ c) c • (CP£) C. On the other hand, i f .T 6 ("Pe)0 i n then f o r . a l l E e £, T(cpE) = («PE)T; . cp[(cp" 1 T ) E ] = <P[E(cp"1T)] , so t h a t CP" 1T e £° and T e ^ ( f t 0 ) , hence <?(£c) = ( ^ e ) 0 • (B . 2 ) tf i s maximal a b e l i a n (MA) i n G i f tf i s an a b e l i a n subalgebra • which i s not:, p r o p e r l y c ontained i n any a b e l i a n s u b a l g e b r a . o f G. tf i s MA i n G i f and o n l y i f tf = tf°. (B .5) I f 5 i s MA i n ^ M ) , then A(tf) cr G($( i i ) , tf) [Dixmier, 1957, page 253] . (B . 4 ) • j ! ^ M u l t i p l i c a t i o n A l g e b r a s . Let ( X , £ , x ) be a-t o t a l l y a - f i n i t e , measure space.. L e t K Q be £ 2(X,£, x) - c o n s i d e r e d as a H i l b e r t space. Let •£ (X,£,x) be the set of a l l complex-valued measurable f u n c t i o n s on X which are bounded X-almost every-, where. This i s a Banach a l g e b r a under the norm l lcpll^ = inf[a>0 : XfxeX: |cp(x.)|>a} = 0] , where cp 6 ^ = ^ ( X ^ x ) . For each cp e ^ d e f i n e the map M^ on j{ : (M f ) ( x ) = c p ( x ) f ( x ) f e J-f 4 3 . f o r X-almost a l l x e X. Each i s a bounded l i n e a r operator on y,Q) and the set i s a.. MA sub-vNa of the vNap^'M ). Moreover, J?/^ i s i s o m e t r i -c a l l y *-isomorphic to. ' £ under the correspondence cp <^-?> M^ : iMqJI _= •lollop, % =.(V*.;" The measure \ gives r i s e to a s e m i - f i n i t e normal f a i t h f u l t r a c e X on : ' • x(Mcp)„ =. J cpdx ; C p e 4 ) ' X . The t r a c e X i s f i n i t e i f the measure space (X,2,X) i s t o t a l l y f i n i t e . (B.5) Automorphisms of z Algebras A l l of our examples a r i s e as f o l l o w s . Continuing w i t h the s i t u a t i o n of ( B . 4 ) , l e t G be a group of one-to-one E-preserving p o i n t transformations of X which a l s o preserve X - n u l l s e t s : i f S e then gS e £ f o r a l l g e G ; i f x(S) = 0 , then x(gS) = 0 f o r a l l g € G, In t h i s case the f u n c t i o n \ = X°g- i s . a measure on O 2 which i s equivalent to \ a And so the equation ( B . 6 ) ( u f f f ) =-(r:/|^-J * g - i , where g e G, f e }iQ) • de f ines a u n i t a r y opera tor on The map g —^ U def ines a u n i t a r y r e p r e s e n t a t i o n o f G on V Moreover, f o r a l l C-P e £ and g e G, ( B.7) U M U - * = M . - 1 , ^ 1 J g to g cp=g . ' , where cD°g - 1 e £ o b v i o u s l y . Thus [ U g : g e G] c £ ^ > ! 0 ) , a 7 ) -(B .8) The group o f automorphisms induced by these . U w i l l ac t e r g o d ' i c a l l y (A.9) on ^JfJ i f and o n l y i f G ac t s on. X as ' f o l l o w s : f o r a l l S e £, x(S A gS) = 0 f o r a l l g e G , i f and o n l y i f \ (S ) = 0 or x(X~S) = 0 . (B.9) ' A c o n d i t i o n e q u i v a l e n t to t h i s I s : If- f e £ 2 ( X , £, \ ) , then f = f °g X - a . e . i f and o n l y i f f i s constant X - a . e . [RO I I I , Lemma 3.3.2, page 199, v o l . I l l o f the C o l l e c t e d Works] . ' C: Con t r ac t i ons , and S t rong F i n i t e n e s s . ( C l ) A vNa a-.' i s s t r o n g l y f i n i t e [ASA §6] over i t s ^/ sub-vNa Of i f there' i s a normal f a i t h f u l c o n t r a c t i o n o f & down to , tha t 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 l i n e a r map tt : ^ Q7 such tha t f o r M,N e 37 and B e R, TTN = N TT(MBN) = M ( T T B ) N 4 5. D: S u b s t a n t i a l Subalgebras ( D . l ) In ASA 57 a sub-vNa cj?7 of a vNa G i s s u b s t a n t i a l i n G i f i s r e g u l a r i n G. [Dixmier, 1954]: G = ^ ( G,c%0 - . •G i s s t r o n g l y f i n i t e over as i n ( C . 1) Under these c o n d i t i o n s the c o n t r a c t i o n of G down'-.to i s unique. * • A l l of our examples i n v o l v e a b e l i a n s u b s t a n t i a l sub-a l g e b r a s which are consequently maximal a b e l i a n : MASSA. E: Agreement o f Automorphisms, Freeness. ( E . l ) I f c ^ / 7 i s any a b e l i a n vNa and s , t e A(c2fr), • then s and t agree on E e Q?f, [ASA §4] i f sF = tP for. a l l F zffl2 such t h a t • F _< E. • (E.2) For s , t e MQff) d e f i n e ,y-E ( s 3 t ) = sup{P : s}t agree on P) Then • • _ . ' ( i ) . s , t agree on E(s, :t) . ( i i ) E ( s , t ) = S ( t _ 1 s , e ) ( i i i ) r E ( s , t ) = E ( s r - 1 , t r " 1 ) i f r e k{Q/f) '* ( i v ) E ( s , t ) E ( t , r ) < E ( s , r ) . • . ' (v) I - E ( s , t ) = s u p f P e ^ : ( s P ) ( t ? ) = 0) (E . 3 ) A n ^ ^ - g r o u p ' H act s f r e e l y on i f E(h,e) = 0 4 6 . f o r h e H ~ {e} ( E . 4 ) Revert now to the n o t a t i o n of ( B . 4 , 5 ) . I f f o r a l l n o n - A - n u l l s e t s S e g and a l l . g e G ~{e] there e x i s t s F e £ such that F c S , \F >' 0 , \[F n gF] = 0 , then the group of automorphisms induced by the IT of ( B . 4 ) a c t s f r e e l y on Q77'. Moreover, i n t h i s c'ase the r e p r e s e n t a t i o n g -» i s an isomorphism.. F: C u t t i n g and P a s t i n g Automorphisms ( F . l ) Let 0/7 he an a b e l i a n vNa and l e t C k(Dff). • Suppose we have " . [ E s : s ej) cOTf (not n e c e s s a r i l y a l l non-zero) such t h a t r E = x ; s ( E ) = I . • ~r ~s s ^s v s' • • Then the map " £ c s E a " d e f i n e d [ASA §4] by ( % s E S")(M) = £ g s(E sM) M e ^ In the u l t r a - s t r o n g o p e r a t o r t o p o l o g y i s an automorphism o f ( F . 2 ) ' ( ' E g B E g T ^ ^ s " 1 ( .sEg)' ' . / (F.3) For t € A(c2?7) t "r s E ' = "Ho ( t s ) E_" °s s . s v ' s 47. "r s E " t = *r ( s t ) ( t ^ E j " JS s 6 S x ' v s (note t h a t t can commute w i t h a l l the s € , hut may s t i l l n ot commute w i t h "r s E ''-.) S S ' (P.4) I f we have a = "v. a. E." ' - i x 1 and B € h(37?), then . a p q - 1 = * 2 i k ( a . B a ^ 1 ) ( a ^ ) ( a B ^ E . ) " •j ' -r and f u r t h e r , i f . 8 = "v b' F " then (P.5) In the n o t a t i o n o f (F . 3 ) and (E . 2 ) , E ( t , *zs s E s * ) ' = J] s E s E ( s , t ) • 1 (P.6) . I f j$ c K(D?7) then [^] = {aeA(«%0 :. a = % s E s " , fE s} cJfiP) ~ If jq/^ i s an.c7^7-group, then so i s [-e/?]. As p o i n t e d out i n (F ' 3)y c a n be an a b e l i a n group w h i l e may not be. I f we l e t , f o r a e A(<3/7), E ( a , e / ) = sup( E ( a , s ) : s e ^ / } 48. then [J] = fa e k{j77) : E{a,tf) = 1} Note a l s o t h a t (A.8) • : =*f° " . [</] = [[«/]] ( P . 7 ) ' I f of i s an c ^ 7 -group and then w i l l be c a l l e d a f u l l 377 -group. (P.8) I f -e^ i s a n c T ^ - g r o u p and a e N ( V ) (A. 12), ' then f o r e = ^ ( s e ^ ) s E " e [<>/] we nave a c a " 1 = "?XsesJ) ( a s a - 1 ) (aE )" so that a[^ef ]a 1 c [ j j / ] . ' S i m i l a r l y a ^[j»f]a. c [jj/] so t h a t • a [ ^ ] a _ 1 = [ / ] and a e N( [ff ] ) . Thus N ( / ) c N( [ / ] ) . G: Strong O r t h o g o n a l i t y and Thickness C r i t e r i a . Let <077 "be an a b e l i a n vNa. ( G . l ) I f a e A{3?7) and E\ i s an o ^ - g r o u p , then we :say "a i s s t r o n g l y o r t h o g o n a l t o H on P" or- " a _ l _ H on P," [ASA 511] i f P s.£777^ a n d there i s a m u t u a l l y o r t h o g o n a l 49. f a m i l y {E±} c ( H ° ) P such that P < z± E ± and f o r a l l i , E ± a ( E ± P ) 0 ( G . 2 ) Define a l s o E"'(a,H) = I - s u p f P e ^ : a ^ H on P} Then (F.6) E(a,H) < E'(a,H) ' • '; and a l s o E'(a,H) = E ' ( a j H ] ) = E ' ^ H 0 0 ) (F.6) and (A.11). (G.3) And i n the event t h a t H i s a group of f i n i t e order [or an - group ( J . l 4 ) ] , and E(a,e) = E( a, H) f o r some a e k(QTJ), then a l s o E(a,e) =.E(a JH) = E'(a,H) (G.4). Note a l s o t h a t i f ^ / c A ( o ^ , and 6 J__H on P f o r some 3 ^c/t > then ,. Y a E a H on E Q P 50. Conversely, i f . Y J _ H on P, then . P J _ H on E^P f o r a l l P e cd • (G.5) Hence i f P __[_ H on P f o r a l l P e we have . P < I -E'( Y , H ) ( G . 6 ) And i f . '. ' • P < I - E'(P,H) . P € ^ then a g a i n we have. P < I - E'( Y,H) . ( G . 7 ) Thickness C r i t e r i o n [ASA § 1 2 ] . Let J77 be MASSA i n G . L e t H be anQ77~group f o r which E'(a,H) = E(a,e) f o r a l l " a e S , a set such t h a t [S-] = &(g,D77) > ( F . 6 ) .and (A. 6 ) , then H° i s t h i c k -',(1.1) i n G and ( H ° ) c = Off. ( G . 8 ) Thickness C r i t e r i o n [ASA § 1 2 ] . Let 0?7 be MASSA i n G - Suppose H i s an 077~group of f i n i t e o rder, or i s an S ^ - group ( J . l 4 ) , and ' :-E(a,h) < E(a,e) f o r a l l . h e H and a l l a e S, a set .such that • [S] = G ( G , ' C / ^ ) , then H° i s t h i c k i n G w i t h ( H ° ) c = 017-H: The C o n s t r u c t i b l e Algebras of ASA- . • -( H . l ) :Any a b e l i a n vNa can be r e p r e s e n t e d on some 51. H i l b e r t space XQ in such a f a s h i o n tha t QT/ i s MA on !HC I f G i s any f u l l -group (F.7)j then we have G c G ^ > ! o ) , ^ 7 ) So we can l e t ^ he the u n i t a r y group ^ = M 0 ) ,«3^) : U induces, some ' geG] Then [ASA s9L t h e r e . i s a H i l b e r t space an isomorphism cp [cp(l) = I ] . "of Q?7 i n t o X ) , and a group isomorphism .f o f ^ i n t o (f^-{ )i), $0^) such that ( § ! 7 Z ( < ^ ) ) = [*\U{dV)) G = a J fU(</ ) u $277} (H.2) The a l g e b r a . 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 ) i n G. (H.4) ? ( ^ ) = Uw,Oih (H.5) G(G J cPo%) •= W " 1 (H.6) ' h^G^dft) = cpN(G)cp_1 [see (A.12) ] And moreover -(H.7) The normal f a i t h f u l c o n t r a c t i o n 7r o f G down to cpc^7 i s g iven by : 52. • TTA = cp( ff*A^), A e n where <jfr i s an isometry of }iQ onto a c l o s e d subspace o f x [ ^ * A ^ e c^7/ f o r a l l A e G]. T h i s subspace >^ \\Q i s c y c l i c f o r G ' and hence i s s e p a r a t i n g f o r G : i f .A [ ^ 1lQ] = {0} f o r A e G , then A = 0 [Dixmier., 1957, page 5, d e f i n i t i o n 3 ] - , . Hence we have (H.8) Lemma I f x i s a s e p a r a t i n g v e c t o r f o r Q/7• s then ^ x i s a s e p a r a t i n g v e c t o r f o r G« (H . 9 ) Comparison with, von Neumann's C o n s t r u c t i b l e Algebras • [Bures, 1963, pages 171-2] The c o l l e c t i o n [0%'&Q,GQ, g~*Ug ] i s s a i d t o be a C-system i f (1) <)TI i s MA i n ^ ( M 0 ) (2) G^ i s a group (3) g -» Up. i s a r e p r e s e n t a t i o n o f G Q i n [as i n (bA}5), where G Q i s taken as a group o f p o i n t t r a n s -formations of a measure space]. Let • •G . be a H i l b e r t space w i t h a complete ortho-normal b a s i s {g: geG Q}. Represent- G Q i n 11. (o&(GQ)): t n e e q u a t i o n ' vg h = ^ h T h e G Q f o r g e G- d e f i n e s a. u n i t a r y V on G f o r a l l g e G Q . Define G o = \ Z G Q C ^ € U f W S e G o ^ ] ' ) • Now the C-system i s s a i d to be f r e e i f an ^ u g ^ = fo} g * G o ~/fe} v .,. • This i s e q u i v a l e n t t o the requirement t h a t the (Off&C) - group induced by the • U £>V a c t s f r e e l y ' ( E . 4 ) . And i n t h i s case g g G Q i s isomorphic to t h i s $>,<£) - group: •g(M<&l) = (U g®V g)(M®l)'(U gfcV g)* = ( U g M U g * ) ® I f o r g e G Q, M e.3l7. We then have (H.3)' 07)<£ i s MASSA i n G Q -( H . 5 ) ' G ( G O ^ O ='[G Q] Nov;, the added f e a t u r e o f the ASA s9 c o n s t r u c t i o n i s the a s s e r t i o n ( H . 6 ) , p r o o f of which r e q u i r e s t h a t (H . 5 ) be proved f o r the case where G may not be the " f u l l c l o s u r e " of the G Q of some f r e e C-system. On the other hand, the ASA §9 c o n s t r u c t i o n l o s e s c o n t r o l ; o f the dimension of the H i l b e r t space i n which G operates. Hence i n the s p e c i a l case ( 4 . 2 0 ) we must us.e (H . 8 ) to show that ).{ can be taken to be s e p a r a b l e . 54. J : Group Type [ASA §8] Let 0?7 b e a n a b e l i a n vNa and l e t - G be any f u l l 07? - group. ( J . l ) A tra c e x on 0?7 i s c a l l e d a G - t r a c e • i f I t i s G - i n v a r i a n t : T(gM) = T ( M ) . M ejrf' , g e G . I f x i s a f i n i t e : normal G-trace, .then we w i l l w r i t e "T i s G-FNT." Analogous expressions w i l l . a p p e a r : SFNT, G-SFNT f o r s e m i - f i n i t e n e s s ; • FNFT, SFNFT, G-FNFT, G-SFNFT f o r f a i t h f u l n e s s . ( J . 2 ) ' The group G i s of f i n i t e type i f f o r . a l l M > 0 i n Ol? there i s a G-FNT t such that TM > 0. Analogously we define "G i s of s e m i - f i n i t e type." G i s of p r o p e r l y . i n f i n i t e type i f there are no G-FNT on Q?7 and of type I I I i f there are no G - SFNT oaOTy. ( J . 5 ) Theorem I f ; 0 7 7 i s MASSA i n G ( w i t h c o n t r a c t i o n TT), and v i s a normal G( G,3D77) - t r a c e o n ^ j then .(v'°^ ) i s a normal t r a c e on G , i t extends v, and i t Is f i n i t e or semi-f i n i t e a c c o r d i n g l y as v i s . That i s , G i s f i n i t e or semi-f i n i t e a c c o r d i n g l y as G( C.30?7) i s . Conversely: I f Off i s MA . i n . G and .p.. i s an SFNFT on G , then ()x\J7?) -is an SFNFT on J77. G can a l s o be c l a s s i f i e d w i t h regard to d i s c r e t e n e s s : (JA) E e 07?V i s G - a b e l i a n i f (gP)(E-P) ; - 0 . ;-f o r a l l g e G and a l l P eJffF such tha t F < E , ( J . 5 ) Theorem. Since . . ' G i s a f u l l 0/7-group, E o.Qff^ i s G - a b e l i a n i f and o n l y i f . :\ E < E (g , e ) ' - - ; r: .-f o r a l l g e G such tha t gE = E . -( J . 6 ) Theorem I f Q77 i s MASSA i n G , and E i s G ( G , ^ ) -a b e l i a n , then• G e i s a b e l i a n , i . e . ' E ' i s an a b e l i a n p r o j e c t i o n of Cx i n the u s u a l sense [Dixmie r , 1957, page 123, d e f i n i t i o n 3,] • ( J . 7 ) The <077 ~grouP G i s s a i d to be of type I i f every non-zero p r o j e c t i o n o f 077 major izes a non-zero G - a b e l i a n p r o j e c t i o n . ( J . 8 ) ' The p r o j e c t i o n s E , P e 077 A R E G - e q u i v a l e n t , E ~ F ( G ) , i f there i s a f a m i l y ( P s : s e G } ' c ^ P such tha t s s PS = E • S s S ( p s ) = P % ( J . 9 ) • Theorem I f 3?7 i s MASSA i n G and . E , F are G ( h.,D??) - equ iva l en t p r o j e c t i o n s o f c?7'/3 then E and F are e q u i v a l e n t i n G i n the u s u a l sense [Dixmie r , 1957, page 225, 5 6 , D e f i n i t i o n 1 ] . ( J . 1 0 ) G i s o f type I , where n i s a c a r d i n a l number, i f t h e r e i s a f a m i l y ( E i : i e n ] C-Qtff o f G - e q u i v a l e n t G - a b e l i a n p r o j e c t i o n s such t h a t ' 7 ( i e n ) E. = I . 1 ( J . l l ) The p r o j e c t i o n E e Q?7 i s G-continuous i f i t major-i z e s no non-zero • G - a b e l i a n p r o j e c t i o n s . G I s o f c o n t i n u o u s  type i f I i s a G-continuous p r o j e c t i o n . ( J . 1 2 ) Summary , • Type o f the f u l l Qjy - Group G D e f i n i t i o n I . Non-zero Off P - members m a j o r i z e non-z e r o G - a b e l i a n p r o j e c t i o n s , I n I i s the sum o f n G - e q u i v a l e n t G-a b e l i a n p r o j e c t i o n s . I i ^ 'NG has c o n t i n u o u s f i n i t e typ.e, " c o 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 semi-f i n i t e t y p e . '. . • I I I There are no n o n - t r i v i a l . G-SPNT on OV, 57. (J.13) The Main Theorem o f ASA $ 8 . I f Off i s MASSA i n a, then we have a n o' = G(G,C1?7). o a has the same vNa - type as' the type of the f u l l Oft-group Gt(Q.30ft). (J.14) Some Type - 1^ Groups (2 _< n _< oo ) Let QT7 be an a b e l i a n vNa. An cZ/7-gTO\xo G i s an S n - group i f ( i ) G i s c y c l i c o f order n : G - (g) g n = e P ( i i ) - There Is a f a m i l y (F^ : ien} cQT? such t h a t g ( F i ) ' = F ( i + l ) m o d n ••• 1 e n and 2 i e n F i = 1 . In t h i s case [ASA §13], the F i are G°° - e q u i v a l e n t [ s i n c e G a G°° (:A.ll). ] and a l s o G°° - a b e l i a n . Hence G°° i s i f type I [or, i f G° i s t h i c k , G° has d e f i c i e n c y type I n (K.10)]. "\ r (J.15) I t a l s o t urns out. t h a t [G] = G°° f o r any S - group G. (J.16) A C r i t e r i o n f o r Type - TTT Groups [Dixmier, : 1957, page 13^ , p r o p o s i t i o n 4]- . 53. I f 077 Is MASSA i n G , i f X i s an SFNT on (J77. and i f Gx{a€G(c,a#) : X(aM) = X(M), M €C^7 + } i s an e r g o d i c a l l y a c t i n g p r o per sub-group of G(G,3L%7), then both G(Gy2?) and the f a c t o r G have type I I I . . Some Non-Ergodic Type-III Groups [ASA § 1 2 , § 1 3 - ] . (J.17) For i = 1 ,2 l e t 377^ be an a b e l i a n vNa and l e t a^ € A(377^). Then [Dixmier, i 1957, page 60, p r o p o s i t i o n 2] there i s a unique automorphism: (a^ ® a 2) e A (<2^ ^077^) such that ( a 1 ® a 2) (M x ® M 2) = ( a ^ j ^ a Mg) f o r M. I f G ± i s an (077^ - group, then G-L & G 2 w i l l , denote the [077^ %377^) - group G l (gG 2 = fg-^gg : g ± 6 G ±, I = 1,2} . -(J.18) I f G 2 i s a n . e r g o d i c a l l y a c t i n g 077^-group, then ( {e} ® G 2 ) ° = ® ( T 2 • (<J.19) I f o ^ 7 ^ i s count a b l y decomposable, I f G 2 i s a' f u l l type. - I I I 07?2_ ~ group, and. i f G^ i s an e r g o d i c type - II-, sub-group o f G 2, then any f u l l [CTTT^ ® ~ g*"oup c o n t a i n i n g {e} ® G.2 i s o f type I I I . (J . 2 0 . ) I f 077 i s cou n t a b l y decomposable and i f off/ is isomorphic 59. t o t h e ^ o o ^ 0 ; 1 ] ~ a l g e b r a , t h e n t h e (o#? $)D77r) - g r o u p (J/^© (L h a s t y p e I I I . K : A u t o m o r p h i c H u l l s a n d D e f i c i e n c y T y p e [ A S A § 1 0 ] L e t ft. b e a t h i c k s u b - a l g e b r a o f t h e v N a IR. ( K . l ) T h e a u t o m o r p h i c h u l l o f ft i s , < a u t o .£. = {Bei??!: s B = B f o r a l l s e A ( f l ) s u c h t h a t S E = E f o r a l l E e P.) N o w i t , i s e a s y t o s e e t h a t ( K . 2 ) ft.c a u t o ft c ft° A l s o , i f s € . . A ( K O • a n d s E = E f o r a l l E e ft, t h e n s ( f t C ) = ftC, s o t h a t s I n d u c e s a n a u t o m o r p h i s m t e A ( f t ) : . t = ( s | e c ) e ftcn A ( a 5 f t c ) H e n c e , s i n c e ft i s t h i c k , - • (K.3) auto e = [£°n Ad?,?, 0)] 0 , (KA) T h e T S A ft i s s a i d t o b e a u t o m o r p h i c a l l y t h i c k i f a u t o ft i s MA. i n 8 , t h a t . I s , b y ( K . 2 ) , i f a n d o n l y i f (K.5) a u t o ft = g c , o r e q u i v a l e n t l y . '" (K - .6 ) e ° n A(s,e c') = [ e ] ( K . 7 ) T h e T S A e i s s a i d t o b e a u t o m o r p h i c a l l y n o r m a l i f 66. • (K.8) auto £; = .-£' ( K . 9 ) Since f o r a 6 A(e) we have ( a g ) c ' = a (e c ) auto(ae) = a(auto e) automorphic n o r m a l i t y and automorphic t h i c k n e s s are i n v a r i a n t under eq u i v a l e n c e ( 1 . 5 ) . (K.10) The d e f i c i e n c y type o f the TSA. £ i s the type (J.12) c o of the f u l l ' e - group £ . . I f £ and tf are . TSA w i t h £° = rr C^ and i f £ and x are e q u i v a l e n t (1.5), then there i s ' f3 e A( B3 £ C ) such t h a t (1.6) h o l d s . C o n s u l t a t i o n o f the t a b l e (J.12) shows t h a t £ and tf have the same d e f i c i e n c y type. L: M u l t i p l i c i t y i n 11^ - F a c t o r s The f o l l o w i n g i s a sk e t c h o f ASA §1. • Let £ be.'an a b e l i a n sub-vNa of the 11-^ - f a c t o r G . ( L . l ) For P e ( £ ° ) P we have the u£-support" o f P , C(P) = i n f (Ee£ P: E > P] f o r which we have ( I ) C(EP) = EC(P)- E e £ ( i i ) C(suptP) =-sup{C(P): Pej£>] ; f o r tPc:(£C)P.': (L.2) L e t now dim(•) denote the no r m a l i z e d dimension f u n c t i o n 6 1 . P o f G , and then f o r non-zero E e e d e f i n e r ( E ) = S U p { ^ ^ i :P e(e c) P 5.0<P<E] m(E) =;inf'{r(F) : Fe£ P,0<F<E} Alx^ays we have mE _< r E . The f u n c t i o n r i s i n c r e a s -r\ P i n g , w h i l e m Is d e c r e a s i n g , and i f • ^ c e i s an o r t h o g o n a l f a m i l y ( L . 5 ) r ( s u p @ ) = sup[rE : E e SL } \ m(sup&) .= i n f [mE : E e 0. } ( L . 4 ) A non-zero p r o j e c t i o n E'e £ has u n i f o r m m u l t i p l i c i t y x i f r E = mE = x.. The sub-algebra £ has uni f o r m m u l t i p l i c i t y x i f I has u n i f o r m m u l t i p l i c i t y x. ( L . 5 ) 'As an example o f t h i s - [ASA § 1 3 ] , l e t Q?7 be MASSA i n the 1 1 ^ f a c t o r G, and l e t G a k(Qff) be an S n-group ( J . l 4 ) such t h a t G° Is TSA i n G»- Then G° has u n i f o r m m u l t i p l i c i t y n. ( L . 6 ) F i n a l l y note t h a t , s i n c e the normalized dimension f u n c t -ion dim(-) i s i n v a r i a n t under a l l automorphisms of G , t h e u n i -form m u l t i p l i c i t y , i f i t e x i s t s , i s i n v a r i a n t under eq u i v a l e n c e ( 1 . 5 ) . .62. BIBLIOGRAPHY D. Bures [ASA] A b e l i a n .Subalgebras o f von Neumann A l g e b r a s : p r e - p r i n t . [1963] C e r t a i n Fac to r s Cons t ruc ted as I n f i n i t e Tensor Products (Composi t io Mathematica, v o l . 15 , F a s c . 2, pp . • 169 - 191, 1963) . J . D ixmie r ' . [195^] Sous-anneaux abe l i ens max-imaux dans l e s f ac t eu r s de type f i n i . (Annals of Mathematics , v o l . 59, no . 2, pp. 279-286, 1954) . . 1 [1957] Les a lgebres d 'opera teurs dans 1 1 espace h i l b e r t i e n ( P a r i s , 1957) . R. V... Kad i s on Normalcy i n opera tor a l g e b r a s . (Duke Math. J o u r n a l , v o l . 29, pp. 459 - ' 4 6 4 , 1962) . J . von Neumann [RO I I I ] On Rings o f Operators I I I (Annals o f Mathematics , v o l . 41 , pp . 9 4 - 1 6 1 , 1940) . [RO I V ] ' On Rings o f Operators IV ( w i t h F . J . Murray) - v (Annals o f Mathematics , v o l . 44, pp . 4 l 8 - 808, 1943) . 

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