"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Nielsen, Ole A."@en . "2011-08-24T17:30:46Z"@en . "1968"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "We are concerned with constructing examples of maximal abelian von Neumann subalgebras (MA subalgebras) in hyperfinite factors of type III. Our results will show that certain phenomena known to hold for the hyperfinite factor of type 11\u00E2\u0082\u0081 also hold for type III factors.\r\nLet M and N be subalgebras of the factor \u00CE\u00B1 . We call M and N equivalent if M is the image of N by some automorphism of \u00CE\u00B1 . Let N(M) denote the subalgebra of \u00CE\u00B1 generated by all those unitary operators in \u00CE\u00B1 which induce automorphisms of M, and let N\u00C2\u00B2(M), N\u00C2\u00B3(M),... be defined in the obvious inductive fashion. Following J. Dixmier and S. Anastasio, we call a MA subalgebra M of \u00CE\u00B1 singular if N(M) = M, regular if N(M) = \u00CE\u00B1, semi-regular if N(M) is a factor distinct from \u00CE\u00B1, and m-semi-regular (m \u00E2\u0089\u00A5 2) if N(M),. . .N(m-1)(M) are not factors but N(m)(M) is a factor.\r\nThe MA subalgebras of the hyperfinite 11\u00E2\u0082\u0081 factor \u00CE\u00B2 have received much attention in the literature, in the papers of J. Dixmier, L. Pukanszky, Sister R. J. Tauer, and S. Anastasio. It is known that \u00CE\u00B2 contains a MA subalgebra of each type. Further, \u00CE\u00B2 contains pairwise inequivalent sequences of singular, semi-regular, 2-semi-regular, and 3-semi-regular MA subalgebras.\r\nThe only hitherto known example of a MA subalgebra in a type III factor is regular. In 1956 Pukanszky gave a general method for constructing MA subalgebras in a class of (probably non-hyperfinite) type III factors. Because of an error in a calculation, the types of these subalgebras is not known.\r\nThe main result of this thesis is the construction, in each of the uncountably many mutually non-isomorphic hyperfinite type III factors of R. Powers, of: (i) a semi-regular MA subalgebra (ii) two sequences of mutually inequivalent 2-semi-regular MA subalgebras 1 (iii) two sequences of mutually inequivalent 3-semi-regular MA subalgebras.\r\nLet \u00CE\u00B1 denote one of these type III factors and let \u00CE\u00B2 denote the hyperfinite 11\u00E2\u0082\u0081 factor. Roughly speaking, whenever a non-singular MA subalgebra of \u00CE\u00B2 is constructed by means of group operator algebras, our method will produce a MA subalgebra of \u00CE\u00B1 of the same type.\r\nH. Araki and J. Woods have shown that \u00CE\u00B1 \u00E2\u008A\u0097 \u00CE\u00B2 \u00E2\u0089\u0085 \u00CE\u00B1, and it is therefore only necessary to construct MA subalgebras of \u00CE\u00B1 \u00E2\u008A\u0097 \u00CE\u00B2 of the desired type. We obtain MA subalgebras of \u00CE\u00B1 \u00E2\u008A\u0097 \u00CE\u00B2 by tensoring a MA subalgebra in \u00CE\u00B1 with one in \u00CE\u00B2. In order to determine the type of such a MA subalgebra, we realize \u00CE\u00B2 as a constructible algebra and then regard \u00CE\u00B1 \u00E2\u008A\u0097 \u00CE\u00B2 as a constructible algebra; this allows us to consider operators in \u00CE\u00B1 \u00E2\u008A\u0097 \u00CE\u00B2 as functions from a group into an abelian von Neumann algebra.\r\nAs a corollary to our calculations, we are able to construct mutually inequivalent sequences of 2-semi-regular and 3-semi-regular MA subalgebras of the hyperfinite 11\u00E2\u0082\u0081 factor which differ from those of Anastasio."@en . "https://circle.library.ubc.ca/rest/handle/2429/36864?expand=metadata"@en . "MAXIMAL ABELIAN SUBALGEBRAS OF VON NEUMANN ALGEBRAS \"\u00C2\u00B0y 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 \u00C2\u00AE 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 \u00C2\u00AE 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 \u00C2\u00AE 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 \u00C2\u00AE 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 \u00C2\u00A3(W) to denote the algebra of a l l operators on M , 1^ (or I , when it i s \u00E2\u0080\u00A2 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 \u00C2\u00A3(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 \u00C2\u00A3(&) 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 \u00C2\u00A3(&) 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; \u00C2\u00A72]): 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 \u00C2\u00A3(\u00C2\u00BB). 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 \u00E2\u0080\u00A2 ''(J) of G onto IB which s a t i s f i e s ,(A*)'= ((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 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\u00C2\u00AB\u00C2\u00B0)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) < \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 ' (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) < \u00C2\u00B0\u00C2\u00B0 (c) f a i t h f u l , i f S.e G + and iu(S) = 0 imply\" \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 , 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: \u00E2\u0080\u00A2 . ( 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 \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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 . \u00E2\u0080\u00A2 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\) \u00E2\u0080\u009E ; n o t i c e t h a t G i s separable'whenever G is' at most count-able. For each g \u00C2\u00A3 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; \u00C2\u00A7 5 . 3 ] ) . A l t e r n a t i v e l y , \u00C2\u00A3 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, \u00C2\u00A3 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\u00E2\u0080\u009E , where G, c G c \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 and n=l n \u00C2\u00B1 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 ) \u00E2\u0082\u00AC i x j i s a n o r*honormal b a s i s f o r U\u00C2\u00AEK . For each j e J we denote by x \u00C2\u00AE t - i o f M i n t o M @ K \u00E2\u0080\u00A2 Given A e \u00C2\u00A3(M\u00C2\u00AEK) , 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 \u00C2\u00A3\u00C2\u00AEK 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\u00C2\u00AEU) , ffl * t V < * ! B V \u00E2\u0080\u00A2 the sum converging i n the strong topology on. X (M) I f ( A a ) a G D a n e ^ ^(^\u00C2\u00AE 30 which converges weakly to an A e \u00C2\u00A3(3i\u00C2\u00A9K) , 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 \u00C2\u00ABj)* A ^ '. Proof: Simple c a l c u l a t i o n s (see [6; pp. 23-24] or ' [12; \u00C2\u00A72.4] )..\u00E2\u0080\u00A2 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\u00C2\u00A9IA)(U0.\u00C2\u00AE'V ) , M e 1U and g e G , form a ^-algebra on \" fc> to' Jf\u00C2\u00A9G (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 *\u00E2\u0080\u0094algebra. \u00E2\u0080\u00A2 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 , * A h = *h_i A <{> e and ( i ) ( i i ) 8. i * A * A * T A ~ [M : g \u00E2\u0082\u00AC G] and B ~ [N : g e G] . For a l l g,h e G o g and M \u00C2\u00A3 7ft : ( i ) (J)* AB g(M8I)(Uh\u00C2\u00AEVh) $ e U* = 6 g^ hM ( i v ) ^ A(U h\u00C2\u00AEV h) ^ U* = M ^ . i \" (v) Vs (U hOT h) A * e U* = U h M h . l g y* . \u00E2\u0080\u00A2 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 -\u00C2\u00BB U ] f o r some f r e e C-g system [#t,M,G,g - U ] . -9. Pro-position 1.4 ( [ 3 ; \u00C2\u00A74] and [4; \u00C2\u00A7 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 \u00E2\u0082\u00AC 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.\u00C2\u00AE<\u00C2\u00A3A 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 <\u00C2\u00B0 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 \u00E2\u0080\u00A2 frC' \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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 _ \u00E2\u0082\u00AC 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.. || | \u00C2\u00AE f. of 1 1 \u00E2\u0082\u00AC 1 _ i e l 1 <\u00C2\u00A7 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 ( \u00C2\u00A9 f. , @ g. ) -\u00C2\u00BBTT (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 \u00C2\u00AE (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 \u00C2\u00A3 Q-adic incomplete d i r e c t product. Note that \u00C2\u00AE (M.,f?) r e a l l y \u00E2\u0080\u00A2 i e l 1 1 11. depends on @ Q , and not on the p a r t i c u l a r (f\u00C2\u00B0)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 (:\u00C2\u00A3\"?)- 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 \u00C2\u00AE (34','. , f ? ) i e l 1 1 j e J , l e t f , = \u00C2\u00AE f ^ x ) .\" Then ( f . ) . . T i s J a rT 1 0 and l e t J! = \u00C2\u00AE (MT,f,\u00C2\u00B0) . I f T e \u00C2\u00A3(M, ) , there i s a unique i e l x 1 x o 12. a. (T) \u00C2\u00A3 \u00C2\u00A3(3f) which s a t i s f i ? 1 o (T)][ \u00C2\u00AE^ f \u00C2\u00B1 ] = (_\u00C2\u00AE_^ f_ ^ f \u00C2\u00B1 ) \u00C2\u00AE (T f \u00C2\u00B1 ) 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 _ \u00E2\u0082\u00AC i _ \u00E2\u0080\u00A2 . 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 \u00C2\u00AE (G_. ,f?) denotes the von Neumann algebra ft(G. : i e I) i e l x _ 1 on 34 ; we c a l l \u00C2\u00AE (G ,f\u00C2\u00B0) the i n f i n i t e tensor product of i e l - 1 o \u00E2\u0080\u00A2 the G i r e l a t i v e to (f\u00C2\u00B1)\u00C2\u00B1\u00C2\u00A3j P r o p o s i t i o n 1.8 ( [ 3 ; \u00C2\u00A7 3 ] ) . Let ) i e I >\u00E2\u0080\u00A2 ( f i ) i e i a n d (G, ). j be as above, and l e t 34 = \u00C2\u00AE (34. , f ? ) , G.= \u00C2\u00AE (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 = \u00C2\u00A3(34) i f each G\u00C2\u00B1 = \u00C2\u00A3-(M\u00C2\u00B1) .' 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 \u00E2\u0080\u00A2[n1,V\u00C2\u00B19G1,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. ~ \u00C2\u00A9 ( M V ? ) , Ik = \u00C2\u00AE (W. 1,^) , G = \u00C2\u00AE ( G 1 , ^ . \u00C2\u00AE(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 \u00E2\u0080\u009Eo i\A\u00E2\u0080\u00A2 \u00C2\u00AE (M 1 \u00C2\u00AE ( G X ) A , f ? \u00C2\u00AE(e 1).) i e l ' 1 onto 34 \u00C2\u00AE G w i t h >( \u00C2\u00AE'(f \u00C2\u00B1 \u00C2\u00AE ( g 1 ) \" ) ) = ( \u00C2\u00AE f i)\u00C2\u00AE(.(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 \u00E2\u0082\u00AC 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). \u00E2\u0080\u00A2 With the n o t a t i o n of the previous paragraph, [7FT,34,G,g -\u00C2\u00BB 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 -\u00C2\u00BB: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 \u00C2\u00A3(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 . \u00E2\u0080\u00A2 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 \u00C2\u00B0 ( 7 U ) = 7K Nm(?7i) = &(U e G : U u n i t a r y and UNm _ 1 (7n) U* = N m - \u00C2\u00B1 (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 \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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),.; \u00E2\u0080\u00A2 . j N ^ f y p a r e not f a c t o r s but N \u00E2\u0084\u00A2 ^ ) i s a factor. 16. .' D e f i n i t i o n 2 .5 Let 7^ 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 \u00C2\u00A3Q 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 . \u00E2\u0080\u00A2' 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. \u00E2\u0080\u00A2 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- \u00E2\u0080\u00A2 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 \u00C2\u00A3 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 ^ \u00E2\u0080\u00A2 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 \u00C2\u00A3(#). ' \u00E2\u0080\u00A2\u00E2\u0080\u00A2Lermiia-2; 4\u00E2\u0080\u0094'\u00E2\u0080\u0094Let- -\u00E2\u0080\u0094fl~--b-e~a--Hilbert- space of dimension at l e a s t two, and l e t VI be a MA subalgebra of \u00C2\u00A3(31) such t h a t there i s a f a m i l y ( E i ) j _ \u00E2\u0082\u00AC 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^ = cp^ 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 \u00C2\u00A3(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 \u00E2\u0080\u00A2shows th a t each E^ A E^ e ?ty . - And as \u00E2\u0080\u00A2 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 \u00C2\u00A3(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 )\u00E2\u0080\u00A2 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 ,\u00E2\u0080\u00A2 and hence u l\u00C2\u00AB1\u00C2\u00ABl(y i ; I)*.-v l,^(v l i l)\u00C2\u00AB. M .Therefore ft(U e \u00C2\u00A3(34) U u n i t a r y and U? f t u * = Wl) t=> P R(U \u00C2\u00B1 J., V \u00C2\u00B1 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,, . \u00E2\u0080\u00A2 \ i j I J \u00C2\u00AB Suppose th a t A e \u00C2\u00A3(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 = \u00C2\u00A3 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 \u00C2\u00A3 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 \u00E2\u0080\u00A2 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 % \u00E2\u0080\u00A2 Lemma 2.5 Let (X,\u00C2\u00A3,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,\u00C2\u00A3,|i)' , the r e l a t i o n ( \ f )(x) = c p ( x ) f (x) f e L2(X,\u00C2\u00A3,n) and x e X 2 ~~ defines an e \u00C2\u00A3(L (X,\u00C2\u00A3,n)) , and cp - i s an i s o m e t r i c ^-isomorphism of L\u00C2\u00B0*(X,\u00C2\u00A3,u) onto a von Neumann algebra which i s MA i n \u00C2\u00A3(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 \u00C2\u00A3 . 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 \u00C2\u00A3(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\u00C2\u00BBT 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 \u00E2\u0082\u00AC M and cp \u00E2\u0082\u00AC L \u00C2\u00B0 * ( X , \u00C2\u00A3 , 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)) = \u00E2\u0080\u00A2>}\u00E2\u0080\u00A2\u00E2\u0080\u00A2, = (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 \u00E2\u0082\u00AC 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 \u00E2\u0082\u00AC C y . \u00E2\u0080\u00A2'\u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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=-\u00C2\u00AB\u00C2\u00B0 , complex numbers. Then :. . S \u00E2\u0080\u00A2 a cp = Acp = e. *wxnz u A cp m =_*\u00C2\u00BB mn vm ^ n ^n ' = e-27rinr -m=-oo m n m . v mn m^ \u00E2\u0080\u00A2 m=-o\u00C2\u00BB As r i s i r r a t i o n a l , e 2 7^-(m-n)r ^ \u00C2\u00B1 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 = \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 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 \u00C2\u00A3 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 \u00C2\u00A7 as a mapping from Z i n t o A ; as such, \u00C2\u00A7 i s not onto , - but every member of A i S equivalent t o a member of A i n the range of \u00C2\u00A7 . .' 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 . - \u00E2\u0080\u00A2 . ^ a i X E . \"\".^ a i X * ( E . ) . ~ a i \u00E2\u0082\u00AC . 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\u00C2\u00B0*(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 \u00E2\u0080\u00A2 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\u00C2\u00B0\u00C2\u00B0( [0,1],-$, X)} . This -is e s s e n t i a l l y ... due to von Neumann, and i s well-known,.although an' e x p l i c i t proof does not seem to appear i n : the \ l i t e r a t u r e . I t can be deduced from the general Maharam c l a s s i f i c a t i o n theory of measure algebras (-cf. [26; C o r o l l a r y 5.1]). Our proof-avoids t h i s ' deep theorem, u s i n g i n s t e a d a weaker c l a s s i f i c a t i o n \u00E2\u0080\u00A2\u00E2\u0080\u00A2. theorem. Let 1*1 be a MA subalgebra of \u00C2\u00A3(\u00C2\u00AB) . I f 7YI s a t i s f i e s the hypothesis of Lemma 2 . 4 , set e(7ty) = 0 ; otherwise, set c(fl[) = 1 . Let n($l) be the maximal number \u00E2\u0080\u00A2' 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 \u00C2\u00BB ) . 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\u00C2\u00B0D(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 \u00E2\u0080\u00A2 Let ( E \u00C2\u00B1 ) 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 ) = \u00C2\u00A3(F(M)> . The c a n o n i c a l isomorphism of 14 onto E(M,)\u00C2\u00A9 F(14) induces an isomorphism of 7H onto T^\u00C2\u00A9?)^- [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 \u00C2\u00A3(E(W)\u00C2\u00A9 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\u00C2\u00A9 V \"is a u n i t a r y operator on E(\u00C2\u00AB)\u00C2\u00ABP(M) w i t h (U\u00C2\u00A9V) (THg\u00C2\u00A9^) (U\u00C2\u00A9 V) * = W E \u00C2\u00A9 7 ? l F . 27. Therefore N(ffl E) \u00C2\u00A9 N ^ ) <= N ^ \u00C2\u00A9 ^ ) . Conversely, suppose tha t W i s a u n i t a r y operator on E(3f) \u00C2\u00A9 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$0)W* = E#0 and.'W(0\u00C2\u00ABF)W* = 0\u00C2\u00AE-F . Therefore W = U$ 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 \u00E2\u0080\u00A2 = \u00C2\u00A3(E(M))e\u00C2\u00A3(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 G0) to-denote the subalgebra of the group operator algebra \u00C2\u00A3 r generated by {V : g e G } .\u00E2\u0080\u00A2 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\u00C2\u00BB1 Let GQ be a subgroup- of G . Then / W p \u00C2\u00AE / 3l(p,0,G o) i s a MA subalgebra of G p \u00C2\u00AE 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 % \u00E2\u0080\u00A2 \" Then . N(73l.p \u00C2\u00AE ^ ( p , G , G O ) ) = G p \u00C2\u00AE ft(p,G,N(G0)) N(G p \u00C2\u00AEft(p,G,GQ)) = G p \u00C2\u00AE 7l(p,G,N(G 0)) . Theorem 5-3 For a subgroup G of G , C - \u00C2\u00AE 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 % .\u00C2\u00AEfll.(G,G0) i s a MA subalgebra of G. \u00C2\u00AE 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\u00C2\u00AB1. 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 \u00E2\u0080\u00A2 ((3) of Theorem 3*2, then N(7R p \u00C2\u00AE m(G,G Q)) = G p \u00C2\u00AE 7n(G,N(G-))\" N(G \u00C2\u00AE 7H.(G,GQ)) = G \u00C2\u00AE tH(G,N(.G0)) p \u00C2\u00BB ^ ' o \" p Theorem 5-6 For. a subgroup G Q of G G p \u00C2\u00AE 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 \u00E2\u0080\u00A2 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 \u00C2\u00AE I&)(U \u00C2\u00AE V ) : M e THQ .and g \u00E2\u0082\u00AC 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 \u00C2\u00A7 O o ( i i ) M = 0 whenever g e G - G g o Proof. Let (? = { E (IM \u00C2\u00AE I A ) ( U \u00C2\u00AE V j : each M e 7)1 and u geF e> \u00C2\u00AB & e> e> o F c G f i n i t e } o 91 = fc( (M \u00C2\u00AE I ^ ) ( U g \u00C2\u00AE 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 = *(M 9 I ) ( U \u00C2\u00AE V ) , cp* T cp K k _ 1 g h g K K k 1 g k 1 g n = cp*(M \u00C2\u00AE l ) ( U k \u00C2\u00AE V k) T cph = * T(M \u00C2\u00AE l ) ( U k \u00C2\u00AE V k) (ph = * g T * k h ^ M 0 I ) ( U k \u00C2\u00AE V *h = . . 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 g \u00E2\u0080\u00A2 n keG s k x g k ^g \u00E2\u0080\u00A2\u00E2\u0080\u00A2 1 o 9 = E * T * A ck keG s k h h ' y g Y h \u00E2\u0080\u00A2\u00E2\u0080\u00A2' 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 \u00E2\u0080\u00A2 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 \u00C2\u00AE 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\u00E2\u0080\u009E. e 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 \u00E2\u0080\u00A2 (tp^ )^ -\u00E2\u0084\u00A2 \u00E2\u0080\u00A2 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. \u00E2\u0080\u00A2 TUg = { a F g + b F S . a j b e e } t \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 Define a u n i t a r y r e p r e s e n t a t i o n n -\u00C2\u00BB of on by' s e t t i n g U S ? \u00C2\u00A7 = ? n ^ m f o r a l l n,m e Z g . Then 33-( 3 . 1 ) . F|* e A \u00E2\u0080\u00A2 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 . \u00E2\u0080\u00A2 P 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. = \u00C2\u00BB \u00C2\u00AE H and l e t = 7 K \u00C2\u00AE 7 H . . Then ? H i s a MA von Neumann algebra on \u00C2\u00A3 , (cp \u00C2\u00AE cpfl) . . V! \u00E2\u0080\u00A2 u p cijptii i s an orthonormal b a s i s f o r S ,' and c p Q \u00C2\u00AE 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 = { \u00C2\u00A3_ c P : c Q e \u00C2\u00A3 and a e A} i s a Y ( g ) = l ^ p T l 1 ? ! Y g \u00C2\u00B0 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 ^ p S. LX = 5- c a d a a 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 . \u00E2\u0080\u00A2 - & 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 -\u00C2\u00BB 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* cp^ = U M 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 = \u00E2\u0080\u0094 > 0 , and 6 = e( || Mcp l | - 6) . 1 +.e o \u00E2\u0080\u00A2 3 * By Lemma J . l l , there i s an S = E c P e J such t h a t a || 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* = . \u00C2\u00A3 c cZ P P R = E da P a a,8 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 ] \u00E2\u0080\u00A2 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 .\u00E2\u0080\u00A2 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\u00C2\u00AB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , and each G i s a n=l n . d \u00E2\u0080\u00A2 nf i n i t e subgroup of G . For each n , l e t B n = a((M \u00C2\u00AE I ^ ) ( U g \u00C2\u00AE 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 \u00C2\u00AE U g . There i s a unique l i n e a r isometry y .of. 34 \u00C2\u00AE ^ \u00C2\u00AE 34 \u00C2\u00AE G onto 34 \u00C2\u00AE \u00C2\u00A7 with; T(cp a \u00C2\u00AE A \u00C2\u00AE cpR \u00C2\u00AE g) =.cpa\u00C2\u00AE cpR \u00C2\u00AE ( y , g ) A f o r a l l ' a,\u00C2\u00A3 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 \u00C2\u00A7 - 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 \u00C2\u00AE a ( p , G ) onto G . Noti c e t h a t f o r M,N e and a = (a,g) e * ( U h \u00C2\u00AE V h) B cpe U* = k h - 1 e F otherwise = U K ,&,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\u00C2\u00AE C M M\u00E2\u0080\u009E = 0 a a e {0} x G o otherwise , and ( i i ) -cf (G Q) i f and only i f K e m \u00C2\u00AE C * a e A x G, M a = 0 o otherwise . Proof. Using [6; p. 57] and-(J.4) / ? ( G o ) = ? [fa\u00C2\u00AE C\u00C2\u00A3 V*&(p,G-,G 0)]7 -1 =7 R(M \u00C2\u00AE l\u00C2\u00A3 \u00C2\u00AE U g \u00C2\u00AE V g : M e 7H , g e G Q ) 7 _ 1 S \u00E2\u0082\u00AC G Q) a n d \" - ? ( G j = ? G ^ \u00C2\u00AE 7 t f p , G , G j 7 - 1 P = 7 R((M \u00C2\u00AE I ^ ) ( U a \u00C2\u00AE V a) \u00C2\u00AE (U \u00C2\u00AE V ) : M e ^ , a e A , g e G Q ) T -1 = a ( ( M \u00C2\u00AE I ^ I j ) ( U M \u00C2\u00AE 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 \u00E2\u0080\u00A2 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 < \u00C2\u00A3 || 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\u00E2\u0084\u00A2 =\u00E2\u0080\u00A2 \u00C2\u00A3 M. N, and N\u00E2\u0080\u009E = E N. N* . As \u00E2\u0080\u00A2 i e F 1 1 \u00E2\u0080\u00A2 '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 \u00E2\u0082\u00AC P 1 \u00E2\u0080\u00A2 47. || Mrp x|| 2 = (M p X,X) 1- 2 (M- M* x,x) = S || M. x| \u00E2\u0080\u00A2 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\u00E2\u0080\u009E = U_ \u00C2\u00AE V . Lemma 3.19 ( c f . [20; Lemma 15] )\u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 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) \u00E2\u0080\u00A2 1 ( i ) . || MbcpQScpo - Nbcpo\u00C2\u00AEcpo|| < | f o r a l l b e \u00C2\u00A3 ; . ( i i ) i f V.= S (N, \u00C2\u00AEIA)(U, \u00C2\u00AEV. ) , then f o r a l l betf D 9 a D c,d \u00E2\u0082\u00AC {0} x G , (3.9) || cp* [tJ T d U* - V T d V*] U* cpQ \u00C2\u00A9 cpjl < . .'. ' y Proof. F i x c. and d i n [0] x G . By Lemma 1.2, .(5.10) Ig = cp* tf U* % = 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 ( \u00C2\u00A3 TA M* cp \u00C2\u00AE cp , cp \u00C2\u00AE cp ) = \u00C2\u00A3 II M cp \u00C2\u00AE cp II 2 < ^ a*3 a a 0 \u00C2\u00B0 0 \u00C2\u00B0 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 \u00C2\u00A5 = S (M^ \u00C2\u00AE I ) ( U \u00C2\u00AE 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* _ \u00C2\u00B1 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 \u00E2\u0080\u00A2 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) * W T W*

* [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 \u00C2\u00AE cp a \u00C2\u00AB ^ ' a c c 1 a d c ^ o 4 9 . < II 2 , M a U M i U cp \u00C2\u00AE cp || + ae^-^ c c ad c \u00C2\u00B0 - \u00C2\u00B0 : + II ? S a 5 c C - l * K % \u00C2\u00AE ^o'H a|3= c ad c \u00C2\u00B0 \u00C2\u00B0 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 aeff-ff c ad c \u00C2\u00B0 9 ' \"+( Z l|M acp o\u00C2\u00AEcp o[| 2)* . a$ 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. \u00C2\u00AE I ) (U. \u00C2\u00AE V, ) , we. \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 . be3 D D D have th a t ( c f . (5112)) 50. . ! U * A T D..\u00C2\u00A5* - V T d V*]- ^ U* cpQ \u00C2\u00AE cpj| = = ||. S (M U M*_! U* - N U N*_! . U*)\u00C2\u00ABp \u00C2\u00AEep || ae3 c a d c a d 1 I' E , ^ a ^ c , \" J 5 c *o \u00C2\u00AE % H + ae3 a c c ad . c ad. c . \u00C2\u00B0 \u00C2\u00B0 \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 . C ^ || M a U c ( M * . l a d - N*. l a d) U* % \u00E2\u0080\u00A2 cpo || + - + S ,11 U c N * _ l \u00E2\u0080\u009E U*(M - N ) cp \u00C2\u00AE cp || ae3 c c a d c a a o o < S II '(M*_! - N*_ x ) cp \u00C2\u00AE cp !|| + . ae3' c ad c -'-ad o o + \u00C2\u00A3 , 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 \u00C2\u00B0 \u00C2\u00B0 a,0eA a p 1 \ where the 8 (a; a,\u00C2\u00A9) 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 \u00C2\u00AE cp = 0 ; as cp \u00C2\u00AE 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 \u00C2\u00AE P p) Mft cpQ \u00C2\u00AE cpQ = P \u00C2\u00AE P s 9 (a; Y,0) cp \u00C2\u00AE cp -yeA 52. = .( 2 e(a; Y , 0 ) P a cp ) \u00C2\u00AE cp ,. YeA . T H and t h e r e f o r e M a e 7*1 \u00C2\u00AE . 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 \u00C2\u00A3j 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) \u00E2\u0080\u00A2 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 \u00C2\u00AE 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\u00C2\u00B1>a2 i . \" \u00E2\u0080\u00A2 From (3.8)^ (3-18) | > || cpo \u00C2\u00AE cpQ - \ cpQ \u00C2\u00AE 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* ( P8 P6 \u00C2\u00AE PY Phr, *o V o ' 1 \u00C2\u00AE Vha2 *\u00C2\u00B0 \u00C2\u00AE ^ = = .(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 \u00C2\u00AE cpo , I 9 P a 2 + h a 2 cpQ \u00C2\u00AE cpQ) \u00C2\u00A3 |c(b; S , a 0 ) | be3' B 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 \u00C2\u00AE Cs, f o r s i m i l a r reasons. In any case, > ||Tcpo \u00C2\u00AE cpQ - Scpo \u00C2\u00AE cp 0H I I (ScpQ \u00C2\u00A9 cp Q , cp Q \u00C2\u00AE 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 \u00C2\u00AE c p Q - S c p c \u00C2\u00AE c p o || > !(Tcpc \u00C2\u00AE cpQ - ScpQ \u00C2\u00AE cpQ , cpo \u00C2\u00AE V ^ ^ l 56. = |(Scp0 \u00C2\u00AE cp , I \u00C2\u00AE P a 2 + h a 2 * 0 \u00C2\u00AE G,G 0) such th a t s ( V j = U \u00C2\u00AE 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 \u00C2\u00AE G w i t h ri(cp\u00E2\u0080\u009E \u00C2\u00AE g ) = cp \u00C2\u00AE 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 \u00C2\u00AE 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 .\u00C2\u00A3(34 \u00C2\u00AE 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 , \u00C2\u00A7 J V j m \u00C2\u00AE n = TI(I \u00C2\u00AE V\u00E2\u0080\u009E) r\ 1 m \u00C2\u00AE n o g a l v g ex = cp g a\u00C2\u00AE(gh) A = (U \u00C2\u00AE V\u00E2\u0080\u009E)(cpa \u00C2\u00AE n ) > and t h e r e f o r e $(-(V ) = U \u00C2\u00AE 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 \u00E2\u0082\u00AC 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^ = \u00C2\u00A7 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 \u00C2\u00AE B(p,G) such t h a t f o r any subgroup G Q of G , ^ \u00C2\u00A7 ( m p \u00C2\u00AE1\u00C2\u00BBl(G,G0)) = THp \u00C2\u00AE71(P,G,G0) \u00C2\u00A7(G p \u00C2\u00AE J l ( G J G Q ) ) = G p \u00C2\u00AE ? 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 ~ $\u00C2\u00A3((> Proof. As 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 \u00C2\u00A7'(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 \u00C2\u00A7' 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 \u00C2\u00AE 7 i ( p , G , G ) i s MA i n G P \u00C2\u00AE (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 \u00C2\u00AE & ( 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 \u00C2\u00AE B ( p , G ) ., f o r 59-7'N(7L)7\" 1 = V R(U : U e G p \u00C2\u00AE e(p,G) and u n i t a r y , and U?lU* =70)^-1 = ^(T^T'1 : U E G P \u00C2\u00AE ^ ( 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 \u00C2\u00AE 7 l ( p , G , N(G Q))' , and s i m i l a r l y . N ( G P \u00C2\u00AE7l(p,G,G O)) = G p \u00C2\u00AE?l(p,G, N(G Q)) Proof of Theorem 3 .3 As G i s a f a c t o r , G p \u00C2\u00AE 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 \u00C2\u00AE-7H.(G-,GQ)' i s MA i n G p \u00C2\u00AE \u00C2\u00A3 Q i f and only i f 1U(G,G ) i s MA i n l(G,GO) U*) . = ^ \u00C2\u00AEft(p,G,G_) or $(U) G p \u00C2\u00AE7L( P JG,G o)\u00C2\u00A7(U)* = $(U G p \u00C2\u00AE 7 H ( G , G o ) U * ) = G p \u00C2\u00AE71(p,G,G O) . By Lemma 3--20, $(U) e G p \u00C2\u00AE7l(p,G, N(G Q)) , and t h e r e f o r e U e G p \u00C2\u00AE 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 \u00C2\u00A3 (\u00C2\u00AE>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\u00C2\u00B0(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 \u00C2\u00AE &(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 \u00E2\u0080\u00A2 - ' (4 .1) G P \u00C2\u00AE 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 \u00C2\u00AE 7 1 ( P J G J g 0 ) i s a MA subalgebra of G \u00C2\u00AE e(p,G) w i t h 64. normalizer G P \u00C2\u00AE 7l(p,G,G) , a f a c t o r d i s t i n c t from G \u00C2\u00AE 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 <-\u00E2\u0080\u00A2 {(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} . \u00E2\u0080\u00A2 . 65. i s a normal subgroup of G . The subgroup G q has p r o p e r t y \u00E2\u0080\u00A2(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 \u00C2\u00B1 ) i = 1,...,n . Let H = {(1 - c ^ ) \" 1 ^ - 6-j) : a \u00C2\u00B1 / 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 \u00C2\u00B1 g Q g\" 1 = g Q , then ( l , B o ) = (a\u00C2\u00B1^\u00C2\u00B1)(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' =\u00E2\u0080\u00A2 1 , and.thus P i = 8^ , i . e . , g\u00C2\u00B1 = g^ . \" . Proof of Theorem 4.2 F i x a~ p e (0,|) and an i n t e g e r m >_ 2 . \u00E2\u0080\u00A2 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 \u00E2\u0080\u00A2 O ) of Theorem 3 .2 \u00E2\u0080\u00A2(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> $lp \u00C2\u00AE Tl(p*G,GQ) i s a 2 -semi-regular MA subalgebra of G p \u00C2\u00AE &(p,G) of improper- l e n g t h m and ffl \u00C2\u00AE$L(G,G ) i s a 2 -semi-regular MA subalgebra of G \u00C2\u00AE @ 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 ) \u00C2\u00A7 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 \u00E2\u0080\u00A2 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--?) \u00E2\u0080\u00A2 i n G w i t h \u00E2\u0080\u00A2 g l 2 = s 2 3 = s34 \u00E2\u0080\u00A2 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 . \u00E2\u0080\u00A2 ( 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 \u00C2\u00AE U i s a u n i t a r y operator i n G_ \u00C2\u00AE B(p,G) such that ( I \u00C2\u00AE U)7Up \u00C2\u00AE7l(p,G,GQ) ( I \u00C2\u00AE U ) * = ^ Jtp \u00C2\u00AEU(P^G,G O) . According to Theorem 3-2, t h i s f o r c e s I \u00C2\u00AE U- e G p \u00C2\u00AE7l(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 ) ) . \u00E2\u0080\u00A2 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 ) , -\u00E2\u0080\u00A2 N(7l (p,G,G Q)) = 7l(p,G,N(G O)) =. \u00C2\u00A3 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 ) . 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