UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
[if-you-see-this-DO-NOT-CLICK]
UBC_1968_A1 N54.pdf [ 3.65MB ]
Metadata
JSON: 1.0080538.json
JSON-LD: 1.0080538+ld.json
RDF/XML (Pretty): 1.0080538.xml
RDF/JSON: 1.0080538+rdf.json
Turtle: 1.0080538+rdf-turtle.txt
N-Triples: 1.0080538+rdf-ntriples.txt
Original Record: 1.0080538 +original-record.json
Full Text
1.0080538.txt
Citation
1.0080538.ris

Full Text

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. BURES, C e r t a i n f a c t o r s constructed as i n f i n i t e tensor products, Compositio Math., 15 (1963) , . pp. 169-191. BURES, A b e l i a n subalgebras of von Neumann algebras, p r e - p r i n t . ' ' . DIXMIER, Sous-anneaux ab e l i e n s maximaux dans l e s f a c t e u r s de type f i n i , Ann. of Math., .59 (1954), pp. 279-286. ' DIXMIER, Les algebres d'operateurs dans l'espace H i l b e r t i e n , G a u t h i e r - V i l l a r s . , P a r i s , 1957. DUNFORD and J. T. SCHWARTZ, L i n e a r Operators, V o l . I I , I n t e r s c i e n c e P u b l i s h e r s , New York, 1963. A. DYE, On groups of measure p r e s e r v i n g transformations I I , Amer. J. Math., 85 (1963) , pp. 551-576. R. HALMOS, Measure Theory, D. van No strand, ,. New York, 1950. R. HALMOS, .Introduction to H i l b e r t space and the theory of s p e c t r a l m u l t i p l i c i t y , Chelsea, New York, 1951. ' ' ' R. KERR, A b e l i a n von Neumann algebras, M.A. Thesis, U n i v e r s i t y of B r i t i s h Columbia, 1965- [4] . D. [5] J. [6] J.' [7] N. [83 • H. [9] p. [10] p. [11] c. 71. .'"[12] P. J. MURRAY and J. von NEUMANN, On r i n g s of operators, ^ Ann. of Math., 37 (1936), pp. 116-229- [13] F. J. MURRAY and J. von NEUMANN, On r i n g s of operators IV, Ann.of Math., 44 (1943), pp. 716-808. ' [14] J. von NEUMANN, Zur Algebra der Funktionaloperatoren und t h e o r i e der normalen Operatoren, Math. Ann. 102 (1929)^ pp. 370-427. [15] J. von NEUMANN, On i n f i n i t e d i r e c t products, Compositio ' Math., 6 (1938),"pp. 1-77. - [16] J.' von NEUMANN, On r i n g s of operators I I I , Ann. of Math., 41 (1940), pp. 94-161. [17] J. von NEUMANN, On r i n g s of operators. Reduction Theory, Ann. of Math., 50 (1949), pp. 401-485. Cl8] R. T. POWERS, Representations o f . u n i f o r m l y h y p e r f i n i t e algebras and t h e i r a s s o c i a t e d von Neumann r i n g s , p r e - p r i n t . . • [19] R. T. POWERS, Representations of u n i f o r m l y h y p e r f i n i t e algebras and t h e i r a s s o c i a t e d von Neumann r i n g s , B u l l . Amer. Math. S o c , 73- (1967),' PP• 572-575. ; [20] L. PUKANSZKY, Some examples of f a c t o r s , P u b l i c a t i o n e s Mathematicae, Debrecen, 4 (1956), pp. 135-156. . • [21] • L. PUKANSZKY, On maximal a b e l i a n subrings of f a c t o r s of ••'-.. • type I I , Canadian J . Math., 12 ( i 9 6 0 ) , pp. 289-296. [22] . T. SAITS and J. TOMIYAMA, Some r e s u l t s on the d i r e c t product of W*-algebras, Tohoku Math. J . , 12 ' S (I960), pp. 455-458. . . •" '•"[23] S. .SAKAI, The theory of W*-algebras, Lecture Notes, Yale U n i v e r s i t y , 1962. - 72. [24] J . SCHWARTZ, Two f i n i t e , n o n-hyperfinite,. non- isomorphic f a c t o r s , Comm. Pure Appl. Math., 16 (1963), pp. 19-26. [25] J. SCHWARTZ, Non-isomorphism of a p a i r of f a c t o r s of type I I I , Comm.Pure Appl. Math., 16 (1963), pp. 111-120. ) [26] I. E. SEGAL, Equivalences of measure spaces, Amer. J. Math., 73 (1951), PP. 275-313. [27] I. E. SEGAL, Decompositions of operator algebras I I , Memoirs of Amer. Math. S o c , 9 (1951), pp. 1-66. [28] SISTER R. J. TAUER, Maximal a b e l i a n subalgebras i n f i n i t e f a c t o r s of type I I , Trans. Amer. Math. S o c , 114 (1965), pp. 281-308. [29] SISTER R. J. TAUER, Semi-regular maximal a b e l i a n sub- algebras i n h y p e r f i n i t e f a c t o r s , B u l l . Amer. Math., S o c , 71 (1965), pp. 606-608. [30] SISTER R. J. TAUER, M-semi-regular subalgebras i n - • h y p e r f i n i t e f a c t o r s , Trans. Amer. Math. S o c , 129 (1967), pp. '530-541. - 1

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
China 7 0
United States 6 0
France 6 0
Czech Republic 1 0
City Views Downloads
Beijing 7 0
Unknown 6 0
Ashburn 3 0
Mountain View 2 0
Redmond 1 0
Prague 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080538/manifest

Comment

Related Items