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UBC Theses and Dissertations

Maximal abelian subalgebras of von Neumann algebras 1968

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MAXIMAL ABELIAN SUBALGEBRAS OF VON NEUMANN ALGEBRAS "°y OLE A. NIELSEN B.Sc.j U n i v e r s i t y of B r i t i s h Columbia, A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY , i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the re q u i r e d standard . THE UNIVERSITY OF BRITISH COLUMBIA May, 1968. In presenting this thesis in partial fulfilment of the requirements for an Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his represen- tatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. advanced degree at the University of British Columbia, I agree that the Department nf hflTHE M/4T/CS The University of British Columbia Vancouver 8, Canada Supervisor: D. J. Bures. ABSTRACT We are concerned w i t h c o n s t r u c t i n g examples of maximal a b e l i a n von Neumann subalgebras (MA subalgebras) i n h y p e r f i n i t e f a c t o r s of type I I I . Our r e s u l t s w i l l show tha t c e r t a i n phenomena known to hold f o r the h y p e r f i n i t e . f a c t o r of type 11^ a l s o hold f o r type I I I f a c t o r s . Let 7?L and tl be subalgebras of the f a c t o r G . We c a l l 7KL and 7L equivalent i f ??L i s the image of % by some automorphism of G . Let N(7)t) denote the sub- algebra of G generated by a l l those u n i t a r y operators i n G which induce automorphisms of %t ',. and l e t N2(7)t) , N^(tH),... be defined i n the obvious i n d u c t i v e f a s h i o n . F o l l o w i n g J . Dixmier and S. Anastasio, we c a l l a MA sub- algebra 7K. of G s i n g u l a r i f N(?H) = ?VL , r e g u l a r i f N(M) = G , semi^regular i f N(?H) i s a f a c t o r d i s t i n c t from G , and m-semi-regular (m >_ 2) i f ' N(7H),. . .Nm"1(?30 are not f a c t o r s but N^OM) i s a f a c t o r . The MA subalgebras of the h y p e r f i n i t e 11^ f a c t o r IB have r e c e i v e d much a t t e n t i o n i n the l i t e r a t u r e , i n the papers of J . Dixmier, L. Pukanszky, S i s t e r R. J . Tauer, and S. Anastasio. I t i s known that & contains a MA subalgebra of each type. Further, B contains p a i r w i s e i n e q u i v a l e n t sequences of s i n g u l a r , semi-regular, 2 -semi-regular, and 3 - semi-regular MA subalgebras. i i i . The only h i t h e r t o known example of .a MA.subalgebra i n a type I I I f a c t o r i s r e g u l a r . In 1956 Pukanszky gave a general method f o r c o n s t r u c t i n g MA subalgebras.in a c l a s s of (probably n o n - h y p e r f i n i t e ) type I I I f a c t o r s . Because of an e r r o r i n a c a l c u l a t i o n , the types of these subalgebras i s not . known. The main r e s u l t of t h i s t h e s i s i s the c o n s t r u c t i o n , i n each of the uncountably many mutually non-isomorphic h y p e r f i n i t e type I I I f a c t o r s of R. Powers, of: ( i ) a semi-regular MA subalg-ebra ( i i ) two sequences of mutually i n e q u i v a l e n t 2- semi-regular MA subalgebras 1 ( i i i ) two sequences of mutually i n e q u i v a l e n t 3- semi-regular MA subalgebras. Let G denote one of these type I I I f a c t o r s and l e t B -denote the h y p e r f i n i t e 11^ f a c t o r . Roughly speaking, when-' ever a non-singular. MA subalgebra of 8 i s constructed by means of group operator algebras, our method w i l l produce a -.MA subalgebra of G of the same type. H. A r a k i and J . Woods have shown that G ® B = G , _ and i t i s the r e f o r e only necessary to con s t r u c t MA subalgebras of G <8> B of the d e s i r e d type. We o b t a i n MA subalgebras of G ® B by tensoring a MA subalgebra i n G w i t h one i n 8- . In order to determine the type of such a MA subalgebra> we r e a l i z e , B as a c o n s t r u c t i b l e algebra and then regard G ® e as a c o n s t r u c t i b l e algebra; t h i s allows us to consider i v . operators i n G ® iB as f u n c t i o n s from a group i n t o an a b e l i a n von Neumann algebra. As a c o r o l l a r y to our c a l c u l a t i o n s , we are able to construct mutually i n e q u i v a l e n t .'sequences of 2 -semi-regular and 3 -semi-regular MA subalgebras of the h y p e r f i n i t e 11-̂ f a c t o r which d i f f e r ' f r o m those of Anastasio. TABLE OF CONTENTS REVIEW OF VON NEUMANN ALGEBRAS MAXIMAL ABELIAN SUBALGEBRAS: DEFINITIONS AND SOME KNOWN RESULTS THE MAIN CONSTRUCTION EXAMPLES OF MAXIMAL ABELIAN SUBALGEBRAS REFERENCES ACKNOWLEDGMENT I t i s a pleasure to acknowledge the h e l p f u l s u p e r v i s i o n of Dr. D. Bures as w e l l as the f i n a n c i a l a s s i s t a n c e of both the N a t i o n a l Research C o u n c i l of Canada and the Mathematics Department of the U n i v e r s i t y o f . B r i t i s h Columbia during the p r e p a r a t i o n of t h i s ' .thesis. 1 REVIEW. OF VON NEUMANN ALGEBRAS In g e n e r a l , our n o t a t i o n and terminology i s th a t of Dixmier's hook [ 6 ] . A H i l b e r t space M i s a non-zero v e c t o r space over the complex numbers C together w i t h an i n n e r product x,y -* (x,y) such t h a t M i s complete w i t h respect to the norm x -\ || x || = (x,x) 2~ . By an operator on W we mean a bounded ( e q u i v a l e n t l y : norm-continuous) l i n e a r t r a n s - formation of M i n t o M . We use £(W) to denote the algebra of a l l operators on M , 1^ (or I , when it i s • understood) to denote the i d e n t i t y operator on & , and to denote the s c a l a r m u l t i p l e s of 1^ . I f "tf c K 3 ["#] i s the smallest c l o s e d l i n e a r subspace of M c o n t a i n i n g and p r [ ^ ] i s the (orthogonal) p r o j e c t i o n onto t h i s subspace I f G c , G' i s the set of a l l those B e £(Jf) such t h a t AB = BA f o r a l l A e G ; G' is ' c a l l e d the commutant of G . A von Neumann algebra (or r i n g of operators) on ){ i s a *-algebra of operators on # s a t i s f y i n g G" = G I f Q a £(&) i s a r b i t r a r y , ft(G) , the smallest von Neumann^ algebra on U c o n t a i n i n g G , i s e a s i l y seen t o be (G U G*)" . This a l g e b r a i c d e f i n i t i o n o f a von Neumann algebra (which i s used by Dixmier 'in h i s book [6]) i s equiv- a l e n t to the t o p o l o g i c a l one o r i g i n a l l y employed by von Neumann: G c £(&) i s a von Neumann algebra i f G i s a weakly c l o s e d *-algebra c o n t a i n i n g Ijj . The equivalence 2. of these two d e f i n i t i o n s i s a p a r t of the f o l l o w i n g more ' general r e s u l t , known as the Double Commutant Theorem (see [6; p . .44] , [7; p.885] , or [14; §2]): i f G i s a -x--algebra of operators on 3i which contains 1^ , then R,(G) . = G" i s the c l o s u r e of G i n each of the f o u r t o p o l - ogies: . weak, strong, ultraweak, and u l t r a s t r o n g - on £(»). Let G and B be von Neumann algebras on the H i l b e r t spaces it and K , r e s p e c t i v e l y . . An i s o - morphism of G onto IB i s a l i n e a r and m u l t i p l i c a t i v e map • ''(J) of G onto IB which s a t i s f i e s ,<J>(A*)'= (<j>(A))* f o r i a l l A e G . I f there i s an isomorphism of G onto B we say that G and B are isomorphic, and we w r i t e G = IB . I t turns out that an isomorphism of G onto B i s n e c e s s a r i l y u l t r a w e a k l y and u l t r a s t r o n g l y bicontinuous [6; p.57] . An isomorphism <j> of G onto B i s c a l l e d s p a t i a l i f there i s a l i n e a r isometry "f of W onto X 'such t h a t <{>(A) = T A T 1 f o r a l l ' A e G' . Let G be a von Neumann algebra on U . A t r a c e on G + = {A e G : A >_ 0} i s a mapping uu : G + - [0J«°)u{*p} which s a t i s f i e s the f o l l o w i n g : ( i ) f o r a l l S,T e G + , uu(S + T) = u).(S) + u>(T) ( i i ) f o r a l l S e G + and a l l X >_ 0 , w(XS) = Xuj(S) (where the convention 0-oo = 0 i s used) ( i i i ) f o r a l l S e G and a l l u n i t a r y U e G , uu(USU*) = u>(S) . 3. The t r a c e u) on G i s c a l l e d . \ (a) f i n i t e , i f u>(l) < • • • ' (b) s e r a i - f i n i t e , i f , given T e G + - {0} , there i s an S e G + w i t h - 0 <' S <_ T and u)(S) < °° (c) f a i t h f u l , i f S.e G + and iu(S) = 0 imply" • S = 0 (d) normal, i f , whenever 3? i s an upwardly-directed set i n G w i t h l e a s t upper bound T e G • , then cu(T) = sup {m(S) : S1 e 3} '. A f a c t o r on . M i s a von Neumann algebra G on H w i t h GA G' = . I t i s the f a c t o r s t h a t have r e c e i v e d the most a t t e n t i o n In the l i t e r a t u r e . . Their extreme non- .commutativity a c t u a l l y makes them r e l a t i v e l y easy to study; moreover, every :von Neumann algebra looks l o c a l l y l i k e a f a c t o r , 'and i n f a c t i s b u i l t up from factors' by means of the d i r e c t i n t e g r a l [ 17] . The comparison theorem ( [ 6 ; p. 338] or [12; Theorem V I ] ) i m p l i e s t h a t i f uu - i s a normal trace.-on G +, where >.,G .is. a f a c t o r , then one of the following- must be the case: • . ( i ) ' u)(A) = 0 f o r a l l A e G+- ' ( i i ) tu(A) f o r a l l A e G.'+ - {0} - ( i i i ) w i s f a i t h f u l and s e m i - f i n i t e . Moreover, t o w i t h i n a p o s i t i v e m u l t i p l e , there i s at most one n o n - t r i v i a l normal t r a c e on G + . ' A f a c t o r G such t h a t there i s no normal non-zero s e m i - f i n i t e t r a c e on G + i s s a i d t o b e of type I I I . I f a f a c t o r G. " i s not of type I I I 4. there i s a normal f a i t h f u l s e m i - f i n i t e t r a c e uo on G which, i n some n o r m a l i z a t i o n , must s a t i s f y one of: ( i ) * ( c p ) = {0,1,...,n} f o r some i n t e g e r n >_ 1 ( i i ) u j ( G P ) = {0,1, . . . ,<*>} ( i i i ) . ( C P ) = [0,1] ( i v ) U j ( G P ) = [0,OP)u{-} , where dP i s the set of p r o j e c t i o n s i n G ". In-case ( i ) , G • i s s a i d to be of type I ; i n t h i s case G i s isomorphic to the algebra of a l l nxn matrices w i t h complex e n t r i e s . In. case ( i i ) , G . i s s a i d to be of type 1^; there i s a unique i n f i n i t e C a r d i n a l a such t h a t G i s isomorphic to" the algebra-, of a l l bounded l i n e a r operators on an a-dimensional H i l b e r t space. . I f ( i i i ) h olds, G . i s of type 11^, and i f . ( i v ) holds, of type 11^. I t i s c l e a r t h a t the n o t i o n of a f a c t o r and i t s type a r e . i n v a r i a n t under isomorphisms. Given th a t f a c t o r s of each type e x i s t , o n separable H i l b e r t spaces,' the .tensor product enables one 'to c o n s t r u c t f a c t o r s of each type on a r b i t r a r y i n f i n i t e - d i m e n s i o n a l H i l b e r t spaces. i ' Let 51- be separable i n f i n i t e - d i m e n s i o n a l H i l b e r t " space. .At present, three [two] non-isomorphic f a c t o r s on & of type 1 ^ [11^] are known ([23; .p. ' 3-85], [ 2 4 ] ] ) . In • t h i s t h e s i s , the'only f a c t o r - o f type 11^ ,which i s o f . i n t e r e s t i s the h y p e r f i n i t e one. In general, a f a c t o r G on W i s c a l l e d h y p e r f i n i t e i f i t i s generated by an i n c r e a s i n g sequence (G'n) w i t h each G n a f a c t o r of type I n . • Murray and von Neumann showed that a l l h y p e r f i n i t e f a c t o r s of type I I , on M are Isomorphic [13; Theorem XIV] (see a l s o [6; p.291]); hence one can speak of the h y p e r f i n i t e 11^ f a c t o r on & . Recently, Powers [19] announced the existence of an uncount- able number of p a i r w i s e non-isomorphic h y p e r f i n i t e f a c t o r s of type I I I on M ( f o r the proof, s e e [ l 8 ] ; i n [2] A r a k i and Woods give a d i f f e r e n t proof of t h i s r e s u l t ) . I t i s these f a c t o r s t h a t .we s h a l l be p r i m a r i l y concerned w i t h i n t h i s t h e s i s . Two non-isomorphic n o n - h y p e r f i n i t e f a c t o r s of type I I I have been constructed on M , one by Pukanszky [20] and one by Schwartz [25]- The remainder of t h i s s e c t i o n discusses the three methods which we employ t o o b t a i n von Neumann algebras. These c o n s t r u c t i o n s - the group operator a l g e b r a , the con- s t r u c t i b l e a l g e b r a , and the i n f i n i t e tensor product - are a l l due t o Murray and von Neumann. Let G be a group w i t h i d e n t i t y e . We use G to denote the H i l b e r t space w i t h orthonormal b a s i s (g\) „ ; n o t i c e t h a t G i s separable'whenever G is' at most count- able. For each g £ G there .is a unique u n i t a r y operator V on G s a t i s f y i n g (1.1) V n = ( g h ) A ' f o r a l l h e G . This defines a u n i t a r y r e p r e s e n t a t i o n g - V of G on G . g The group operator algebra over the group G i s the von Neumann algebra. Qr = B(V : g e G) on G ( f o r a complete d i s c u s s i o n of the group operator a l g e b r a , see. e i t h e r [6; pp. 301-303] or [13; § 5 . 3 ] ) . A l t e r n a t i v e l y , £ r can be described as the set of a l l those operators T on G w i t h (T g, n) = (T e, ( h g _ 1 ) A ) f o r a l l g,h e G . The algebra i s a. f a c t o r i f and only i f G has the i n f i n i t e con- jugate c l a s s p r o p e r t y , i . e . , whenever (1 .2) {hgh~"L : h e G} - i s i n f i n i t e whenever g ̂  e i n t h i s case, £ G i s n e c e s s a r i l y of type 11^ . I f G i s separable and i f 6q i s a f a c t o r , then i s h y p e r f i n i t e whenever G i s h y p e r f i n i t e , i . e . , G - tj G„ , where G, c G c • • • and n=l n ± d ' ( 1 .3) each G n i s a f i n i t e .subgroup; of G . Before proceeding to the c o n s t r u c t i b l e a l g e b r a , we w i l l b r i e f l y c onsider the tensor product of two H i l b e r t spaces. Let 2i and K be H i l b e r t spaces xvith orthonormal b a s i s ( c ? i ) i e i a n d ( ^ j ) j e J > r e s P e c t i v e l y . Then. (cpi j ) € i x j i s a n o r*honormal b a s i s f o r U®K . For each j e J we denote by <j). the canonical, embedding J x -> x ® t - i o f M i n t o M @ K • Given A e £(M®K) , the u (j>*. A V (which are operators on W) are c a l l e d the m a t r i x . J elements of A r e l a t i v e to the orthonormal b a s i s (f.) . T i an operator on £®K i s completely determined by i t s m a t r i x elements. Lemma 1.1 With the notation" of the preceding paragraph 7. f o r each j , k e J and A,B e Z(ll®U) , ffl * t V < * ! B V • the sum converging i n the strong topology on. X (M) I f ( A a ) a G D a n e ^ ^(^® 30 which converges weakly to an A e £(3i©K) , then f o r each j , k e J , (^J A a ^ ) a e D c o n v e r S e s weakly to «j)* A ^ '. Proof: Simple c a l c u l a t i o n s (see [6; pp. 23-24] or ' [12; §2.4] )..• C o n s t r u c t i b l e algebras were f i r s t considered by Murray and von Neumann i n [12] and [ 16] , and f u r t h e r developed by Dixmier i n [6; pp. 127-137]; our n o t a t i o n and terminology i s taken from Bures [3]- The system [?7l,y,G,g -> U ] i s g c a l l e d a C-system i f i s a maximal a b e l i a n von Neumann algebra on the H i l b e r t space U , i f G i s a group, and 1 i f g -> U i s a u n i t a r y r e p r e s e n t a t i o n of G on M w i t h U..7YI TJ* =7tl f o r a l l g e G .'- Let [TfV,W,G,gU ] be a g to g C-system. F i n i t e l i n e a r combinations of the operators (M©IA)(U0.®'V ) , M e 1U and g e G , form a ^-algebra on " fc> to' Jf©G (V as i n ( l , l ) ) j . . . w e use G[?7l,M,G,g - U 1 t o denote to g the von Neumann algebra on M@G generated by t h i s *—algebra. • I f A e G[TU,^,G,g - U ] , the matrix elements of A s r e l a t i v e to the orthonormal b a s i s (g) >, f o r G are such th a t f o r a l l g,h e G , <j> * A <J> h = <l> *h_i A <{> e and ( i ) ( i i ) 8. i * A <b U* <s Tfi, . S e t t i n g M_ = <j>* A <b U* , we o b t a i n a v g ' e . & & g y g xe g 5 f a m i l y (M ) r i n lYl which completely determines. A , and we w r i t e A ~ [M : g e G] . A l t e r n a t i v e l y , the algebra G[?)I,M,G,g -» U ] can be described as the set of a l l those T e such that f o r some f a m i l y (M ) r i n % , g g£^ <|>* T <L = M _nU _q f.or a l l g,h e G . b 1 gh gh Lemma 1.2 Let [??l,W,G,g -+ U ] be a C-system and o l e t A and B be operators i n G[?H ,M,G,g -+ U .] w i t h §> A ~ [M : g € G] and B ~ [N : g e G] . For a l l g,h e G o g and M £ 7ft : ( i ) (J)* AB <J) U* = E M _i U _i N. U* _ x', where the g e g keG Sk x gk 1 k gk ± sum converges i n the•strong topology on 7U ( i i ) ( i i i ) <f>g(M8I)(Uh®Vh) $ e U* = 6 g^ hM ( i v ) ^ A(U h®V h) ^ U* = M ^ . i " (v) Vs (U hOT h) A * e U* = U h M h . l g y* . • Proof. Simple c a l c u l a t i o n s . D e f i n i t i o n 1.3 The C-system [ftt,tt,G,g - U ] i s c a l l e d : ( i ) f r e e , i f 771 n U ' W = {0} f o r a l l g e G - {e} ( i i ) ergodic, i f % n {U^ : g e G}' = . A von Neumann algebra i s c a l l e d c o n s t r u c t i b l e i f i t i s s p a t i a l l y isomorphic t o G[#t ,3i,G,g -» U ] f o r some f r e e C- g system [#t,M,G,g - U ] . - 9. Pro-position 1.4 ( [ 3 ; §4] and [4; § 7 3 ) . The C-system [?U,W,G,g - U ] i s f r e e i f and only i f , f o r each g e G - {e}, there e x i s t s a f a m i l y ( E j _ ) i € i o f P r o j e c t i o n s i n 7f{ such E E. that E , = 1 and E, U E, U* = 0 f o r a l l i 6 I P r o p o s i t i o n 1.5 ( [ 6 ] ) . Let [?ft,W,G,g - U ] be a f r e e C-system, and l e t G = G[7?[,K,G,g - U ] . . Then to' 7H.®<£A i s a maximal a b e l i a n sub algebra- of G , and G i s a f a c t o r i f and only i f [?72,34,G,g -* U ] i s ergodic. I f to G i s a f a c t o r , then: ( i ) G i s of type I i f and only i f ?U contains a . minimal p r o j e c t i o n ; I f n i s the c a r d i n a l i t y of a maximal f a m i l y of p a i r w i s e orthogonal minimal p r o j e c t i o n s i n Kfl 3 then G i s of type I ( I i ) G i s f i n i t e ( i . e . , of type 11^ or I , n <° a) i f and only i f there i s a normal f i n i t e f a i t h f u l t r a c e w on % + w i t h . u>(U M U*) = u>(M) f o r a l l g e G and a l l M e % + .. ( i i i ) G i s of t y p e . I l l i f and only i f there does,not e x i s t a normal s e m i - f i n i t e f a i t h f u l t r a c e u) on W + w i t h u)(U_ M U*) = to(M) f o r a l l g e G and a l l M e • frC' • P r o p o s i t i o n 1.6 ( [ 8 ] , [13; Lemma 5 . 2 . 3 ] ) . Let [M,3i,G,g. - U_] be a f r e e C-system, and suppose that' to • G[ftl,!t 3G,g - U0.] Is a f a c t o r of type I I . .. If." G i s to -1- a.belian, then G[7?l,M,G,g - U ] i s h y p e r f i n i t e . g 10. In [15] a complete d i s c u s s i o n of the i n f i n i t e tensor product o f von Neumann algebras can be found. Let I be an i n f i n i t e i n d e x i n g s e t , and l e t ( ^ i ) j _ € j b e a f a m i l y of H i l b e r t spaces. A f a m i l y ( f - ) - T w i l l be c a l l e d a C 0-sequence i f each f. e W. . and i f L | 1 - || f.. || | <oo. i e l 1 Two C Q-sequences ( ^ j J i e i a n d ( g i ^ i e l a r e c a l l e d equivalent i f 2 I 1 - ( f . ,g. )| <°° 1 t h i s i s an equivalence r e l a t i o n i e l 1 1 . • on the set of a l l C Q-sequences. Let ( ^ ^ i e i ^e a f i x e d C Q-sequence, and l e t £ • denote the equivalence c l a s s determined by ( f j ^ i g j •  • F o r ea,ch ( f - ) . T e $ , l e t ® f.. denote the map i e l ( g i ) i € l -*TT (fi^§i) o f @ 0 i n t o c • D e f i n i n g f i n i t e i e l ' l i n e a r combinations of the ® f. i n the obvious manner, i e l 1 we o b t a i n a v e c t o r space V .' The map ( f . ) . T -> ® f. of 1 1 € 1 _ i e l 1 <§ i n t o V i s c l e a r l y m u l t i - l i n e a r . " " The .form ( © f. , @ g. ) -»TT (f n-jg n-) extends t o a s e s q u i - l i n e a r i e l 1 i e l 1 i e l 1 1 - form on V which can be shorn to be an i n n e r product. We product, which we denote by ® (M^,f?) , as the i n f i n i t e w i l l r e f e r t o the completion of "V r e l a t i v e to t h i s i n n e r ,0 i e l t ensor product of the M. r e l a t i v e to ( f ? ) . x ; von Neumann, r e s e r v i n g the phrase i n f i n i t e tensor ( d i r e c t ) product' f o r a much l a r g e r H i l b e r t . s p a c e , c a l l e d t h i s space the £ Q-adic incomplete d i r e c t product. Note that ® (M.,f?) r e a l l y • i e l 1 1 11. depends on @ Q , and not on the p a r t i c u l a r (f°)j[ei G ^ 0 s e l e c t e d . The f o l l o w i n g r e s u l t f a c i l i t a t e s working w i t h the i n f i n i t e tensor product space. P r o p o s i t i o n 1.7 ([15; Lemma 4 . 1 . 4 and Theorem V I I ] ) Let I be a n ^ i n f i n i t e i n d e x i n g s e t , l e t (K.). ̂  be a f a m i l y .of H i l b e r t spaces, and f o r each I e l , l e t f ? be a u n i t v e c t o r i n . ( i ) For each i e l , choose an orthonormal b a s i s J (:£"?)- T w i t h 0 e J . f o r M. . Let J be :he set of a l l those j e TT J . w i t h j ( i ) = 0 i e l x ! f o r a l l but f i n i t e l y many i e l , and f o r each .pj(i) i e l an orthonormal b a s i s f o r ® (34','. , f ? ) i e l 1 1 j e J , l e t f , = ® f ^ x ) ." Then ( f . ) . . T i s J a rT 1 0 <J£<J ( i i ) L et I = U I, be a d i s j o i n t Linion. Then there keK K i s a unique l i n e a r isometry ( c a l l e d the a s s o c i a - t i v i t y isomorphism) of ® (34.,f.) onto i e l 1 x ( © (34. ,f°), ® f ? ) which c a r r i e s ® f. keK i e l , i e l , i e l k L  \ i n t o ® ( ® f.) f o r each C -sequence ( f . ) . T keK i e l . ' 1 o - ^ i ' i e l k equivalent t o ( f ? ) - T' Let ( " i i ) i e j a n d ( ^ i ^ i e i ' o e a s i n P r o p o s i t i o n 1.7> and l e t J! = ® (MT,f,°) . I f T e £(M, ) , there i s a unique i e l x 1 x o 12. a. (T) £ £(3f) which s a t i s f i ? 1 o (T)][ ®^ f ± ] = (_®_̂  f_ ^ f ± ) ® (T f ± ) O" 1 o i e l x ' i e l - C i } ~ ~o f o r each CQ~sequence ( ^ j ^ i g j e quivalent to ( ^ i ) j _ € i _ • . r i t i s e a s i l y seen t h a t a.. i s a- isomorphism; f o l l o w - o i n g the us u a l n o t a t i o n , we w r i t e T f o r a. (T) . I f o G^ i s a von Neumann algebra, on 34 , then o o G^ = {T : T e G. } i s a von Neumann algebra on 34 . I f , o 1 o f o r each 1 e I , G. i s a von Neumann on 3-4. , then ® (G_. ,f?) denotes the von Neumann algebra ft(G. : i e I) i e l x _ 1 on 34 ; we c a l l ® (G ,f°) the i n f i n i t e tensor product of i e l - 1 o • the G i r e l a t i v e to (f±)±£j P r o p o s i t i o n 1.8 ( [ 3 ; § 3 ] ) . Let ) i e I >• ( f i ) i e i a n d (G, ). j be as above, and l e t 34 = ® (34. , f ? ) , G.= ® (G.,f?) Then: i ( i ) . G is,maximal a b e l i a n on 34 i f each G^ i s maximal a b e l i a n on 34. i ( i i ) G i s a f a c t o r i f and only i f each G^ i s a f a c t o r ( i i i ) G = £(34) i f each G± = £-(M±) .' Let I be an i n f i n i t e i n d e x i n g set. For each i e l , l e t G 1 be a group w i t h i d e n t i t y e 1 , l e t Q1 = G C f l l ^ a V ^ g - u j ] , where •[n1,V±9G1,S - u j ] ' i s a 13. f r e e C-system, and l e t f ? be a u n i t v e c t o r i n M 1 . Set U. ~ © ( M V ? ) , Ik = ® (W. 1,^) , G = ® ( G 1 , ^ . ®(e i) A) , i e l i e l i e l and l e t G- be the weak d i r e c t product of the G 1 . For each g = ( g 1 ) . T -e G , l e t U = TT 11% (a f i n i t e produce l e l s i e l g i n which the f a c t o r s commute). From P r o p o s i t i o n 1.7 i t f o l l o w s t h a t there i s a l i n e a r isometry Y of ,i\A „o i\A• ® (M 1 ® ( G X ) A , f ? ®(e 1).) i e l ' 1 onto 34 ® G w i t h >( ®'(f ± ® ( g 1 ) " ) ) = ( ® f i)®(.(g 1) i e I) A -i e l 1 ' i e l 1 < i e l whenever ( ^ j J i g j i s a G 0~sequence equivalent to ( ^ i ^ i e l and ( g 1 ) i e I € G . ' P r o p o s i t i o n 1 .9 ( [ 3 ; P r o p o s i t i o n 4.1] and P r o p o s i t i o n ^ 1.4). • With the n o t a t i o n of the previous paragraph, [7FT,34,G,g -» U ] i s a.' f r e e C-system which i s ergodic i f and only i f each [Wl 1,34 1,G 1,g -* U^] i s ergodic. The map A -»:7'AT~"L i s an isomorphism of G onto G[W2,34,G,g - Ii ]. . 2 MAXIMAL ABELIAN SUBALGEBRAS: DEFINITIONS AND SOME KNOWN RESULTS Only separable H i l b e r t spaces w i l l be considered i n the' remainder of t h i s t h e s i s . The f i r s t p a r t of t h i s s e c t i o n c o n s i s t s of the b a s i c d e f i n i t i o n s which, to some extent, serve t o c l a s s i f y the maximal a b e l i a n (MA) subaigebras of a f a c t o r . Next, a p summary'of the known r e s u l t s concerning MA subalgebras of the h y p e r f i n i t e 11^ f a c t o r i s given. We conclude t h i s s e c t i o n w i t h a complete c l a s s i f i c a t i o n of the MA subalgebras of £(3i) ; although t h i s r e s u l t xvas known to von Neumann, i t s proof does not seem to appear e x p l i c i t l y i n the l i t e r a t u r e . I f 7H and % are subalgebras .of a von Neumann algebra G , we say th a t tt[ and .71 are equivalent i n G (or simply e q u i v a l e n t , i f G i s understood) i f there i s an automorphism of G which c a r r i e s IK . onto % . This defines an equivalence r e l a t i o n on the c o l l e c t i o n of a l l subalgebras of G . . One problem i n the s t r u c t u r e theory of von Neumann algebras i s to c l a s s i f y up to equivalence a l l of the sub- algebras of a given von Neumann algeb r a , i . e . , the determination of a l l equivalence c l a s s e s of subalgebras. This problem i s , of course, extremely d i f f i c u l t . • The m u l t i p l i c i t y t h e o r i e s of Halmos'[lO] and of Segal [2?] give s o l u t i o n s t o the c l a s s i f i c a t i o n up to equivalence of the a b e l i a n subalgebras 15. of a f a c t o r of type I a c t i n g on a H i l b e r t space of a r b i t r a r y dimension. . For f a c t o r s of type I L ^ , the analogous problem has been examined and some r e s u l t s have been obtained by Bures [ 4 ] , R e c a l l t h a t a subalgebra ^ of a von Neumann algebra G i s MA i n G i f and only i f 711* G =. D e f i n i t i o n 2.1 Let W be a subalgebra of the von Neumann algebra G . For each i n t e g e r m >_ 0 , we i n d u c t i v e l y define subalgebras Nm (7ft) of G by: N ° ( 7 U ) = 7K Nm(?7i) = &(U e G : U u n i t a r y and UNm _ 1 (7n) U* = N m - ± (T)l)' } . m > 1 . We w i l l w r i t e N(fll) i n s t e a d of N 1 ^ ) , and we c a l l , t h i s the normalizer of 7?l ( i n G) . Notice t h a t ( ^ ( W ) ^ 1 i s a n e x P a n d i n S sequence of subalgebras of G . . . . D e f i n i t i o n 2.2 • I f Vl i s a MA subalgebra of f a c t o r G , we c a l l Vfi : ( i ) r e g u l a r i f N-(1U) = G ( i i ) semi-regular, i f • N(Tt) i s a f a c t o r d i s t i n c t from G ( i i i ) s i n g u l a r , i f = %, ( i v ) m-semi-regular (m >_ 1 and an'integer),, i f W l . N(7K),.; • . j N ^ f y p a r e not f a c t o r s but N ™ ^ ) i s a factor. 16. .' D e f i n i t i o n 2 .5 Let 7̂ 1 be a MA subalgebra of a von Neumann algebra G , and l e t m >_ 1 be an i n t e g e r . We say that 7H has: ( i ) proper l e n g t h m , i f N m" 1(?n) ^ G but Nm(??0 = G ( i i ) improper l e n g t h m , i f N^CW) f N m(m) = N m + 1(?n) f G . The d e f i n i t i o n s of r e g u l a r , semi-regular and s i n g u l a r MA subalgebras were f i r s t given by Dixmier [ 5 ] , while the no t i o n of m-semi-regularity i s due t o Ana s t a s i o [ 1 ] . D e f i n i t i o n 2 .3 i s a refinement of Tauer's l e n g t h of a MA sub- algebra [ 2 8 ] . I t i s easy t o see th a t i f Tfi and U are equivalent subalgebras of a von Neumann algebra G , then so "are N(7W) and N(7l) . Consequently, each of the p r o p e r t i e s of D e f i n i t i o n s 2.2 and 2 .3 i s an i n v a r i a n t of the equivalence c l a s s determined by a MA subalgebra. The study of MA subalgebras of the ' h y p e r f i n i t e 11^ f a c t o r was i n i t i a t e d by Dixmier i n h i s seminal paper [ 5 ] . Let G be a group, and consider the group operator a l g e b r a £Q on G . I f G Q . i s a subgroup of G , l e t N.(G )' be the normalizer of G Q i n G , and l e t G Q) = R(V g : g e G Q) c f G . Dixmier showed t h a t , under c e r t a i n c o n d i t i o n s on G and G Q ^ ( G Q ) i s a MA sub- algebra of. - ' d G and N(ft?(G )) = 7fl(N(G )) . Using these r e s u l t s and choosing s u i t a b l e groups G and subgroups G Q he constructed examples of a r e g u l a r , a semi-regular and a 17. s i n g u l a r MA subalgebra of the h y p e r f i n i t e 13^ f a c t o r . The groups used by Dixmier i n these c o n s t r u c t i o n s may be described as f o l l o w s . Let P be a countably i n f i n i t e f i e l d which i s the i n c r e a s i n g union of a sequence of f i n i t e s u b f i e l d s ( i n p a r t i c u l a r , we may take f o r F the a l g e b r a i c completion of a f i n i t e f i e l d ) , and l e t K be the m u l t i p l i c a t i v e group o f non-zero elements of F . •' The* set K x F becomes a group under the operation (a,b)(c,d) = (ac,ad + b) The group K x F i s h y p e r f i n i t e and has the i n f i n i t e con- jugate c l a s s property (see the proof of Theorem 4.1). The subgroup K x {0} of K x F i s i t s own normalizer and 77l(K x {0}) i s a s i n g u l a r MA subalgebra of ^ j ^ x p > whi l e {1} x F i s a normal subgroup and 7 ^ l({l} x F) i s a r e g u l a r MA subalgebra. , I t i s a b i t more d i f f i c u l t to o b t a i n a semi-regular MA subalgebra. • Let H -be the group of a l l non-singular 2 x 2 matrices over F and l e t L be the normal subgroup of H c o n s i s t i n g of a l l s c a l a r m u l t i p l e s of the i d e n t i t y matrix. Let G = H/L , l e t H Q and be the subgroups of ; H w i t h t y p i c a l elements b ^ 0 , r e s p e c t i v e l y , and l e t G = H /L . Then the normalizer o f - 13. G Q i n G i s H^/L , and % ( G o ) i s a semi-regular MA- • subalgebra of (?Q . Let F and K be as above. Pukanszky has shown tha t f o r some subgroups K Q of K , 7A (K X {0}) i s a s i n g u l a r MA subalgebra of £ K x F [21]. By v a r y i n g F and o K Q a p p r o p r i a t e l y , he constructed a sequence of p a i r w i s e i n e q u i v a l e n t s i n g u l a r MA subalgebras of the h y p e r f i n i t e I L ^ • f a c t o r . The mutual inequivalence of these subalgebras \ia.s e s t a b l i s h e d by means of the m u l t i p l i c i t y theory of Segal. Using group operator algebras over groups of m a t r i c e s , Anastasio constructed i n f i n i t e sequences of p a i r - wise i n e q u i v a l e n t 2-semi-regular and J>-semi-regular MA sub- algebras of the h y p e r f i n i t e 11^ f a c t o r [ l ] . The i n v a r i a n t of proper l e n g t h was used to e s t a b l i s h the mutual ine q u i v a l e n c e of these subalgebras. In the proofs of Theorems 4.2 and 4.3 the groups, used w i l l be described. Tauer's c o n s t r u c t i o n s of MA subalgebras of the h y p e r f i n i t e 1 1 ^ . f a c t o r are based on a d i f f e r e n t method. For each i n t e g e r p >_ 1 , l e t M^ denote the algebra of a l l 2^ x 2 P matrices w i t h complex e n t r i e s . Embedding Mp i n M_ i n a s u i t a b l e manner and u s i n g the normalized t r a c e on V i each M^ , M = U M becom'es a p r e - H i l b e r t space.;' l e t p p=l p 34. denote i t s completion. We can regard M as a set of 1 9 . operators on M by l e t t i n g each element of M act on M by l e f t m u l t i p l i c a t i o n . The von Neumann algebra G on U generated b y M i s the h y p e r f i n i t e 11^ f a c t o r . Tauer .constructs examples to' show t h a t : ( i ) f o r each i n t e g e r m >_ 2 , G contains m . p a i r w i s e i n e q u i v a l e n t semi-regular MA subalgebras of proper l e n g t h m ( [ 2 8 ] , . [ 2 9 ] ) ( i i ) f o r each i n t e g e r m >_ 2 , G contains an m-semi- r e g u l a r MA subalgebra [ 3 0 ] . The remainder of t h i s s e c t i o n i s taken up w i t h the c l a s s i f i c a t i o n of the MA subalgebras of £(#). ' ••Lermiia-2; 4—'—Let- -—fl~--b-e~a--Hilbert- space of dimension at l e a s t two, and l e t VI be a MA subalgebra of £(31) such t h a t there i s a f a m i l y ( E i ) j _ € j o f minimal p r o j e c t i o n s i n ? t with-, 2 E.' = I .. Then 7*1 i s r e g u l a r . Proof. As 7h -is MA on ' W , each E^ must be of rank one. Hence we can s e l e c t an orthonormal b a s i s (cp.). T f o r . M such t h a t E^cp^ = cp̂  f o r each I e l . I n p a r t i c u l a r , I must c o n t a i n at l e a s t two elements. . Suppose t h a t an A e £(W) commutes w i t h each E^ . As each E^ i s a minimal p r o j e c t i o n , a simple c a l c u l a t i o n •shows th a t each E^ A E^ e ?ty . - And as • A = E E. A = E E. A E. i e l 1 i e l 1 1 2 0 . i n the weak topology on £(34) A e TA For d i s t i n c t elements i and j of I 3 define u n i t a r y ' o p e r a t o r s U. . and V. . on 34 by s e t t i n g r cp < k k / k = i k * j . k ^ i , j k = i k = J )• f o r a l l k e I . .  Given an A e Vl, i t i s easy t o v e r i f y t h a t each U. . A(U. . ) * and each V. . A(V. . ) * commute w i t h every ,• and hence u l«1«l(y i ; I)*.-v l,^(v l i l)«. M .Therefore ft(U e £(34) U u n i t a r y and U? f t u * = Wl) t=> P R(U ± J., V ± j. : i , j e I and i ^ j ) , and so i t s u f f i c e s to show tha t i f an A,e X(34) commutes wi t h each U. . and each V. ." , then A e C,, . • \ i j I J « Suppose th a t A e £(34) commutes w i t h each U. and each V. For each k e I we can w r i t e 21. Acp = £ a,, cp. . where the a,, are complex numbers, -f e l " F i x i , j e I w i t h i ̂  j . Then £ a. . cp, = Acp. = U. .Acp . = E a, . U. . cp. J j . V *k " A t p x - V i j *>J -' " k j V i j *k • On comparing c o e f f i c i e n t s i n these two expansions, we see that a.. = a.. , a. . = a., and a. . = -a.. , and t h e r e f o r e n 33 13 3~^~ ^ J J A e % • Lemma 2.5 Let (X,£,u) be a f i n i t e measure space, where E is. a a-algebra of subsets of X . For each cp e L**0 (X,£,|i)' , the r e l a t i o n ( \ f )(x) = c p ( x ) f (x) f e L2(X,£,n) and x e X 2 ~~ defines an e £(L (X,£,n)) , and cp - i s an i s o m e t r i c ^-isomorphism of L°*(X,£,u) onto a von Neumann algebra which i s MA i n £(L2(X,Z,g)) . Proof. Easy c a l c u l a t i o n s (see e.g. [6; pp. 117-118] or [11;.pp. 6-14]). Lemma 2.6 Let X = [0,1] , l e t . I. be the B o r e l sub- sets of X , and l e t \ be Lebesgue measure on £ . Let M = L 2(X,E afc) and l e t = [u^..: cp e L~(X,E,\)} Then ^ i s a re g u l a r MA subalgebra of Z(tt)... Proof. By Lemma 2.5, Tfi i s a MA subalgebra o f £(l!) . Let r e (0,1) be a f i x e d i r r a t i o n a l number, and l e t T : X X . be a d d i t i o n by r modulo 1 . I t i s c l e a r t h a t .the map of f - f»T i s a u n i t a r y t r a n s f o r m a t i o n , say U , of H . Moreover, U fllu* = IK , f o r i f . f € M and cp € L ° * ( X , £ , X ) are a r b i t r a r y , . ; U M^ U* f = U M ^ f o T " 1 ) = U(cp.(f.T" 1)) = •>}••, = (cp.T).f = M ^ f . To show that 7*1 i s r e g u l a r , i t x v i l l s u f f i c e to show t h a t i f an A € commutes w i t h U and w i t h each u n i t a r y operator i n tH then A € C y . •'• For each n e 2 , l e t cp n(x) = e 2 7 r i n X , x e X ; i t i s well-known that ( C * ) N ) N E 2 i s 3 X 1 o r ^ n o n o r m a l b a s i s f o r -M ' . A simple c a l c u l a t i o n shows th a t Lto .= e • cp f o r n n each i n t e g e r n . Now suppose th a t an operator A e V commutes w i t h U and w i t h each u n i t a r y i n 7k.. . For each n e.Z we can w r i t e Acpn = Z a m n cp^ , where the are m=-«° , complex numbers. Then :. . S • a cp = Acp = e. *wxnz u A cp m =_*» mn vm ^ n ^n ' = e-27rinr - m=-oo m n m . v mn m̂ • m=-o» As r i s i r r a t i o n a l , e 2 7^-(m-n)r ^ ± u n l e s s m = n . comparing c o e f f i c i e n t s and u s i n g t h i s remark, we see that = 0 unless m = n . Consequently, there i s a f a m i l y " ((CL.V,, of complex numbers such t h a t Ao) = a cp_ f o r each a iic ci n n n 23. n e Z . Now f o r each i n t e g e r n , M i s a u n i t a r y n op e r a t o r i n and t h e r e f o r e a c p = A c p = M Acp = a M cp = a cp . o o ^o cpn -n -n cp^ -n -n o Thus a Q = a + 1 = a + 2 = ••• 3 and so A e Lemma 2.7 Let H i . be a MA von Neumann algebra on 3i which possesses no minimal p r o j e c t i o n s . Then i s re g u l a r . Proof. As 34 i s separable, there i s a u n i t v e c t o r x e 34 which i s separating , f o r 1 H , i . e . , M e 7fy and M x = 0 imply M = 0 [27; Lemma 2.5]. A simple c a l c u l a t i o n [6; p.6] shows t h a t x i s c y c l i c f o r = W[ , i . e . [ I H x ] =34 Applying now [27; Lemma 1.2], there i s a compact Hausdorff space X , a re g u l a r measure \i -on the a - f i e l d £ generated by the compact subsets of X w i t h M-(X) = 1 , and a l i n e a r isometry of J! onto L 2(X,E,|i;) c a r r y i n g 7 H onto { M c p : cp e i f ^ X ^ u ) } . - As D t does not possess minimal p r o j e c t i o n s , the measure algebra of (X,S,u) i s non-atomic. L e t ' ( f R ) be an. everywhere-dense sequence i n L (X,Z,)i) , and.for.each n. , l e t E n = ( x e X :' |f (x) - 11 <_ ..." C e r t a i n l y each .E e 2 . Given E e E and e > 0 , there i s an i n t e g e r . n ' such that 2 4 . - .'. > J | f n ( K ) - 1| 2 dji(x)'+ J | f n ( x ) | 2 dM(x) E-E n E n-E , > i n(E A E n) , where A denotes symmetric d i f f e r e n c e . Hence the measure algebra of (X,E,|i) i s separable. By a c l a s s i f i c a t i o n theorem of Halmos and von'Neumann ( c f . [9; p. 173]), there i s an isomorphism $ of the measure algebra of .(X,E,u) onto th a t of ( [ 0 , 1 ] , ^ , X) , where A i s the B o r e l subsets of [0,1] and X i s Lebesgue measure on A . In an obvious manner, we can regard § as a mapping from Z i n t o A ; as such, § i s not onto , - but every member of A i S equivalent t o a member of A i n the range of § . .' I t i s r o u t i n e t o check t h a t modulo the equivalence r e l a t i o n "equal almost everywhere", the map n n . - • . ^ a i X E . "".^ a i X * ( E . ) . ~ a i € . C a n d E i 6 E 2 i s w e l l - d e f i n e d , l i n e a r , , and i s an L -isometry of the set of simple f u n c t i o n s on (X,E,u) onto the set of simple f u n c t i o n s o rl_- (C9A],-df , X)_;_ hence the map extends'to a l i n e a r isometry of L 2(X,E,u) 'onto L 2 ( [0,1],-cf,- X) . I t i s readily', seen that t h i s isometry c a r r i e s [M^ .: cp e L°*(X,E,|a)} onto {M^ : cp e L ~ ( [0,1], A ,X)} . Therefore a c t i n g on M i s s p a t i a l l y isomorphic to [VL ,: cp e L e e( [0,1], tf ,X)3 a c t i n g on L 2 ( [0,1] ,4 , X) . 0 25- 'As the l a t t e r i s r e g u l a r (Lemma 2 . 6 ) , so i s the former. Remarks (1) Our proof of Lemma 2.4 does not make use of the assumption t h a t W i s a separable H i l b e r t space. (2) Segal has shown th a t Lemma 2.5 holds provided only that the measure space- i s s e m i - f i n i t e ( i n the sense t h a t every set of i n f i n i t e measure contains sets of a r b i t r a r i l y l a r g e f i n i t e measure) and l o c a l i z a b l e ( i . e . the measure algebra i s complete as a p a r t i a l l y • ordered set) [ 2 6 ] . . (j5) Lemma 2.7 c o n s i s t s e s s e n t i a l l y i n showing t h a t a MA von Neumann algebra without minimal p r o j e c t i o n s on a separable H i l b e r t space i s s p a t i a l l y isomorphic to {M^ : cp e L°°( [0,1],-$, X)} . This -is e s s e n t i a l l y ... due to von Neumann, and i s well-known,.although an' e x p l i c i t proof does not seem to appear i n : the \ l i t e r a t u r e . I t can be deduced from the general Maharam c l a s s i f i c a t i o n theory of measure algebras (-cf. [26; C o r o l l a r y 5.1]). Our proof-avoids t h i s ' deep theorem, u s i n g i n s t e a d a weaker c l a s s i f i c a t i o n ••. theorem. Let 1*1 be a MA subalgebra of £(«) . I f 7YI s a t i s f i e s the hypothesis of Lemma 2 . 4 , set e(7ty) = 0 ; otherwise, set c(fl[) = 1 . Let n($l) be the maximal number •' of p a i r w i s e orthogonal minimal p r o j e c t i o n s i n T^L (0 <_ n(7)1) < _ o » ) . The combination c(fl) = 0 , n(%) = 0 i s i m p o s s i b l e , whil-e examples of a l l other combinations can be r e a l i z e d as L°D(X,E,u) a c t i n g on L 2(X,E,n) under p o i n t - wise m u l t i p l i c a t i o n f o r some f i n i t e measure space (X,E,|a) . Theorem 2.8 Let ̂  be a MA'von Neumann, algebra on Jt 771 i s r e g u l a r i f c(flt) = 0 or i f c 0 H ) = 1 and n(JM) = 0 ; f o r a l l other p o s s i b l e combinations, does not f a l l i n t o any of the cl a s s e s of D e f i n i t i o n 2.2. Proof. Lemma 2.4 [Lemma 2.7] shows t h a t tK i s re g u l a r i f c(tt[) = 0 [c(TH) = 1 and ri(?ft) = 0]. Now suppose t h a t c(k) = 1 and n(M) >. 1 • Let ( E ± ) i j be a maximal f a m i l y of p a i r w i s e orthogonal minimal p r o j e c t i o n s i n tyk. , and l e t E = E E. ,. F = I - E . i e l Then both E and F are non-zero p r o j e c t i o n s i n 7 f t . Notice t h a t % E [ W F ] i s a MA von Neumann algebra on E(Jt) [F(JI)'] ' s a t i s f y i n g the hypothesis of Lemma 2.4 [Lemma 2.7] and t h e r e f o r e N(WE) = X(E(3*)) , N ( W p ) = £(F(M)> . The c a n o n i c a l isomorphism of 14 onto E(M,)© F(14) induces an isomorphism of 7H onto T^©?)^- [6; p. 22], and so i t s u f f i c e s to show, tha t T ^ ^ ^ p i s a semi-regular subalgebra of £(E(W)© P(M)) . Let U and V be u n i t a r y operators on E(14) and F(M) , r e s p e c t i v e l y , such that. U %^ U* = %^ and vTJlp V* =7flp . Then U© V "is a u n i t a r y operator on E(«)«P(M) w i t h (U©V) (THg©^) (U© V) * = W E © 7 ? l F . 27. Therefore N(ffl E) © N ^ ) <= N ^ © ^ ) . Conversely, suppose tha t W i s a u n i t a r y operator on E(3f) © F(X) w i t h wflYlgeTRjJW* = /f/[ . As automorphisms of a von Neumann algebra map minimal p r o j e c t i o n s i n t o minimal p r o j e c t i o n s , . W(E$0)W* = E#0 and.'W(0«F)W* = 0®-F . Therefore W = U$ V , where U and V are u n i t a r y operator on E(W) and F(H), r e s p e c t i v e l y , such that U* = 7 ^ and v W p V* = 7 ^ . This shows t h a t • = £(E(M))e£(F(M)) , which i s not a f a c t o r . Theorem 2.9 Two MA subalgebras W l and "TL on l i are equivalent i n i f and only i f c(1H) = c ( 7 l ) and n(7Kl) = nCTU) . Proof. The proof of t h i s theorem i s contained i n the .proofs of the preceding r e s u l t s . 3 THE MAIN CONSTRUCTION Throughout t h i s s e c t i o n , p w i l l denote a f i x e d p o i n t i n (0,^) and G w i l l denote a f i x e d countably . i n f i n i t e group w i t h i d e n t i t y e . We begin w i t h a summary of t h i s s e c t i o n . . Our f i r s t task i s to c o n s t r u c t a type I I I f a c t o r G c o n t a i n i n g a r e g u l a r MA subalgebra ', a type 1 ^ f a c t o r <B(p,G) , and, f o r each subgroup G of G , a subalgebra 7l(PiG,G Q) of iB(p,G) . For a subgroup GQ of G , we w i l l use 'Dl(G.>G0) to-denote the subalgebra of the group operator algebra £ r generated by {V : g e G } .• R e c a l l t h a t \a g O N(G Q) denotes the normalizer of a subgroup G Q of G . Our second task i s to prove the f o l l o w i n g s i x theorems, which c o n s t i t u t e the main r e s u l t s of t h i s s e c t i o n : Theorem 3»1 Let GQ be a subgroup- of G . Then / W p ® / 3l(p,0,G o) i s a MA subalgebra of G p ® R(p,G) i f and only i f - (a) : G ' i s a b e l i a n and {g^-'g g - 1 : g^ e G } i s v o o o o o i n f i n i t e whenever g e G -. G . Theorem 3-2 Suppose th a t G Q i s a subgroup o f ' G s a t i s f y i n g (3) : gi'ven a f i n i t e subset F of G and a g e G , there are i n f i n i t e l y many g Q e G Q such t h a t : 29- ( i ) h,k e F and h g Q k "L= g Q imply' h = k ( i i ) i f ' , g $ N(G Q) , then g g Q g _ 1 k % • " Then . N(73l.p ® ^ ( p , G , G O ) ) = G p ® ft(p,G,N(G0)) N(G p ®ft(p,G,GQ)) = G p ® 7l(p,G,N(G 0)) . Theorem 5-3 For a subgroup G of G , C - ® 7 l(p,G,G ) i s a f a c t o r i f and only i f G Q has the i n f i n i t e conjugate c l a s s p r o p e r t y (see (1.2)) . Theorem 5-.4 Let G q be a subgroup of G . Then % .®fll.(G,G0) i s a MA subalgebra of G. ® 6Q-. i f and only i f G Q s a t i s f i e s c o n d i t i o n (a) of Theorem 3«1. Theorem 3-5- I f G Q i s a subgroup of G s a t i s f y i n g c o n d i t i o n • ((3) of Theorem 3*2, then N(7R p ® m(G,G Q)) = G p ® 7n(G,N(G-))" N(G ® 7H.(G,GQ)) = G ® tH(G,N(.G0)) p » ̂  ' o " p Theorem 5-6 For. a subgroup G Q of G G p ® 7 l l ( G Q ) i s a f a c t o r i f and only i f G Q has the i n f i n i t e conjugate c l a s s property. The algebra • G and i t s subalgebra 7ty are defined i n the t e x t preceding Lemma 3.10 whil e B(pyG) and the 7l(p,G,G ) are defined a f t e r Lemma 3-13 and i n D e f i n i t i o n 3.15, r e s p e c t i v e l y . The proofs of the s i x theorems are given at the end of t h i s s e c t i o n . 30. Before proceeding to the a c t u a l c o n s t r u c t i o n s , we f i r s t e s t a b l i s h a t e c h n i c a l r e s u l t . Lemma 3-7 Let [Jf[ ,W,G,g - U ] be a C-system, and g l e t G = G[7Yl,H,G-,g - U ] . Let TV) be a subalgebra of VK, g o l e t . G Q be a subgroup of G , and suppose th a t U g ? ^ 0 U g = 7KQ f o r a l l g e G Q . Then R( (M ® I&)(U ® V ) : M e THQ .and g € G Q) c o n s i s t s of a l l those operators A e G w i t h A ~ [M : g e G] s a t i s f y i n g : ( i ) M e "Pi whenever g e G § O o ( i i ) M = 0 whenever g e G - G g o Proof. Let (? = { E (IM ® I A ) ( U ® V j : each M e 7)1 and u geF e> « & e> e> o F c G f i n i t e } o 91 = fc( (M ® I ^ ) ( U g ® V g ) : M e VLQ and g e G ) ( ? 2 = [A e G : A s a t i s f i e s ( i ) and ( i i ) } . Observe th a t QQ c (J (Lemma 1.2) and t h a t , by the c o n t i n u i t y of m a t r i x elements (Lemma l . l ) ' , l ? 2 i s a von Neumann algebra. A simple c a l c u l a t i o n together w i t h an a p p l i c a t i o n of the double commutant theorem shows tha t <PQ i s a weakly dense sub«*-algebra of 6^ , and th e r e f o r e t h a t <P c S?2 . To - 31. show th a t ^ 2 a i i : w i l l - s u f f i c e to show t h a t Q'Q c Suppose t h a t T e , i . e . , T i s an operator on M ® G which commutes w i t h each (M ® I ) ( U ® V ) , M e 7K g g and . g e G .- For any k e G Q , g,h e G , and M e 7 R o , M U <p* T <|> = <f>*(M 9 I ) ( U ® V )<j> , cp* T cp K k _ 1 g h g K K k 1 g k 1 g n = cp*(M ® l ) ( U k ® V k) T cph = <J>* T(M ® l ) ( U k ® V k) (ph = * g T * k h ^ M 0 I ) ( U k ® V *h = <b* T <J>. . M U. Y g Y k h k Let A e (? p w i t h A ~ [M- : g e G] he given. For a l l g,h e G 3 4>* A T cj>, = E <p* A <p _ T cj> g • n keG s k x g k ^g •• 1 o 9 = E <j>* T <k . Ms U, k e G  Y g Y k h k k o • = S <p* T k , <j>* A ck keG s k h h ' y g Y h ••' 32. where the sums a l l converge i n the. weak topology. As an operator i s completely determined by i t s m a t r i x elements, T e (Pg • C o r o l l a r y 3 .8 Let G,g - U J be a C-system, l e t G = G[T^t,M,G,g - U ] , and l e t G be a subgroup of G . Then R(U g ® V g : g e G Q) c o n s i s t s of a l l those operators A e G w i t h A ~, [M : g e G] s a t i s f y i n g : ( i ) M„. e <DV whenever g e G A ' g ft ° o ( i i ) M = 0 whenever g e G - G §> o For each g e G , l e t M s be 2-dimensional H i l b e r t ..space with.'orthonormal b a s i s • (tp̂ )̂ -™ • The vectors i i l i t T o = + a / 1 - p form a second orthonormal b a s i s f o r #̂  . L e t Fg = p r |] n e Z 2. • TUg = { a F g + b F S . a j b e e } t • • Define a u n i t a r y r e p r e s e n t a t i o n n -» of on by' s e t t i n g U S ? § = ? n ^ m f o r a l l n,m e Z g . Then 33- ( 3 . 1 ) . F|<U g)* - ••" n,me Z 2 U g ^ g ( U g ) * = 'ftg n 6 z a f r e e and ergodic C-system Lemma 5-9 For each g e G , [ 7*l g,M g,Z 2 Jn - U g] i s Proof. I f A e ( 7 t g ) ' , then A ^ g = F g , n e Zg 3 tfhich i m p l i e s t h a t 971 g i s MA on M g . Hence w [ 1 H s J)i g,Z 2,n-.- U g] i s a C-systc- P r o p o s i t i o n 1 .4 , the p r o j e c t i o n s F g and F g , and (3 .1 ) imply that the C-system [ f t t g , M s , Z 2 , n - U g] i s f r e e . To show e r g o d i c i t y , suppose th a t a F g + b F g (a,b e l ) commutes w i t h U g . Then a F^ + b F g = U^(a F 6 + b P^)(U^) o 1 1 v o 1 ' v 1 ' = a F s + b F g , 1 o which i m p l i e s t h a t a = b . As F g and F g are minimal p r o j e c t i o n s i n w i t h F g + F f = I , each G g = G[$}g, X g , Z 2 , n - U g] i s a f a c t o r of type I ^ on the 4 -dimensional H i l b e r t space ttg ® % 2 . ' * Let. A be the set of a l l f u n c t i o n s from G i n t o Z 2 which have f i n i t e support. Under component-wise a d d i t i o n ^ A i s an' a b e l i a n group; we use 0 to denote the i d e n t i t y i n A 34. For each g e G , l e t y be the element of A defined by V*(h) = 1 ' h = g 0 h e G - {g} Given a,B e A , vie define elements a A 0 and a v 0 of A s e t t i n g (a A 0)(g) = min ( a ( g ) , 6(g)} g e G (a v 0)(g) = max ( a ( g ) , 0(g)} g e G (we consider to be ordered i n the n a t u r a l way, i . e . , 0 _< l ) . For a,0 e A , we w i l l w r i t e . a <_ 0 whenever a A B = a . Let li - ® (M g , cp?) , ® ( f l a g , cp s), and geG ° geG . ° f o r each a e A , l e t cp = ® <P^„\ and l e t u = TT a . geG a ^ g ; a geG °^ s ; (a f i n i t e product i n which the factor's commute). From P r o p o s i t i o n s 1.7 and 1.9 we know tha t ( c P a ) a e ^ ^ s a n ortho- normal b a s i s f o r M , that [?H,W,A,o; -* U ] i s a f r e e and ergodic C-system, and tha t G = G[7S|,34,A,a -» U ] i s s p a t i a l l y isomorphic to ® (G g,cp g ® 6) . G i s t h e r e f o r e a hyper- geG p • f i n i t e f a c t o r a c t i n g on a separable H i l b e r t space; moreover, i t f o l l o w s from [3; Prop. 5-5] t h a t G p i s of type I I I . As the group G has served merely'as an i n d e x i n g set i n t h i s c o n s t r u c t i o n , G^ Is a c t u a l l y independent of the p a r t i c u l a r choice of G . Let ^ = 1V[ ® <C£ 3 5 . Lemma 5.10 T^p i s a r e g u l a r MA subalgebra of G p . Proof. That 7H. i s a MA subalgebra of G i s p a r t of P P P r o p o s i t i o n 1 . 5 . For a l l u n i t a r y operators U e T(l and a l l a e A .} . ( u ® I)7H ( u ® I ) * = u 7H u * ® CA = p u p As . G„ = R(U ® I , U„ ® V„ : a e A , U a u n i t a r y e % ), *fyp i s r e g u l a r . ! For each g e G , l e t i P Y . ' - ( ¥ ) ' F o - ( A ) . F i s . and f o r each a e A , l e t I M a = 0 TT p.. otherwise . U ( g ) = l . Y g (a f i n i t e product i n which the f a c t o r s commute). Notice th a t - - . ( 5-2) P a P p = P a A p + a v ^ a A P f o r a 1 1 a>* e A • Each P i s a s e l f - a d j o i n t operator on W s a t i s f y i n g 3 6 . • P <p_ = cp . As ?>t i s MA on H 3 cp i s both c y c l i c and Li O Li O - sepa r a t i n g f o r 7VL . Let S. = » ® H and l e t = 7 K ® 7 H . . Then ? H i s a MA von Neumann algebra on £ , (cp ® cpfl) . . V! • u p cijptii i s an orthonormal b a s i s f o r S ,' and c p Q ® c p Q i s both c y c l i c and sepa r a t i n g f o r TVt . Lemma 3-11 ( i ) f = { £_ c P : c Q e £ and a e A} i s a<a a s t r o n g l y dense sub-*-algebra of c o n t a i n i n g 1^ . j . ^ = {a%<a P « ^ P P : C ^ 6 e C ' a n d . 5 € A} i s a s t r o n g l y dense sub-*-algebra of c o n t a i n i n g Iz . Proof. I t i s c l e a r t h a t 1^ e and t h a t ^ i s a l i n e a r space c l o s e d under the ̂ -operator. For each g e G , F g e J , for. . ' - - v = ( ¥ ) * r f - f e ) * ( p o - ^ ) ' g ( 3 . 3 ) FS - V P U - P ) (P Y + P„) • To show t h a t i s c l o s e d under m u l t i p l i c a t i o n , l e t a,Be A be given; from (3-2) and the observation t h a t .(UAB + avB) A (aA0) = 0 , i t i s s u f f i c i e n t to show t h a t f o r each y e A , 37. P 2 = E c P Y a a a<Y f o r some c e C . I f Y = 0 t h i s i s obvious, and i f Y e A - {0} , Y Y(g)=l Y g • Y(g)=l P ° ^ ° ° = I T [ ^ = = v + p o ] > Y ( g ) = l ^ p T l 1 ? ! Y g ° which i s of the r e q u i r e d form ( i n t h i s c a l c u l a t i o n we. used ( 3 . 3 ) ) . This shows t h a t t i s a sub-*-algebra of , which contains I and generates By the .double commutant theorem, t h i s proves ( i ) . The proof of ( i i ) i s s i m i l a r . . Lemma 3-12 I f S = cJP ' . T = E :e are a<a a<a such t h a t a < a , c • 0 and d„ ^ 0 imply a = 0 , then (STcpQ,cpo) = '(Scpo,cpo)(Tcpo,cpo) . '•' 7" ""' Proof. A simple c a l c u l a t i o n : (STcp ,co ) = E c Q d '(P cp , PQco ) • u> ̂  p S. LX = 5- c a d a a<a .7 38. = c d o o For each a e A and g e G we define an element ga of A by s e t t i n g (ga)(h) = a ( g - 1 h ) h e G. Notice t h a t (gh)a = g(ha) f o r a l l g,h e G and a e A . For each g e G , the r e l a t i o n U cp = cp a e A g v a v g a defines a unique u n i t a r y operator U on 34 and the map g g -• U i s a u n i t a r y r e p r e s e n t a t i o n of G on 34 ; moreover, ' u g * 5 u g = * r f % h e G > a e A u g u * = Dn g e. G . Lemma 3.13 ' ( i ) I f an M e H s a t i s f i e s U M U* = M f o r a l l . • - & S g e G Q , where G Q i s an i n f i n i t e subgroup of G , then 7 ( i i ) [?k,34,G,g -» U ] i s a f r e e and ergodic C-system. Proof. ( i ) For such an M and f o r a l l g e G Q , Mcprt = U M U* cp̂  = U M <s . o g g o g Y o 0 39. I f MCD = £ c cp , then ° . aeA a £ c cp = U £ c cp = E c cp = E c -i cp„ , . a ̂ a g „ . a „ . a ̂ ga e _ 1 a a ' aeA & aeA aeA aeA » ^ and thus c„ = c _i f o r a l l a e A . As a e A - {0} a g a i m p l i e s t h a t {ga : g e GQ} i s i n f i n i t e , we must have c a = 0 unless a = 0 , and" t h e r e f o r e McpQ = C Q CPq . As cpQ i s sepa r a t i n g f o r TR , M = c Q I . ( i i ) From the preceding we know tha t [Wl,tt,G,g -» U ] i s an ergodic C-system. I f the system i s not f r e e , there i s an M e %. - {0} and a " g e G - {e} such that U M e %.. g • Let e > 0 he f i x e d but a r b i t r a r y ; as cpQ i s sep a r a t i n g e || Mcp || f o r fit , 6 = — > 0 , and 6 = e( || Mcp l | - 6) . 1 +.e o • 3 * By Lemma J . l l , there i s an S = E c P e J such t h a t a<a a a || (S-- M)cpQ|| < '6 . '.Hence || ScpQ|| > || McpJI - 6 > .0 , and consequently - || (S - M)cpo|| < 6 = c(|| Mcpo|| - 6) < e|| Scp0|| . As a has f i n i t e support, we can f i n d an h e G - {e} w i t h a(h) = d ( g - 1 h ) . = 0 . Now S S* = . £ c cZ P P R = E da P a a,8<d a 3 a P a<a a a . f o r some d e C . Applying Lemma 3.12 and (3.-3) , - || ir u so? - u s P h cp || 2 = " O g 'o g o Y o M = || F s " l h Sep - S ? cp || 2 II Q yq O O" = (F«" l h S S* cp o,cp o). + ( F ^ S S* cp - 2 ( ? j F g ~ l h S S* cp ,cp ) v O O ^ 0 * 0 ' = 2(p,- p 2 ) || ScpJI 2 . On the other hand, • || U Sco - U S cp || < 11 o g o g o 4 o 11 — < || P^ U S cp - P^ U M cp II + — " o g ^o o g "o 1 1 + || U M cp - U S ? cp || 11 g o ^o g o o 1 1 < '|| Scpo - Mcpo|| + || McpQ - Scpo|| < 2 e. || ScpJI . Combining .these two c a l c u l a t i o n s y i e l d s * e2|| ScpJI 2 > 2(p - : p 2 ) || ScpJ| 2 2 e 2 >_ p - p 2 . 2 As p - p > 0 , t h i s c o n t r a d i c t s the a r b i t r a r i n e s s of and the system [ T ^ L M^G g - U ] • i s therefore, f r e e . S Let B(p,G) = G[$l,M,G,g - U .] . 41. Lemma 3.14 fc(p,G) i s a f a c t o r of type r e a c t i n g on a. separable. H i l b e r t space. B(p>G) i s h y p e r f i n i t e whenever G i s e i t h e r h y p e r f i n i t e or a b e l i a n . Proof. We use P r o p o s i t i o n 1.5-- That 8(p,G) i s a f a c t o r f o l l o w s from Lemma 3 .13. As M - (Mcp ,cpQ) i s a f i n i t e normal f a i t h f u l t r a c e on s a t i s f y i n g (U M U* cp cp ) = (Mcp ,cp ) f o r a l l M e TK+ and 8(p,G)- i s f i n i t e . And as G i s o f type I I I , 7?L cannot c o n t a i n any minimal p r o j e c t i o n s , which i m p l i e s t h a t B(p,G) i s not of type I .• Therefore B(p,G) i s a f a c t o r of type I I , L l I f G i s a b e l i a n , then (B(p,G) i s h y p e r f i n i t e , by P r o p o s i t i o n 1.6. Suppose now th a t G i s h y p e r f i n i t e , say G = U ' G , where G, c Q c« • • , and each G i s a n=l n . d • n f i n i t e subgroup of G . For each n , l e t B n = a((M ® I ^ ) ( U g ® V g) : g e G n and M e 7 H h f o r some h e G ) ; each Bn i s f i n i t e - d i m e n s i o n a l as a ve c t o r space, and more- over, ' 6(p,G) = R(B : n = 1,2,...) Using [13; Theorem XII] (or [6; p. 299])> we conclude t h a t : j?.(p ,G) i s h y p e r f i n i t e . Let $ = A x G , the group-theoretic d i r e c t p r o d u c t , and f o r a = (a.,g) , l e t U a = U a ® U g . There i s a unique l i n e a r isometry y .of. 34 ® ^ ® 34 ® G onto 34 ® § with; T(cp a ® A ® cpR ® g) =.cpa® cpR ® ( y , g ) A f o r a l l ' a,£ j Y e A and g e G . I t i s s t r a i g h t f o r w a r d to prove ( c f . P r o p o s i t i o n 1 .9) t h a t [ ih § - U ] i s a f r e e and ergodic C-system, and tha t i f G = G,[ifl.3TA3 S ,a. - -U ] , then A - fA?'1 i s an' isomorphism of G P ® a ( p , G ) onto G . Noti c e t h a t f o r M,N e and a = (a,g) e <j , 7 ( ( M ® I£)(U a ® V a) ® (N ®'i G)(U g ® V g ) ) r _ 1 = ( 3 . 4 ) = ((M ® N) ® I ^ ) ( U a ® V a) . D e f i n i t i o n 3-15 For each subgroup. G q of G , define a subalgebra ^ 1 ( P J G , G O ) of ( B ( p , G ) and' subalgebras ( G Q ) and - G ( G Q ) . of G as f o l l o w s : % ( P , G , G O ) = B ( U g ® V g : g e G Q ) . $ . ( G 0 ) = r - ? h p * ? I ( P , G , G O ) ? - 1 3 ( G o ) = r G P ® ? i ( p , G , G o ) r - i .. i : N o t i c e that these subalgebras are a l l proper. 43. Lemma 3 . l 6 Let G Q be a. subgroup of G . The sub- .' algebra 7 l(p,G,G 0) i s MA In' &(p,G) . i f and only i f . G Q , s a t i s f i e s (a) : G i s a b e l i a n and {g g g _ 1 : g e G } i s v ' o . °o o o o i n f i n i t e whenever g e G - G Q . Proof. Suppose t h a t the subgroup G q s a t i s f i e s con- d i t i o n (a). Then ?l(p,G,G )' i s an a b e l i a n a l g e b r a , and to show t h a t . i t i s MA i n &(p,G) , .we mu s t . v e r i f y t h a t 6(P,G) a ( 7 l ( p , G , G o ) ) ' c 7l(p,G,G 0) . Let B e S(p,G) r\ (7l(p,G,G ) ) ' w i t h B ~ [M : g e G ] be o g given. From Lemma 1.2 we have th a t f o r a l l g e G and h * G 0 ( 3 . 5 ) • +*H B ( U H • V H ) £ E U* H = M G (3-6) - <p\ (U, ® V. ) B cp U*. = U, M _ i U* w ' Y g h v h h' y e gh h h gh n (3 ,7) (<P* B B * ^ o ^ o ) = ( 2 M k M k c p o ^ c p o ) = 2 I' M o " 2 ' e e o o k € Q is. keG K 0 where the expressions (3-5) and (3-6) are equal. I f g e G then M = U, M U* f o r a l l h e G , which, by Lemma 3-13, i m p l i e s ' t h a t M e € v .' I f g e G - G ., then f o r a l l g <fi . • o h e G 0 , . II MP 0II = II U h M h. l g h U* cpoM = || M h_ l g hcp o|| ; 44; "by (3-7) and c o n d i t i o n (a) , t h i s means tha t M cpQ = 0 , and consequently that M = 0 . C o r o l l a r y 3 .8 now i m p l i e s t h a t B e T K p ^ G j . ' Conversely, suppose that J[ (p,G,G Q) i s a MA sub- algebra of B(p,G) . I f c o n d i t i o n (a) f a i l s , then, as 7t(p,G,G Q) . a b e l i a n i m p l i e s G Q a b e l i a n , there i s a g e G - G Q such t h a t \ F = (g 0S g^ 1 :.g 0 e GQ3 i s f i n i t e . Let s ,J. ' ' " B = 2 U ® V ; ' i heF n n then B e U5(p,G) and, as g e F , B | 7Z(p,G,G Q) . For any h e G Q and k e G , and . .'• ; • <j>* ( U h ® V h) B cpe U* = k h - 1 e F otherwise = U K <p*_T B <p U -, U* h h x k e h X k . h -{ -1, I h k e F 0 " otherwise , where we used Lemma 1.2. As h - 1 k e F i s equivalent t o •1th"1 e F (h e G , k e G) B commutes w i t h U h ® f o r a l l h € G i . e . , B e (71 (p,G,G Q))' . This i m p l i e s that 7\ ("£>,&,G0) i s not MA i n B(p,G) , which i s a c o n t r a d i c t i o n . Hence c o n d i t i o n (a) must hold. Lemma 5.17 Let G Q be a subgroup of G . An A e G w i t h A ~ [Mn-.': a eG] i s an element of ( i ) # (G ) i f and only i f M a eft® C M M„ = 0 a a e {0} x G o otherwise , and ( i i ) -cf (G Q) i f and only i f K e m ® C * a e A x G, M a = 0 o otherwise . Proof. Using [6; p. 57] and-(J.4) / ? ( G o ) = ? [fa® C£ V*&(p,G-,G 0)]7 -1 =7 R(M ® l£ ® U g ® V g : M e 7H , g e G Q ) 7 _ 1 S € G Q) a n d " - ? ( G j = ? G ^ ® 7 t f p , G , G j 7 - 1 P = 7 R((M ® I ^ ) ( U a ® V a) ® (U ® V ) : M e ^ , a e A , g e G Q ) T -1 = a ( ( M ® I ^ I j ) ( U M ® V M ) : M e T H , a e A , g e G Q ) . The d e s i r e d conclusions now f o l l o w from Lemma 3-7. Lemma 5.18 Let 71 be an a b e l i a n von Neumann algebra on the H i l b e r t space K , and l e t x be a non-zero v e c t o r i n X . Let and j be two f a m i l i e s i n % such t h a t M = E M. N. and N = S N. N* e x i s t i n 71 i n • i e l 1 1 i e l 1 1 the strong t o p o l o g y / a n d suppose t h a t N <_ I . Then II M x|| 2 < £ || M. x|| 2 . i e l Proof. As lH i s a u n i f o r m l y c l o s e d commutative B*- algebra w i t h i d e n t i t y , the Gelfand-Naimark r e p r e s e n t a t i o n theorem (see e.g. [7; p. 876]) gives an i s o m e t r i c *-isomorphism A f ^ of 71 onto C(X), X some compact Hausdorff Space. Let F be an a r b i t r a r y f i n i t e subset of I , and s e t . M™ =• £ M. N, and N„ = E N. N* . As • i e F 1 1 • 'ieF 1 1 ' I >_ N _> N p , 1 >_ f K >_ X I fw -I > a n d consequently F i e F i Passing back to % , |M | 2 < E :|.M, | 2 , and t h e r e f o r e b - i € P 1 • 47. || Mrp x|| 2 = (M p X,X) 1- 2 (M- M* x,x) = S || M. x| • i e F 1 1 i e F 1 . Taking the supremum over a l l f i n i t e P e l , we are done. In order to s i m p l i f y the n o t a t i o n , l e t X denote the i d e n t i t y i n Q and f o r each a e Q , l e t T„ = U_ ® V . Lemma 3.19 ( c f . [20; Lemma 15] )• L e t a u n i t a r y operator U e G w i t h U ~ [M : a e 9 ] and an e > 0 be given. Then there i s a f i n i t e subset 3 of Q such that • o f o r any f i n i t e subset IF~ of - $ c o n t a i n i n g 3 , there i s a f a m i l y ( N b ) b e 3 ; of elements of ? such that:. (3;8) • 1 ( i ) . || MbcpQScpo - Nbcpo®cpo|| < | f o r a l l b e £ ; . ( i i ) i f V.= S (N, ®IA)(U, ®V. ) , then f o r a l l betf D 9 a D c,d € {0} x G , (3.9) || cp* [tJ T d U* - V T d V*] U* cpQ © cpjl < . .'. ' y Proof. F i x c. and d i n [0] x G . By Lemma 1.2, .(5.10) Ig = cp* tf U* <f>% = 2 M a M* , where -.the sum converges s t r o n g l y . Hence there i s a f i n i t e subset $ of ^ such that 48. 4 ( £ TA M* cp ® cp , cp ® cp ) = £ II M cp ® cp II 2 < ^ a*3 a a 0 ° 0 ° a & a 0 0 - 2 5 6  Y o * o F i x a f i n i t e subset 3 of 3 c o n t a i n i n g 3 and l e t ¥ = S (M^ ® I ) ( U ® V b) be3 Again u s i n g Lemma 1.2, we.find t h a t * c 0 *d ̂  h K E [ ^ -1 U T 4 U* _ i ] U _ ! [ U a M*_! U*] U* _ ± a e t ? ca a * ca ca a a a ca E M -I -] U M* -, U* ae9 c a ^ d " 1 c a " 1 c • a n ) = S fi u r _ ! (J*-., ae$ a c c ad c and s i m i l a r l y (3-12) <j>* W T W* <p U* = £ ' M U."M*_i . U* , c a ^ c a e ^ ' a c c -^ad c where the sum i n (3.11) converges s t r o n g l y and 3' = J? A c g d (we use the convention t h a t the empty sum i s zer o ) . By means of (3.10) and Lemma 3-18, we o b t a i n || <j>* [U T d U* 1 W T d W*] i t U* c p Q OP cp. -1 = || E M U M*_-i U cp ® cp a « ^ ' a c c 1 a d c ^ o 4 9 . < II 2 , M a U M i U cp ® cp || + ae^-^ c c ad c ° - ° : + II ? S a 5 c C - l * K % ® ^o'H a|3= c ad c ° ° • • aeff-ff c ad c ° 9 ' "+( Z l|M acp o®cp o[| 2)* . a$<? < ( 2 || M*_i <PQ « <P0II 2 ) * + aetf-I? c ad ° o / J.O I t f o l l o w s from (3.10) tha t each i s i n the u n i t a b a l l o f m . Hence, by Lemma 3.11 and the Kaplansky d e n s i t y theorem [6; p. 4 6 ] , there i s a f a m i l y ( N ^ ^ g ^ °^ e 1 6 1 1 1 6 3 1 * 3 i n the u n i t b a l l of such th a t H l\ *o ® ^o ~ \ Vq® ^ O ' I ± m i n <f > I5n') ' f o r a 1 1 b ' 6 1 5 > where n i s the number of elements i n 3 . In p a r t i c u l a r , (3 .8 ) i s s a t i s f i e d . L e t t i n g V = S ' (N. ® I ) (U. ® V, ) , we. • • . be3 D D D have th a t ( c f . (5112)) 50. . ! U * A T D..¥* - V T d V*]- ^ U* cpQ ® cpj| = = ||. S (M U M*_! U* - N U N*_! . U*)«p ®ep || ae3 c a d c a d 1 I' E , ^ a ̂ c , " J 5 c *o ® % H + ae3 a c c ad . c ad. c . ° ° ••• . C ̂  || M a U c ( M * . l a d - N*. l a d) U* % • cpo || + - + S ,11 U c N * _ l „ U*(M - N ) cp ® cp || ae3 c c a d c a a o o < S II '(M*_! - N*_ x ) cp ® cp !|| + . ae3' c ad c -'-ad o o + £ , J K f i a " V * o * ' o ' l Combining the l a s t two i n e q u a l i t i e s by means' of the t r i a n g l e i n e q u a l i t y gives the estimate (3-9) . Lemma 5.20 ( c f . [20; Lemma 17]). Suppose G i s a subgroup of G s a t i s f y i n g P ($) : given a f i n i t e subset F of, G and a- g e G , \ there are i n f i n i t e l y many g Q e G Q such t h a t : ( i ) h,k e F and h g Q k""1 = g Q imply h = k ( i i ) i f g | N(G Q) , then g g Q g" 1 | G Q . . 5 1 . Then N ( f ( G 0 ) = N ( J J(G Q)) = J ( N ( G Q ) ) . Proof. I t i s easy t o see tha t i ( N ( G o ) ) c N(? ( G Q ) ) , I ( N ( G o ) ) c N $ ( G Q ) ) . i Conversely, suppose t h a t we are given a u n i t a r y operator U e 5 s a t i s f y i n g .one of (3.13) U $ (G Q)U* = (?(G Q) (3.14) U 3 ( G Q ) U * = 1 ( G Q ) . We w i l l be done i f we can show that U e ̂  (N(G Q)) .. Let U ~ [M : a ] > and f o r each a e ^ , l e t a ° ° a,0eA a p 1 \ where the 8 (a; a,©) are complex numbers. Suppose we knew th a t (3.15) 8(aj a,8) = 0 .whenever 6 ^ 0 (3.16) 6(a; a }0) = 0 whenever a | 4 x N ( G Q ) I f a'|'A x N(G ) > then fi cp ® cp = 0 ; as cp ® m i s sepa r a t i n g f o r > M = 0 . And i f a e A x N(G ) , then f o r a l l a_,B e A , M a cp a^p R = ( P a ® P p) Mft cpQ ® cpQ = P ® P s 9 (a; Y,0) cp ® cp - yeA 52. = .( 2 e(a; Y , 0 ) P a cp ) ® cp ,. YeA . T H and t h e r e f o r e M a e 7*1 ® . Lemma 3.17 now i m p l i e s t h a t U e (N(G )) . Hence i t i s s u f f i c i e n t to show t h a t (3.15) and (3-16) hold. F i x an (cx,g) e £j and an ( a ^ o ^ ) e A x A , and l e t e > 0 be given. A p p l y i n g Lemma 3.19 to U and e , we get a f i n i t e subset 3 of J and, w i t h 3 = 3 U ( ( a j g ) } > a f a m i l y ' ( ^ ) - b e g ; o f elements of . tf s a t i s f y i n g ( 3 . 8 ) and (3-9) • By the . f i n i t e n e s s of 3 and\the d e f i n i t i o n of $ (Lemma J. 1 1 ) , there are complex numbers a(b; 3 , Y ) and an a e A such,that (3.17) ' " N. = S a(b; P,Y ) P A ® P V f o r a l l b e 3 ; B,Y<.a p Y without l o s s of g e n e r a l i t y , we may assume t h a t a±>a2 i . " • From (3.8)^ (3-18) | > || cpo ® cpQ - \ cpQ ® cpo|| v > |9(b; B, Y) - a(b; 6,Y)| f o r a l l b e 3 and a l l B,Y _< a . Let F = ( h e G : (B,h) e 3 f o r some M i ] , a f i n i t e subset of G c o n t a i n i n g g . Ap p l y i n g c o n d i t i o n (B) to the set g - 1 F and the element g and u s i n g the f a c t t h a t a has f i n i t e support, we can f i n d a g e G such t h a t o o (3.1-9) a A g g Q g" 1 a = 0 53. (3 .20) h,k e g" 1 F and h g Q k - 1 = g Q imply h = k (3.21) i f g i N(G Q) , then g g Q g _ 1 | G Q . In order to s i m p l i f y the n o t a t i o n , l e t h = g g Q g l e t c = (0,g ) , and l e t d = (0,g g Q g" 1) . Let V be as i n (3-9) of Lemma 3.19 , l e t 3?' = 3 A d 3 c " 1 , and l e t S = <{>* f T c V* <pv U d . • Notice t h a t (a,g) e 3?' , a l s o t h a t 2?' i s not empty. Now ( c f . ( 3 . 1 2 ) ) S = 2 N U, N*_T U* a e 3;' a d d -L A C d = 2 N Uj N? U* .. . , ~ a d b d a,be<5 b=d _ 1ac I f a = (0,k) and b = (y,l) are elements of 3 , the r e l a t i o n cT~1ac = b i m p l i e s t h a t (0,g g" 1 g - 1 k g ) = (y,-t) Hence 0 = y and ( g _ 1 k ) g Q ( g - 1 i ) ' 1 = gQ ; as k,£ e F , (3 .20) may be a p p l i e d , g i v i n g , k = & . Therefore a = b , and the double sum reduces to a s i n g l e sum. ' On s u b s t i t u t i n g (3.17) i n t o t h i s sum we get S = 2 N, U , NT U* - be!?' 1 0 d b = ., 2 a(b; 0,y)a(b; 6,TU ( P Q ® P ) • 0,Y,6-,n<a V P 6 • V 0 S 5 4 . £ / a(b;B, Y)a(b; 6 3ri)(P p®P v)(P 6®U hP T iU*) : b 6c? B,Y,S,n<a £ / a(b;P,Y)a(b 56,TU PpPg ® P yP h T 1 • b £<? B,Y,6,r\<a From (3 .19) and the assumption th a t ' <_ a , i t • f o l l o w s t h a t y,T) <_a and + ha^ = Y + hri together imply- t h a t Y = "H = a 2 • Hence, f o r a l l .3^,6,^ <. ° \> ( P8 P6 ® PY Phr, *o V o '  1 ® Vha2 *° ® ^ = = .(P f lP 6 c p o , c p o ) ( P Y + h T l cpQ , cpQ) . , 1 . 6 = 6 and y = r\ = 0 . otherwise, and t h e r e f o r e (Scpc ® cpo , I 9 P a 2 + h a 2 cpQ ® cpQ) £ |c(b; S , a 0 ) | be3' B<a 2 ;' 1 |a((a,g); a ^ O g ) ! 2 To show tha t (3 .16) holds, suppose t h a t (a,g) £ A x N(G Q) and tha t = 0 . Let T = cp* U T c U* <pt U* \ • 55. As U s a t i s f i e s one of ( 5 .15), ( 3 .14), T = 0 (Lemma 3-17 and ( 5.21)). The i n e q u a l i t y (3-9) now gives Combining.this estimate w i t h ( 3 « l 8 ) , we get | 6 ( ( a , g ) ; a x , 0 ) | < J 0 ( ( a , g ) ; a^O) - t r ( ( a , g ) ; 0^,0)1 + + !c((a,g); 0 ^ , 0 ) 1 i _< e . As e > 0 was a r b i t r a r y , we conclude t h a t 8((a,g); a.^,0) = 0 and t h e r e f o r e that ( 3 - l 6 ) holds. To show t h a t (3.15) holds, suppose that. a'2 ̂  0 . I f g I N(G ) , then,as before, T = 0 ; and i f g e N(G Q) , then T e 7H ® Cs, f o r s i m i l a r reasons. In any case, > ||Tcpo ® cpQ - Scpo ® cp 0H I I (ScpQ © cp Q , cp Q ® cpb) I >. !a((a,g); a ,0)| 2 0 and t h e r e f o r e (using ( 3 . 9 ) ) > II T c p Q ® c p Q - S c p c ® c p o || > !(Tcpc ® cpQ - ScpQ ® cpQ , cpo ® V ^ ^ l 56. = |(Scp0 ® cp , I ® P a 2 + h a 2 * 0 ® <P0)I 1 |a((a,g); 0 ^ , 0 ^ ) 1 As be f o r e , t h i s i m p l i e s that 9(a,g); a^,a^) = 0 . This completes the proof of Lemma 3 . 2 0 . Lemma 3-21 There i s an isomorphism § Q of . CQ onto ?1(P>G,G 0) such th a t s ( V j = U ® f o r a l l g e G-o v g y g .g to $ n ( f l l ( G , G J ) = 71(p,G,G^) for. a l l subgroups G of G o . Proof. Let r\ be the unique u n i t a r y operator on 34 ® G w i t h ri(cp„ ® g ) = cp ® g' f o r a l l a e & , g e G . For any T e C Q ^ l e t $ (T) = T| ( I W ® T ) T I _ 1 . I t i s t r i v i a l t h a t $ Q i s a normal *-isomorphism of C-G i n t o .£(34 ® G) w i t h $ ( I ) e I . I f g e G , then f o r a l l . a e A , h e G , § J V j m ® n = TI(I ® V„) r\ 1 m ® n o g a l v g ex = cp g a®(gh) A = (U ® V„)(cpa ® n ) > and t h e r e f o r e $(-(V ) = U ® V . Using [6; p.57], we have o g g g th a t f o r any subgroup G Q of. G , . 57. * 0 ( 7 H ( G , G 0 ) ) = $ Q ( R ( V g : g € G Q ) ) = a ( 5 Q ( V g ) : g 6 G Q) = 7l(p,G,G Q) ; i n p a r t i c u l a r , t h i s i m p l i e s t h a t *o^G^ = § o ^ G , G ) ) = ft(p,G,G) . Lemma 3 -22 There i s a ^-isomorphism 5 of G p <8> i n t o Gp ® B(p,G) such t h a t f o r any subgroup G Q of G , ^ § ( m p ®1»l(G,G0)) = THp ®71(P,G,G0) §(G p ® J l ( G J G Q ) ) = G p ® ? l(p,G,G 0) . Proof. The r e s u l t f o l l o w s e a s i l y from Lemma 3 . 2 1 and [6; pp. 57 and 6 o ] . Lemma 3 . 2 3 For any subgroup G q of G , CQ ~ $£((> Proof. As G <z G , we may consider G to be a sub-o ' * o space of & . I f T e 7H.(GjG ) and g € G - G , then ( c f . Lemma 3 .21) $'(T) • e 7 l(p,G,G ) and therefore,'using C o r o l l a r y 3 . 8 , 0 = (cp*. r , ( l ® T) r f 1 <Pe cpQ , cpQ) - = ( n(I ® T) r f 1 cpQ ® 'e* , cpQ ® = ( l ® T c p Q ® c e , cp c®g >) = (T ̂  , |) . 58. Now G Q i s i n v a r i a n t under W(G,G ) , f o r i f T e 7>t(G,G0) and g e G Q , then by the above c a l c u l a t i o n , T g" = S (T g, n)n = I ' (T e , ( h g " : L ) A ) n = heG heG = S (T ft) (kg) A keG^ o Hence the r e s t r i c t i o n §'(T) of a T e 7H.(G,G ) to G i s N v o o an operator on 6 Q . I t i s easy to v e r i f y t h a t §' i s a normal *-isomprphism of ?*t(G,Go) i n t o 6^ o Using [6j p. 5 7 ] , *'(7K(G,G 0)) = $ ' ( R ( V : g.6 G Q)) = R ( * ' ( V G ) : g e G Q ) = CQ . o Proof o f Theorem 5-1- As f)/[ i s MA i n - G P , a r e s u l t of S a i t o and Tomiyama [22] i m p l i e s that fYl ® 7 i ( p , G , G ) i s MA i n G P ® (B(p , G ) i f and only i f 71 ( p , G , G Q ) i s MA i n B ( p , G ) . But by Lemma 3.16, t h i s i s the case i f any only i f c o n d i t i o n (a) holds. Proof of Theorem 5 . 2 . As A - f A J - 1 i s a normal ^-isomorphism of G ® & ( P J > G ) "onto G ( c f . the t e x t preceding D e f i n i t i o n 5-15) , J N ( 7 l ) 7 ~ 1 = N ( 7 ft?'""1) f o r any subalgeb r a ? l of G ® B ( p , G ) ., f o r 59- 7'N(7L)7" 1 = V R(U : U e G p ® e(p,G) and u n i t a r y , and U?lU* =70)^-1 = ^(T^T'1 : U E G P ® ^ ( P ^ G ) a j l d u n i t a r y , and U# U* = % ) = R(U : U e 5 and u n i t a r y , and W W 1 Uf = / T I T " 1 ) = N(y 71 T" 1) In p a r t i c u l a r , u s i n g D e f i n i t i o n 3-15 and Lemma 3 . 2 0 , = r i N ( ' ? ( G O ) ) T = 7 " 1 ^ (N(G 0 ) ) 7 = G p ® 7 l ( p , G , N(G Q))' , and s i m i l a r l y . N ( G P ®7l(p,G,G O)) = G p ®?l(p,G, N(G Q)) Proof of Theorem 3 .3 As G i s a f a c t o r , G p ® 71 (p,G,G Q) i s a f a c t o r i f and only i f ft(p,G,GO) i s a f a c t o r [6; p. 3 0 ] . As the pro p e r t y of being a f a c t o r i s preserved.by isomorphisms, Lemmas 3*21 and 3 .23 imply t h a t ' ?1(P JG,G 0) i s a f a c t o r i f and only i f {? & i s a f a c t o r . " 6 0 . But.the group operator algebra C?G i s a f a c t o r i f and only o i f GQ has the i n f i n i t e conjugate c l a s s property. Proof of Theorem 5.4 ( c f . [5; Lemma l ] ) . As i n the proof of Theorem 3 . 1 , 7 H p ®-7H.(G-,GQ)' i s MA i n G p ® £ Q i f and only i f 1U(G,G ) i s MA i n <?G . I f (a) holds, then, by Lemmas 3 - l 6 and 3 . 2 1 , 7l(p,G,GQ) = ® 0(^(G,G Q)) i s MA i n T U p^G) = $0(CG) , and so 7H(G,GO) i s MA i n £ Q . . Conversely, i f Dt(G,G^) i s MA i n (? r , a c a l c u l a t i o n s i m i l a r to t h a t i n the proof of the "only i f " p a r t of Lemma 3 - l 6 shows tha t c o n d i t i o n (a) must be s a t i s f i e d . .Proof of Theorem 5-5 F i r s t of a l l , i t i s c l e a r t h a t G p ®7It(G, N(G Q)) c N(7H p ®?fc(G,G0)) G p ®%(G, N(G Q)) c N(G p ® m(G,G Q)) . - To show th a t the.opposite i n c l u s i o n s h o l d , l e t U be a u n i t a r y ' operator i n G p ®' C Q s a t i s f y i n g • o n e of u f l f p ®7H(G,Go) u* = Kg ® R(G,G O) U G p ®?H(G,Go) u* = G p ® W(:G,GO) . By Lemma 3 -22, $(U) i s a u n i t a r y o p e r a t o r . i n G ® B(p,G) IT such t h a t - e i t h e r • * ( V ) A p ® 1 l(P,G,G o)5.(U)* = 5(U?l( p ®7>l(G,GO) U*) . = ^ ®ft(p,G,G_) or $(U) G p ®7L( P JG,G o)§(U)* = $(U G p ® 7 H ( G , G o ) U * ) = G p ®71(p,G,G O) . By Lemma 3--20, $(U) e G p ®7l(p,G, N(G Q)) , and t h e r e f o r e U e G p ® 7 U(G,N ( G Q ) ) . Proof of Theorem 3-6. S i m i l a r to the proof of Theorem J>. J>. .4 . EXAMPLES OF MAXIMAL ABELIAN SUBALGEBRAS We begin by s t a t i n g i n four theorems the main r e s u l t s of t h i s t h e s i s . A f t e r a b r i e f d i s c u s s i o n of the co n s t r u c t i o n s of the previous s e c t i o n , we t u r n to the proofs of the theorems. Theorem 4.1 Each of the type I I I f a c t o r s G p , 0 < p < | j contains a semi-regular MA subalgebra. Theorem 4.2 For each i n t e g e r m _> 2 and each p e ( 0 , % ) , contains two 2-semi-regular MA subalgebras, one of improper l e n g t h m and one of proper l e n g t h m . Theorem 4.3 For each i n t e g e r m _> 3 and each p e (0,|-) , G contains two 3-semi-regular MA subalgebras, P one of improper l e n g t h m and one of proper l e n g t h m . Theorem 4.4 For each i n t e g e r m >_ 2 , the h y p e r f i n i t e 11-^ f a c t o r contains ( i ) a 2-semi-regular MA subalgebra of improper l e n g t h m ( i i ) a 3-semi-regular MA subalgebra of improper l e n g t h m + 1 . The f a c t o r s G p , p e (0,-|) , were f i r s t studie.d by Pukanszky, who obtained them by a measure-theoretic c o n s t r u c t i o n [20]. In t h i s paper Pukanszky a l s o c o n s t r u c t s , f o r each " P £ (®>W a 1 1^ each countably i n f i n i t e group G , a type I I I 6 3 . f a c t o r G(p,G) and, f o r each subgroup G Q of G , a sub- algebra 0°(p,G,Go) of G(p,G) That 6>(p,G,Go) i s MA i n G(p,G) whenever G Q s a t i s f i e s c o n d i t i o n (a) of Theorem 3.1 i s not d i f f i c u l t to show. I t i s reasonable t o conjecture t h a t N((P (p,G,G Q)) =ff(p,G, N ( G Q ) ) under c o n d i t i o n (6) of Theorem 3 -2; however, Pukanszky's proof of t h i s statement i s not v a l i d . Our algebra G ® &(p,G) i s obtained by modifying the c o n s t r u c t i o n of P . ' Pukanszky's G(p,G). Powers has shown th a t i f 0 < p < q < % , then G ^ and G ^ are non-isomorphic; u n f o r t u n a t e l y , h i s proof ' depends h e a v i l y on C*-algebra techniques ( [ l 8 ] , [ 1 9 ] ) . A r a k i and Woods have given.a proof of t h i s r e s u l t which.uses only methods, of von Neumann algebras [ 2 ] ; i n a d d i t i o n , they" show, th a t • - ' (4 .1) G P ® 8 = G P for' each p e ( 0 , | ) , where 0 i s the h y p e r f i n i t e 11^ f a c t o r . Proof of Theorem 4 .1 R e c a l l the c o n d i t i o n s (a) and (3) of Theorems 3.1 and 3 . 2 , r e s p e c t i v e l y . We f i r s t shoitf t h a t i t w i l l s u f f i c e to c o n s t r u c t a countably i n f i n i t e hyper- f i n i t e group G w i t h the i n f i n i t e conjugate c l a s s p roperty and a normal subgroup G Q of G s a t i s f y i n g c o n d i t i o n s (a) and (6) . For then, by Theorems 3 . 1 , 3 - 2 , and 3-3> ik ® 7 1 ( P J G J g 0 ) i s a MA subalgebra of G ® e(p,G) w i t h 64. normalizer G P ® 7l(p,G,G) , a f a c t o r d i s t i n c t from G ® IB(PJG) . Applying the isomorphism (4.1) and Lemma 3.14, we are done. We now t u r n to the c o n s t r u c t i o n of such a G and G Q . Let F be a countably i n f i n i t e f i e l d which i s the i n c r e a s i n g union of a sequence of f i n i t e s u b f i e l d s ( i n p a r t i c u l a r , we may take f o r the F the a l g e b r a i c completion of a f i n i t e f i e l d ) . The set . G = {(a,p) : a,8 e F and a ^ 0} becomes a group under the oper a t i o n (CX,8)(Y,6) = (a a 6 + 8) . I t i s easy t o see tha t G i s countably i n f i n i t e and hyper- f i n i t e . To v e r i f y that G has the i n f i n i t e conjugate c l a s s p r o p e r t y , l e t a (a,8) e G <-• {(1,0)} be given. F o r " a l l ( Y,S) e,G , ( Y ^ ) ( a , P ) ( Y ^ ) _ 1 = (Y a, Y (3 + S ) ( Y _ 1 , - Y _ 1 6) = (a, -a 6 + Y 3 + 0 . I f a = 1 , then ' 8 ^ 0 , and so - a & + Y S + 6 = Yf3 runs through i n f i n i t e l y many elements as y runs through F - {0}; and i f a ^ l , - a 6 + y P + 5 runs through i n f i n i t e l y many elements as 6 runs through F . I t i s easy t o v e r i f y that _ ' G Q = {(1,8) : 8 e F} . • . 65. i s a normal subgroup of G . The subgroup G q has p r o p e r t y •(a) ., f o r i f (a,B) e G - G Q , then a ^ 1 , and so ( l , Y ) ( a , P ) ( l , Y ) " 1 = (a,B + Y ) ( 1 , - Y ) = (a, -a Y + P + Y) runs through i n f i n i t e l y many elements as Y runs through F . F i n a l l y , we show tha t G Q has property (B) . Let &13"',^n 6 G be given, w i t h , say, g i = ( a i , p ± ) i = 1,...,n . Let H = {(1 - c ^ ) " 1 ^ - 6-j) : a ± / 1 and 1 < i , j < n} , a f i n i t e subset of F . I f g = (1,B Q) f o r some B Q e F > H and i f g ± g Q g" 1 = g Q , then ( l , B o ) = (a±^±)(l^o)(a B . ) - 1 = (a.,a.B +6".) ( a " 1 - , -a" 1^.) v i J iro ' l ' v j > J . J = (a. a " 1 , r-a.a~^"B . + a.B . + B. ) Hence a. = a. , and so 8 = -B. + a.B + B. . I f a. / -I- , x o o j i o ^ i l then B = (1 - a. )~"I"(B. - B.) , a c o n t r a d i c t i o n ; t h e r e f o r e w - L J - jfj 'ai' =• 1 , and.thus P i = 8^ , i . e . , g± = g^ . " . Proof of Theorem 4.2 F i x a~ p e (0,|) and an i n t e g e r m >_ 2 . • Suppose t h a t we had a countably i n f i n i t e h y p e r f i n i t e group G w i t h the i n f i n i t e conjugate c l a s s p roperty and a 6 6 . subgroup G ' of G such that ( 4 . 2 ) ( i ) G s a t i s f i e s c o n d i t i o n (a) of Theorem 3.1 o ( 4 . 3 ) ( i i ) G Q % N(G Q) ̂  v 5 K m ( G o ) = G ' a n d e a c h N K ( G Q ) , 0 <_ k <_ m - 1 , s a t i s f i e s c o n d i t i o n • O ) of Theorem 3 .2 •(4.4) ( i i i ) N(G ) does not have the i n f i n i t e conjugate c l a s s p r operty w h i l e N (G ) does. Then, from S e c t i o n 3> $lp ® Tl(p*G,GQ) i s a 2 -semi-regular MA subalgebra of G p ® &(p,G) of improper- l e n g t h m and ffl ®$L(G,G ) i s a 2 -semi-regular MA subalgebra of G ® @ of proper l e n g t h m . As (B(p,G) and are both hyper- f i n i t e 11-^ f a c t o r s , two a p p l i c a t i o n s of (4.1) completes the proof of the theorem. Hence i t s u f f i c e s to con s t r u c t such a group G 'and subgroup G^ . Again, l e t ' F . be a countably i n f i n i t e ^ f i e l d which i s the i n c r e a s i n g union of a sequence" of f i n i t e s u b f i e l d s . Let G-.. be the group, of a l l (m+2) x (m-i-2) matrices (g. .) over F w i t h ( 4 . 5 ) § n ^ 0 ( 4 . 6 ) g i j L = 1 i = 2,... ,m+2 ( 4 . 7 ) ' & u = 0 - i > j , . and l e t G Q be the subgroup of G c o n s i s t i n g of a l l those matrices' (g^-j) i n G w i t h 6 7 . . g12 = s 2 3 g 2 j. = 0 j = 4,. . . ,m+2 g i ( j = 0 3 < i < J • The group G i s c l e a r l y countably i n f i n i t e and h y p e r f i n i t e . Anastasio: has shown tha t G has the i n f i n i t e conjugate c l a s s ' property and that the subgroup G Q s a t i s f i e s ( i ) , ( i i ) , and . ( i l l ) [ 1 ] . ./ Proof of Theorem 4.3. The proof i s s i m i l a r to that of Theorem 4.2. Let the f i e l d P be as bef o r e , and l e t a p e (0)i) and an i n t e g e r m >_ 3 be f i x e d . Let G be the group of a l l (m+2) x (m+2) matrices (gn-n-) over F s a t i s f y - i n g (4.5), (4.6), and (4.7), and l e t G Q be the subgroup of G c o n s i s t i n g of a l l those matrices (g--?) • i n G w i t h • g l 2 = s 2 3 = s34 • g13 = g24 g 2 j = g 3 j = 0 J* = 5,...,m+2 . . g . . = o 4 < i < j . . Then G i s a countably i n f i n i t e h y p e r f i n i t e group w i t h the : i n f i n i t e conjugate c l a s s property (see [ 1 ] ) ; moreover, 6 8 . • ( i ) GQ s a t i s f i e s c o n d i t i o n (a) of Theorem 3-1 ( i i ) G Q p N(G Q) ̂  5 N m ( G 0 ) = G > a n d e a c h N k ( ? o ) > 0 _< k _< m-1 , s a t i s f i e s c o n d i t i o n (6) of Theorem 3-2 ( i i i ) . N(G ) and N 2(G ) do not have the i n f i n i t e conjugate c l a s s p roperty w h i l e N^(G ) does. As'before, t h i s i s s u f f i c i e n t to e s t a b l i s h our theorem. Before proceeding to the proof of Theorem 4.4, we must f i r s t prove i . Lemma 4.5 Let p be a p o i n t i n (0,^) , l e t G be a countably i n f i n i t e group, and l e t G ' be a subgroup of G . I f G Q s a t i s f i e s c o n d i t i o n (B) of Theorem 3'2, then N C 7 l ( p , G , G o ) ) = 7 l(p,G,N(G 0)) . Proof. That f| (p, G,N( G Q )). c N( 71 (p, G,G Q ).) i s t r i v i a l . For the converse, l e t a u n i t a r y operator U i n B(p,G) , w i t h ; U? L(p,G,G O) U* = 7l(p,G,G O) be given. Then I ® U i s a u n i t a r y operator i n G_ ® B(p,G) such that ( I ® U)7Up ®7l(p,G,GQ) ( I ® U ) * = ̂ Jtp ®U(P^G,G O) . According to Theorem 3-2, t h i s f o r c e s I ® U- e G p ®7l(p,G,N(GQ and t h e r e f o r e U e7l(p,G,N(G_)) . 6 9 . Proof of Theorem 4 . 4 . Let an i n t e g e r m > 2 and a p o i n t p i n (0,1) he f i x e d . Let the f i e l d F , the group G of (m+2) x (m+2) matrices over F and i t s subgroup G Q be as i n the proof of Theorem 4 . 2 . Then fo(p,G) i s the hyper- f i n i t e II-j^ f a c t o r (Lemma 3 .14) and 7l(p,G,G O) i s a MA sub- al g e b r a of 0(p,G) (Lemma 3.16 and ( 4 . 2 ) ) . • By Lemma 4 . 5 and (4 . 3 ) , 71(PJG,G q) has improper l e n g t h m . By Lemma 3-21, Lemma 3 - 2 3 , and ( 4 . 3 ) , -• N(7l (p,G,G Q)) = 7l(p,G,N(G O)) =. £ N ( G ) N 2(-n(p,G,G 0)) = 7 l ( p , G , N 2 ( G 0 ) ) S C N 2 . ! 0 As the n o t i o n of a f a c t o r i s an i n v a r i a n t under isomorphisms, (4 .4 ) shows that <7t(p,G,G O) i s 2 -semi-regular. This proves ( i ) . The proof of ( i i ) i s s i m i l a r , the groups and subgroups from the proof of Theorem 4 . 3 being . employed. REFERENCES [ l ] S. ANASTASIO, Maximal a b e l i a n subalgebras i n hyper- f i n i t e f a c t o r s , Amer. J. Math., 87 (1965), pp. 955-971. [2] H. ARAKI and E. J. WOODS, A c l a s s i f i c a t i o n of f a c t o r s , p r e - p r i n t . [3] D. 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