UBC Theses and Dissertations

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UBC Theses and Dissertations

Flow under a function and discrete decomposition of properly infinite W*-algebras Phillips, William James 1978

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FLOW UNDER A FUNCTION AND DISCRETE DECOMPOSITION OF PROPERLY INFINITE W -ALGEBRAS by WILLIAM JAMES PHILLIPS B.Sc. Queen's U n i v e r s i t y , 1971 M.Sc. Queen's U n i v e r s i t y , 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of M a t h e m a t i c s We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1978 ^ C ^ W i l l i a m James P h i l l i p s In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the 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 representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. M a t h e m a t i c s Department of The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date F e b r u a r y 16, 1978 i i ABSTRACT The aim o f t h i s t h e s i s i s t o g e n e r a l i z e t h e c l a s s i c a l f l o w under a f u n c t i o n c o n s t r u c t i o n t o n o n - a b e l i a n W - a l g e b r a s . We o b t a i n e x i s t e n c e and u n i q u e n e s s theorems f o r t h i s g e n e r a l i z a t i o n . As an a p p l i c a t i o n we show t h a t the r e l a t i o n s h i p between a c o n t i n u o u s and a d i s c r e t e d e c o m p o s i t i o n o f a p r o p e r l y i n f i n i t e W - a l g e b r a i s t h a t o f g e n e r a l i z e d f l o w under a f u n c t i o n . S i n c e c o n t i n u o u s d e c o m p o s i t i o n s a r e known t o e x i s t f o r any p r o p e r l y i n f i n i t e W - a l g e b r a , t h i s l e a d s t o g e n e r a l i z a t i o n s of Connes' r e s u l t s on d i s c r e t e d e c o m p o s i t i o n . i i i TABLE OF CONTENTS Page INTRODUCTION 1 1. CROSSED PRODUCTS 3 2. FLOW BUILT UNDER A CEILING 16 3. UNIQUENESS OF FLOW UNDER A CEILING . . . . 30 4. FLOW UNDER A CEILING AND WEAK EQUIVALENCE . . 47 5. APPLICATION TO PROPERLY INFINITE W*-ALGEBRAS . 56 BIBLIOGRAPHY 64 APPENDIX 65 i v ACKNOWLEDGEMENTS I w i s h t o thank Don Bures f o r s u g g e s t i n g t h i s p r o b l e m and f o r h i s g u i d a n c e d u r i n g t he a n a l y s i s . I am g r a t e f u l t o Cathy Agnew f o r t y p i n g t h i s t h e s i s . - 1 -I n t r o d u c t i o n The p u r p o s e o f t h i s t h e s i s i s t o g e n e r a l i z e t h e c l a s s i c a l " f l o w under a f u n c t i o n " c o n s t r u c t i o n t o n o n - a b e l i a n W - a l g e b r a s . That i s , g i v e n a W - a l g e b r a N, an automorphism 0 of N and a p o s i t i v e s e l f -a d j o i n t o p e r a t o r <j> a f f i l i a t e d t o t h e c e n t r e o f N we show i n a n a l o g y w i t h t h e a b e l i a n c a s e , how t o c o n s t r u c t a c o n t i n u o u s a c t i o n a o f t h e r e a l s on a W - a l g e b r a M (see D e f i n i t i o n 1.1). The c o v a r i a n t system {M,a} (see D e f i n i t i o n 1.1) i s c a l l e d t h e f l o w b u i l t on {N,6} under th e c e i l i n g $ . We o b t a i n " e x i s t e n c e " and " u n i q u e n e s s " theorems f o r t h e r e p r e s e n t -a t i o n o f a g i v e n c o v a r i a n t system o v e r t h e r e a l s as a f l o w b u i l t under a c e i l i n g . As an a p p l i c a t i o n we o b t a i n Connes' d i s c r e t e d e c o m p o s i t i o n theorems ( [ 1 ] theoreme 4.4.1, theoreme 5.3.1 and theoreme 5.4.2) from T a k e s a k i ' s c o n t i n u o u s d e c o m p o s i t i o n theorems ( [ 2 ] theorem 8.1, lemma 8.2 and c o r o l l a r y 8.4) t h e r e b y e l u c i d a t i n g t h e c o n n e c t i o n between t h e s e two r e s u l t s . I n s e c t i o n 1 we s t a t e t h e main r e s u l t s o f T a k e s a k i ' s d u a l i t y t h e o r y f o r c r o s s e d p r o d u c t s . I n s e c t i o n 2 we d e f i n e f l o w b u i l t under a c e i l i n g and g i v e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a c o v a r i a n t system o v e r t h e r e a l s t o be i s o m o r p h i c t o a f l o w b u i l t under a c e i l i n g . S e c t i o n 3 d e a l s w i t h t h e u n i q u e n e s s p r o b l e m . That i s , we show t h e r e l a t i o n s h i p between {N^,0^,cJ>^} and {^^,8^,^^} when t h e c o r r e s p o n d i n g f l o w s a r e i s o m o r p h i c . I n s e c t i o n 4 we d e a l w i t h t h e u n i q u e n e s s p r o b l e m i n c a s e t h e f l o w s a r e o n l y w e a k l y e q u i v a l e n t (see d e f i n i t i o n 1.5). I n s e c t i o n 5 we d e r i v e d i s c r e t e d e c o m p o s i t i o n theorems f o r p r o p e r l y i n f i n i t e W - a l g e b r a s u s i n g T a k e s a k i ' s c o n t i n u o u s d e c o m p o s i t i o n theorem and our - 2 -r e s u l t s on flow b u i l t under a c e i l i n g . The appendix consists of a proof of the r e s u l t s of section 2 i n the s p e c i a l case where the ideas of [3] and [4] may be applied. - 3 -1. C r o s s e d P r o d u c t s The purpose o f t h i s s e c t i o n i s t o c o l l e c t some r e s u l t s on c r o s s e d p r o d u c t s o v e r a b e l i a n g r o u p s . E x c e p t f o r t h e p r o o f o f p r o p o s i t i o n 1.10, t h e s e r e s u l t s and p r o o f s may be f o u n d i n [ 2 ] , [5] and [ 6 ] . I n t h e f o l l o w i n g l e t G denote a l o c a l l y compact a b e l a i n group w i t h d u a l group G . Haar measure i n G i s denoted s i m p l y by d t . We b e g i n w i t h some measure t h e o r e t i c a l r e s u l t s . A f u n c t i o n X f r o m G i n t o a H i l b e r t space . H i s c a l l e d s t r o n g l y ( o r B o u r b a k i ) m e a s u r a b l e i f f i t s a t i s f i e s t h e c o n d i t i o n s : ( i ) t — y ( € ( t ) , n ) i s m e a s u r a b l e f o r n e H . ( i i ) f o r each compact s e t K <= G, t h e r e i s a s e p a r a b l e subspace of H such t h a t £(t) e f o r a l m o s t e v e r y t e K . 2 L (G;H) d e n o t e s t h e v e c t o r space o f s t r o n g l y m e a s u r a b l e f u n c t i o n s £: G —>• H w h i c h s a t i s f y : | C ( t ) | | 2 d t < » . 2 By i d e n t i f y i n g e l e m e n t s o f L (G;rf) w h i c h a r e e q u a l a l m o s t everywhere and by d e f i n i n g a , n ) = ( 5 ( t ) , n ( t ) ) d t 2 2 2 we o b t a i n t h e H i l b e r t space L (G;H) . L (G;<C) i s denoted by L (G) . 2 2 L (G;H) i s i d e n t i f i e d w i t h H 9 1 (G) by mapping h\ % r\ t o t h e f u n c t i o n t —*• n(t)£ . L°°(G) de n o t e s the v e c t o r space o f m e a s u r a b l e f u n c t i o n s f : G — • <C - 4 -w h i c h s a t i s f y s u p { | f ( t ) | : t e G} < °° . By i d e n t i f y i n g e l ements of 00 00 L (G) w h i c h a r e e q u a l l o c a l l y a l m o s t e v e r y w h e r e , we o b t a i n L (G) . CO We make no d i s t i n c t i o n between elements of L (G) and t h e c o r r e s p o n d i n g 2 2 m u l t i p l i c a t i o n o p e r a t o r on L (G) • O p e r a t o r s on L (G;H) a r e u s u a l l y 2 d e f i n e d by d i s p l a y i n g an o p e r a t o r on L (G;rf) . F o r example, i f t —>• x ( t ) i s a f u n c t i o n f r o m G t o B(H) w h i c h s a t i s f i e s : ( i ) f o r each £ e H, t —> x(t)£ i s s t r o n g l y m e a s u r a b l e ( i i ) sup{ I | x ( t ) I I : t e G} < °° , 2 t h e n t —>• x ( t ) d e f i n e s an o p e r a t o r on L (G;H) by t h e f o r m u l a : (x5)(t) = x ( t ) ? ( t ) f o r K e L2(G;H) . We now b e g i n t h e d e f i n i t i o n o f c r o s s e d p r o d u c t s . D e f i n i t i o n 1.1. A c o n t i n u o u s a c t i o n a of G on a W - a l g e b r a M i s a homomorphism t —>- a of G i n t o t h e group of automorphisms * o f M such t h a t f o r each x e M, t h e map t —>• a (x) i s a - s t r o n g c o n t i n u o u s . The p a i r {M,a} i s c a l l e d a c o v a r i a n t system o v e r G . We have t h e u s u a l n o t i o n o f homomorphism ( i m b e d d i n g , isomorphism) K: {M,a} —>- {N,g} . That i s , K i s a c o n t i n u o u s W - a l g e b r a homomorphism ( i m b e d d i n g , isomorphism) of M i n t o N s u c h t h a t K « a = 8 f c ° K, t e G . T y p i c a l examples of c o v a r i a n t systems can be o b t a i n e d i n t h e * f o l l o w i n g way: suppose M i s a W - a l g e b r a a c t i n g on a H i l b e r t space H and t —> U ( t ) i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n o f - 5 -G on H such that f or a l l t e G U(t)M U(t) = M . Then a f c(x) = U(t)xU(t) for t e G and x e M defines a continuous action of G on M (see [2] proposition 3.2). Given a covariant system {M,a} over G with M acting on a H i l b e r t space H, we can define a continuous imbedding TT of M 2 into the bounded operators on the H i l b e r t space L (G;H) by: for x e M and ? e L 2(G;H), (ir (x)£)(t) = « . (x)?(t) . We also have a strongly continuous unitary representation of G on L 2(G;H) defined by: for s e G and 5 e L 2 ( G ; H ) , ( A (s)£)(t) = S(t-s) . Note that A ( S ) T T ( X ) A ( S ) = IT (a (x)) for a l l s e G and for a l l a a a a s x e M . D e f i n i t i o n 1.2. Given a covariant system {M,a} over G with M acting on H, the crossed product W {M,a} i s the W -algebra on 2 L (G;H) generated by TT (M) and ^ ( G ) . In [2] proposition 3.4, i t i s shown that the d e f i n i t i o n of W {M,a} i s independent of the H i l b e r t space n on which M acts. More p r e c i s e l y , i f K i s an isomorphism of {M,a} onto {N,3} then there i s an isomorphism K of W {M,a} onto W {N,g} such that K ^ ( x ) = T T ^ ( K X ) , for a l l x £ M and K A (s) = A ( s ) , for a l l s e G . - 6 -We can r e l a t e t h i s d e f i n i t i o n o f c r o s s e d p r o d u c t t o t h e o r i g i n a l d e f i n i t i o n (see f o r example [7] c h a p i t r e 1, s e c t i o n 9.2) i n t h e f o l l o w i n g way: assume t h a t a i s u n i t a r i l y implemented on t h e H i l b e r t space H, t h a t i s , t h e r e i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n t —> U ( t ) o f G i n H s u c h t h a t a f c ( x ) = U ( t ) x U ( t ) f o r a l l t e G 2 and a l l x e M . We d e f i n e a u n i t a r y o p e r a t o r U on L (G;H) by: (U5)(t) = U ( t K ( t ) f o r K e L2(G;H) . 2 2 Then, w i t h the u s u a l i d e n t i f i c a t i o n o f L (G;r/) w i t h ff 8 L (G) we have: U ir (x)U = x 8 1, f o r a l l x e M a U X ( s ) U * = U ( s ) 8 X(s) , f o r a l l s £ G a V o * where (A(s)£)(t) = 5(t-s) f o r £ £ L (G) . The W - a l g e b r a g e n e r a t e d by M 8 1 and {U(s) 8 X( s ) : s e G} i s the " o r i g i n a l " d e f i n i t i o n o f c r o s s e d p r o d u c t . U s i n g t h e u n i t a r y i m p l e m e n t a t i o n we can g i v e some o p e r a t o r s i n the commutant W {M,a} . Namely, M 1 8 1 c W*{M,a}' and U(s ) 8 X ( s ) e W {M,a}' f o r s e G . We s h a l l use t h i s o b s e r v a t i o n l a t e r . The f i r s t n o n - t r i v i a l r e s u l t c o n c e r n i n g c r o s s e d p r o d u c t s d e a l s w i t h t h e d u a l a c t i o n . We can d e f i n e a c o n t i n u o u s a c t i o n o f G on W {M,a} by l o o k i n g a t the c h a r a c t e r s X p ( t ) = <P»t> f o r p e G . We d e f i n e a - 7 -s t r o n g l y c o n t i n u o u s u n i t a r y 2 L (G;H) by the f o r m u l a : r e p r e s e n t a t i o n p y (p) of G on y ( p K ( t ) = <P,t> Ut), f o r £ e L 2(G;H) . Then y a ( p ) T T a ( x ) y a ( p ) TT (x) , f o r a l l x e M V„(P) A„(s)y„(p) = <P,s> A ( s ) , f o r a l l p e G, s e G . Oi Ut L* CI Hence ap(y) = U a ( p ) y V a ( p ) f ° r y e W {M,a} d e f i n e s a c o n t i n u o u s a c t i o n o f G on W {M,a} . D e f i n i t i o n 1.3. G i v e n a c o v a r i a n t system {M,a} o v e r G, t h e d u a l A c o v a r i a n t s y s t e m {W {M,oc},a} i s d e f i n e d by: A * A a (y) = ya(p)yva(p) f o r p e G and y e W {M,a} . A I t i s c l e a r t h a t ^ ( M ) c {y e W {M,a}: ap(y) = y> f o r a l l p e G}, bu t t h e c o n v e r s e i s n o t so o b v i o u s . To s t a t e t h e n e x t r e s u l t we use the n o t a t i o n Theorem 1.4 ( t h e u n i q u e n e s s theorem). F o r a c o v a r i a n t s y s t e m {M,a} ov e r G we have x, f o r a l l t e G} . W*{M,a} a = TT (M) P r o o f . The f o l l o w i n g p r o o f i s due t o L a n d s t a d [ 6 ] . We have 8 -W {M,a} = W {M,a} n y ( G ) ' so a W*{M,a} a = [W*{M,a}' u u ( G ) ] 1 . We assume t h a t a i s u n i t a r i l y implemented on H by t —> U ( t ) . By our p r e v i o u s remark M 1 8 1 u { U ( t ) * 8 A ( t ) : t e G} c W*{M,a}' . Hence i t s u f f i c e s t o show [M 1 8 1 u' { U ( t ) * 8 A ( t ) : t e G} u y (G) ]' c TT (M) . a a U s i n g t h e o p e r a t o r U as p r e v i o u s l y d e f i n e d , i t s u f f i c e s t o show t h a t [UM' 8 1 U* u A (G) u y a ( G ) ] ' c M 8 1 . 2 S i n c e ^ a ( G ) an^ ^ (G) g e n e r a t e 1 8 B(L (G) i t s u f f i c e s t o show * t h a t i f y e B(rf) and y 8 1 e [UM' 8 1 U ] ' t h e n y e M . S i n c e (Ux 8 1 U*£)(t) = U ( t ) x U ( t ) * ? ( t ) f o r Z e L 2(G;H) and x e B(H) i t i s c l e a r t h a t y must be i n M i f y 8 1 commutes w i t h U M' 8 1 U* . || T h i s theorem may be used t o deduce t h e r e l a t i o n s h i p between {M,a} and {N,f3} under t h e a s s u m p t i o n t h a t t h e d u a l c o v a r i a n t systems a r e i s o m o r p h i c . T h i s r e l a t i o n s h i p l e a d s t o the d e f i n i t i o n : D e f i n i t i o n 1.5. L e t {M,a} and {N,3) be c o v a r i a n t systems o v e r G . ( i ) an a c o c y c l e i s a a - s t r o n g c o n t i n u o u s map t —> u t of G i n t o t h e u n i t a r y group of M w h i c h s a t i s f i e s : u , = u a (u ) , f o r a l l s . t e G . t+s t t s ( i i ) {M,a} and {N,g} a r e s a i d t o be w e a k l y e q u i v a l e n t i f f t h e r e i s an i s o m o r p h i s m K of M onto N and a 3 c o c y c l e t —>• u^ s u c h t h a t A K a f c(x) = u t 3 t ( K x ) u t > f o r a l l t e G, x e M,. P r o p o s i t i o n 1.6. The c o v a r i a n t systems {M,a} and {N,3} o v e r G * a r e w e a k l y e q u i v a l e n t i f f t h e d u a l c o v a r i a n t systems {W {M,a},a} A and {W {N,3},g} a r e i s o m o r p h i c . P r o o f : see [2] p r o p o s i t i o n 3.5 and c o r o l l a r y 3.6 f o r t h e i m p l i c a t i o n = > . The c o n v e r s e i s easy i n v i e w of t h e u n i q u e n e s s theorem. || We now g i v e a (weak) c h a r a c t e r i z a t i o n of c r o s s e d p r o d u c t s : P r o p o s i t i o n 1.7. L e t {N,g} be a c o v a r i a n t system o v e r G . L e t M be a W - s u b a l g e b r a o f N and t u ( t ) a s t r o n g l y c o n t i n u o u s u n i t a r y ' r e p r e s e n t a t i o n o f G i n N s u c h t h a t ( i ) N i s g e n e r a t e d by M and { u ( t ) : t e G} ( i i ) u ( t ) M u ( t ) * = M f o r a l l t e G ( i i i ) M c N 6 and 0 ( u ( t ) ) = <p,t> u ( t ) , f o r a l l t e G, p e G . P We denote by a, t h e c o n t i n u o u s a c t i o n of G on M g i v e n by * a^(x) = u ( t ) x u ( t ) f o r t e G and x e M . Then t h e r e i s an A i s o m o r p h i s m K of {N,(3} w i t h {W {M,a},a} such t h a t : - 10 -K x = TT (x) , f o r a l l x e M a K u ( t ) = * a ( t ) , f o r a l l t e G P r o o f : T h i s p r o o f i s e s s e n t i a l l y due t o T a k e s a k i . We assume N a c t s 2 " 2 on H . L e t F be t h e F o u r i e r t r a n s f o r m mapping L (G) onto L (G) . Then ft x —*• 1 8 F TT D(x) 1 8 F p 2 i s an i s o m o r p h i s m of N i n t o t h e bounded o p e r a t o r s on L (G;r/) . We have 1 8 F TT 0(X) 1 8 F* = x 8 1, f o r a l l x e M p 1 8 F T r D ( u ( t ) ) l 8 F* = u ( t ) 8 A ( t ) , f o r a l l t e G . p 2 Now, l e t U be t h e u n i t a r y o p e r a t o r on L (G;f/) g i v e n by ( U ? ) ( t ) = u(t)£(t) f o r K e L2(G;H) . Then U 1 8 F TT„(X)1 8 F U = TT (x) f o r x e M II* 1 8 F ir„(u(t))l 8 F*U = A ( t ) f o r t e G . 3 a ft ft Set K(y) = U 1 8 F T r R ( y ) l 8 F U f o r y e N . We have K3 (y) = a ( K V ) , f o r a l l y e N, p e G P P P r o p o s i t i o n 1.7 y i e l d s t h e b i d u a l theorem. Theorem 1.8 (The B i d u a l Theorem). L e t {M,a} be a c o v a r i a n t system - 11 -2 o v e r G . L e t 3 be t h e c o n t i n u o u s a c t i o n o f G on M 8 B ( L CG)) d e f i n e d by 3 = « t 8 a d X ( - t ) (Here adu(x) = uxu ) ft ft * 2 {W {W {M,a},a},a} i s i s o m o r p h i c t o {M 8 B(L (G)),3> Then 2 P r o o f : S i n c e P a ( G ) and ^ a ( G ) g e n e r a t e 1 8 B(L (G)) i t f o l l o w s t h a t 7ra(M), x a ( G ) a n d y a ( G ) t o g e t h e r g e n e r a t e M 8 B ( L 2 ( G ) ) .' That i s , w"{M,a}_ and y (G) g e n e r a t e M 8 B ( L 2 ( G ) ) . We a l s o have 3 (x) = x, f o r a l l t e G, x e W {M,a} i B t ( y a ( p ) ) = <P,t> y Q ( p ) . f o r a l l p e G, t e G . A M o r e o v e r , p > ady (p) g i v e s a c o n t i n u o u s a c t i o n o f G on W {M,a} Namely, ^ A A ' * a ^ ( x ) = y a ( p ) x y a ( p ) , f o r a l l x e W {M,a}, p e G . 2 By p r o p o s i t i o n 1.7, {M 8 B ( L ( G ) ) , 3 } i s i s o m o r p h i c t o ft ft >N /\ {W {W {M,a},d},a} . || Remark: The B i d u a l theorem says t h a t t h e b i d u a l c o v a r i a n t s y s t e m i s w e a k l y e q u i v a l e n t t o {M 8 B ( L 2 ( G ) ) , a 8 i d } . I n c a s e M i s p r o p e r l y i n f i n i t e and G i s second c o u n t a b l e , T a k e s a k i 2 has shown (see [2] lemma 4.7) t h a t {M 8 B(L ( G ) ) , a 8 i d } i s w e a k l y e q u i v a l e n t t o {M,a} . Theorem 1.9. L e t {M,a} be a c o v a r i a n t system o v e r a second c o u n t a b l e - 12 -group G . I f M i s p r o p e r l y I n f i n i t e t h e n t h e b i d u a l c o v a r i a n t I ft * ~ ~ .. s y s t e m {W {W {M,a},a},a} i s w e a k l y e q u i v a l e n t t o {M,a} . [| We now p r e p a r e f o r t h e s t r o n g v e r s i o n o f t h e c h a r a c t e r i z a t i o n o f c r o s s e d p r o d u c t s . P r o p o s i t i o n 1.10. .Let {N,g} be a c o v a r i a n t system o v e r G . Suppose t —>- u ( t ) i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n o f G i n N such t h a t 6 p ( u ( t ) ) = <p,t> u ( t ) , f o r a l l t e G, p e G . Then N i s g e n e r a t e d by N and u(G) . P r o o f : We may assume t h a t 3 i s u n i t a r i l y implemented by p —>• V(p) ft on t h e H i l b e r t space H f o r N . We have V ( p ) u ( t ) V ( p ) = <p,t> u ( t ) , f o r a l l t e G, p e G . We want t o show t h a t N = {[N D V ( G ) ' ] u u ( G ) } " . E q u i v a l e n t l y , • N' = [N* u V ( G ) ] " n u ( G ) ' . ft Now, x — • u ( t ) x u ( t ) , d e f i n e s a c o n t i n u o u s a c t i o n 6 of G on t h e W*-algebra [N1 u V ( G ) ] " such t h a t < S t ( x ) = x, f o r a l l x e N', t e G <$ t(V(p)*) = <pTt> V ( p ) * , f o r a l l p e G, t e G . ft M o r e o v e r , x —*• V(p) xV(p) d e f i n e s a c o n t i n u o u s a c t i o n a of G on ft N' . We a p p l y p r o p o s i t i o n 1.7 t o c o n c l u d e t h a t {W {N',a},ct} i s i s o m o r p h i c t o {[N' u V(G)]",<5} . The i s o m o r p h i s m c a r r i e s ^ ( x ) t o - 13 -x f o r x e N' and ^ ( P ) t o V(p) f o r p e G . By theorem 1.4 ( t h e u n i q u e n e s s theorem) W*{N',a} a = TT (N') o r [N* u V ( G ) ] " n u ( G ) ' = N' . || The c h a r a c t e r i z a t i o n theorem f o r c r o s s e d p r o d u c t s now f o l l o w s . Theorem 1.11 (The c h a r a c t e r i z a t i o n t h e o r e m ) . L e t (N,3) be a c o v a r i a n t s ystem o v e r G . Suppose t h a t t —>• u ( t ) i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n o f G i n N s u c h t h a t g ^ ( u ( t ) ) =<p,t> u ( t ) , f o r a l l t e G, p e G . L e t M = = {x e N: 3 (x) = x, f o r a l l P * p e G} . Then a f c ( x ) = u ( t ) x u ( t ) f o r x e M, t e G, d e f i n e s a c o n t i n u o u s a c t i o n o f G on M and t h e r e i s an i s o m o r p h i s m K of {N,3> w i t h {W {M,a},a} s u c h t h a t K x = TT (x) , f o r a l l x e M a K u ( t ) = X ( t ) , f o r a l l t e G . a P r o o f : I n v i e w of p r o p o s i t i o n s 1.7 and 1.10 we need o n l y v e r i f y t h a t u ( t ) M u ( t ) = M f o r a l l t e G . But f o r x e M, t e G and p e G 3 ( u ( t ) x u ( t ) * ) = B ( u ( t ) ) x B ( u ( t ) * ) p P P * = <p,t> u ( t ) x u ( t ) <p,t> * II = u ( t ) x u ( t ) . || We have t h e f o l l o w i n g s p e c i a l c a s e as a c o r o l l a r y . - 14 -C o r o l l a r y 1.12. L e t {M,a} be a c o v a r i a n t system o v e r G . Suppose t h a t p —> u( p ) i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n of G i n t h e c e n t r e o f M such t h a t o t t ( u ( p ) ) = <p,t> u ( p ) f o r a l l t e G - oo and a l l p e G . L e t xp> f ° r P e G, denote t h e c h a r a c t e r i n L (G) g i v e n by X p ( s ) = < P > S > • L e t t —>• denote t h e c o n t i n u o u s a c t i o n of G on L°°(G) g i v e n by ( a t ( f ) ) ( s ) = f ( s - t ) f o r f e L°°(G) . Then t h e r e i s an i s o m o r p h i s m K of {M,ct} w i t h {M 8 L ( G ) , i d 8 a} s u c h t h a t a K x = x 8 1 f o r x e M K u ( p ) = 1 8 x p f ° r p e G . P r o o f : S i n c e u ( p ) i s i n t h e c e n t r e o f M, t h e a c t i o n 3 o f G on Ct M g i v e n by u ( p ) i s t r i v i a l . By theorem 1.11 t h e r e i s an i s o m o r p h i s m K of {M,a} ont o {W*{M°\3),3} such t h a t K-x = TT 0(X) f o r a l l x e M01 1 3 K n u ( p ) = X D ( p ) f o r a l l p e G . 1 p ct S i n c e 3 i s t h e t r i v i a l a c t i o n , ^QCX) = x 8 1 f o r x e M . Moreover 3 (XQ(p)£)(q) = 5(q-p) f o r E, e L2(G;H) (as u s u a l , H i s t h e H i l b e r t space o f M). U s i n g t h e F o u r i e r t r a n s f o r m 2 " 2 mapping L (G) —> L (G) we o b t a i n an i s o m o r p h i s m K of M i n t o rv oo M 8 L (G) such t h a t ct K x = x 8 1, f o r a l l x e M K u ( p ) = 1 8 Xp» f ° r a H p e G . - 15 -S i n c e (x p: p e G} g e n e r a t e s L ( G ) , K i s onto. M o r e o v e r , K ° a o K ^ = id 8 0, f o r t e G . II - 16 -2. F l o w b u i l t under a c e i l i n g We f i r s t r e v i e w t h e c l a s s i c a l " f l o w under a f u n c t i o n c o n s t r u c t i o n (see [3] and [ 4 ] ) . L e t (fi,A,y) be a c o m p l e t e o - f i n i t e measure s p a c e . L e t T: fi —>- fi be a b i m e a s u r a b l e b i j e c t i o n s u c h t h a t y ° T i s e q u i v a l e n t t o u . L e t 9: fi — > • [0,°°) be A m e a s u r a b l e and assume t h a t t h e r e i s a p a r t i t i o n oo fi = u fi of fi i n t o T i n v a r i a n t m e a s u r a b l e s e t s and numbers e > 0 - i n n n= l f o r each n such t h a t d>(w) > e f ° r to e fi and n = 1,2,3,... — n n We denote by (R,L,m), Lebesgue measure on (R . Set fia = {(o),s) e f i x R : 0 < s < <Kto)} . <P — L e t ( f i , , A , , p . ) be t h e c o m p l e t i o n o f t h e r e s t r i c t i o n of A x L t o fi, (f) 9 9 <P w i t h r e s p e c t t o y x m . Note t h a t f o r each to e fi and r >_ 0, t h e r e i s a u n i q u e i n t e g e r n >^  0 such t h a t 9(w) + <J>(Tio) + . . . + 9 ( T n _ 1 w ) £ r < 9 ( 1 0 ) + <j>(Tu)) + ... + <J>(Tnco) . S i m i l a r l y , f o r each u e fi and r < 0, t h e r e i s a u n i q u e i n t e g e r n < 0 such t h a t -9 (T oi) - <j>(T to) £ r < -<j)(T co) - . . . - 9(T to) . So, i f we s e t * n ( o > ) = \ 9(co) + . . . + 9 ( T n 1to) f o r n > 0 0 f o r n = 0 [ - 9 ( T _ 1 o ) ) - ... - 9(T nto) f o r n < 0 th e n f o r each (w,r) e fi x El, t h e r e i s a u n i q u e i n t e g e r n such t h a t - 17 -(T nto, r - 4> (to)) e fi, . T h i s o b s e r v a t i o n a l l o w s us t o d e f i n e , f o r e ach n 9 T ch t e IR, a mapping W ^ of fi^ by: W T , ( ! >(a),s) = ( T V S + t - <f> (oi)) t n where n i s t h e u n i q u e i n t e g e r such t h a t (T nto, s + t - 9 (to)) e fi, . n 9 The mappings W T^ s a t i s f y t h e f o l l o w i n g p r o p e r t i e s : (See [3] and [ 4 ] ) : X (i) 1) f o r each t e R, W i s a b i m e a s u r a b l e b i i e c t i o n of fi, ' t J 9 X cb s u c h t h a t p, « W ' i s e q u i v a l e n t t o p, 9 t 9 2) f o r each s , t e R, W^ '* = W1'* . W1'* t+s t s 3) i f we e q u i p fi, * IR w i t h t h e c o m p l e t i o n o f A± x L w i t h r e s p e c t t o p. x m t h e n (to,s,t) —*• vF'^(u>,s) i s a m e a s u r a b l e 9 t mapping. X d) The a c t i o n of IR on fi^ g i v e n by t —*• W ^ i s c a l l e d t h e f l o w b u i l t on t h e t r a n s f o r m a t i o n T, under t h e c e i l i n g f u n c t i o n <f> . Theorem 2 of [3] and theorem 3.1 of [4] g i v e c o n d i t i o n s under w h i c h an a c t i o n o f IR on a measure space may be r e a l i z e d i n t h i s way. I n t h e a p p e n d i x we show t h a t t h e arguments of [3] and [4] a r e v a l i d under weaker c o n d i t i o n s . Our p u r p o s e i n t h i s s e c t i o n i s f i r s t l y t o r e c o g n i z e t h e " f l o w under a f u n c t i o n " c o n s t r u c t i o n i n o p e r a t o r a l g e b r a i c t e r m s , s e c o n d l y t o g e n e r a l i z e the c o n s t r u c t i o n t o n o n - a b e l i a n W - a l g e b r a s and t h i r d l y , t o g i v e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s under w h i c h a c o v a r i a n t s y s t e m o v e r IR may be r e a l i z e d i n t h i s way. X ob Note t h a t p r o p e r t i e s 1 ) , 2) and 3) of W ^ a l l o w us t o d e f i n e a * oo c o n t i n u o u s a c t i o n a of IR on t h e a b e l i a n W a l g e b r a L ( p ^ ) by: - 18 -a t ( f ) = f o W ^ * , f o r f e L°(y ) . We a l s o have t h e a c t i o n 0 of Z ( t h e i n t e g e r s ) on t h e a b e l i a n oo a l g e b r a L (y) d e f i n e d by: 0 ( f ) = f • T' 1, f o r f e L°°(y) . CO We would l i k e t o v i e w the c o v a r i a n t system {L (y ),a} as a r i s i n g f r o m <P oo {L (y),6} w i t h o u t r e f e r e n c e t o t h e measure s p a c e s . F o r t h i s we i n t r o d u c e some n o t a t i o n . L e t t —> denote t h e c o n t i n u o u s a c t i o n CO oo of (R on L (m) (= L (JR.)) g i v e n by CO (a f ) ( s ) = f ( s - t ) , f o r f e L (m) . f i r s 0 0 F o r s e IR, l e t x g he t h e c h a r a c t e r X g ( r ) = e i n L (m) . ~ 00 oo Lemma 2.1. There i s a ( u n i q u e ) automorphism 6 of L (y) 8 L (m) s a t i s f y i n g : ~ 00 ( i ) 0 ( x 8 1) = Gx 8 1, f o r a l l x e L (y) ( i i ) 6(1 8 x ) = 6(e 1 S < f') 8 x » f o r a l l s e R . s s oo The c o v a r i a n t s y s t e m {L (y ),a} i s i s o m o r p h i c t o t h e r e s t r i c t i o n o f OO 00 ~ i d 8 a t o t h e f i x e d s u b a l g e b r a of L (y) 8 L (m) under 8 . P r o o f : L e t (Q, x IR, A x L, y x m ) d e n o t e t h e c o m p l e t i o n of A x L w i t h r e s p e c t :to t h e measure y x m . D e f i n e a mapping T of 9. x (R by T(u),r) = (To), r - $(<*>)) • - 19 -Then T i s a bimeasurable b i j e c t i o n and y x m » T i s equivalent to _ ~ CO — y x m,. This gives an automorphism 0 of L (u x m) by ~ _ 1 oo 0f = f ° T for f e L (y x m ) . CO — oo CO Identifying L (y x m ) with L (y) 8 L (m) y i e l d s properties ( i ) and ( i i ) of 0 . Note that the sets T n f i ^ for n e Z are d i s j o i n t and 00 t h e i r union i s Q x (R . Hence L (y.) i s isomorphic to the fi x e d 00 — ~ subalgebra of L (y x m) under '0 . Moreover, under t h i s i d e n t i f i c a t i o n , a corresponds to the r e s t r i c t i o n of i d 8 . || We now propose to take the conclusion of lemma 2.1 as the d e f i n i t i o n * of "flow under a function" f o r non-abelian W algebras. For the existence of 0 we need Lemma 2.2. Let G and H be l o c a l l y compact abelian groups. Let {M,a} be a covariant system over G and l e t (g,q) —• v(g,q) be a strongly continuous map from G x H into the unitary operators i n the centre of M s a t i s f y i n g : ( i ) v(g 1+g 2,q) = v ( g 1 , q ) a g ( v ( g 2 > q ) ) f o r a l l g±,g2 e G and q e H . ( i i ) v(g,q 1+q 2) = v(g,q ] L)v(g,q 2) f o r a l l g e G and q i 5 q 2 e H . ~ CO Then there i s a (unique) continuous action a of G on M 8 L (H) such that - 2 0 -a (x 8 1 ) = a x 8 1 , f o r a l l g e G, x e M g g a ( 1 8 x ) = v ( g , q ) 8 x f o r a l l g e G, q e H (where x i s the c h a r a c t e r X q ( h ) = <q,h> ) . Mo r e o v e r , i f L(H) ft 2 d e n o t e s t h e W a l g e b r a on L (H) g e n e r a t e d by s h i f t t h e n t h e r e i s a s t r o n g l y c o n t i n u o u s map g —>• v of G i n t o t h e u n i t a r y o p e r a t o r s i n t h e c e n t r e o f M 8 L(H) such t h a t V * 2 = V g i 0 i d ( V ' f o r a 1 1 § 1 ' S 2 £ G ft oo a (x) = v a 8 i d ( x ) v , f o r a l l g e G, x e M 8 L (H) g g g g * 2 " P r o o f : L(H) i s t h e W a l g e b r a on L (H) g e n e r a t e d by t h e u n i t a r y o p e r a t o r s A(q) f o r q e H where ( X ( q ) O ( p ) = S(p-q) f o r £ £ i . 2 (H) . 2 2 " The F o u r i e r t r a n s f o r m F: L (H) —> L (H) c a r r i e s Xq t o ^(q)» i . e . ft ft CO ~ F x q F = A(q) . We a l s o have F L ( H ) F = L (H) . Hence, i t s u f f i c e s t o e x h i b i t a s t r o n g l y c o n t i n u o u s map g —> w of G i n t o t h e u n i t a r i e s i n t h e c e n t r e o f M 8 L (H) such t h a t w = w a 8 i d ( w ) , f o r a l l e , , g 0 e G S l 8 2 g l 8 1 G 2 w ( 1 8 A(q))w* = v ( g , q ) 8 A ( q ) , f o r a l l g e G, q e H . o o 00 ^ F o r t h i s we assume t h a t M a c t s on n . Then M 8 L (H) a c t s on 2 " L (H;hO . S i n c e (g,q) v ( g , q ) i s s t r o n g l y c o n t i n u o u s we can d e f i n e CO ^ w i n t h e c e n t r e o f M 8 L (H) by: g (w„?)(p) = v ( g , P K ( p ) f o r Z £ L 2 ( H , H ) . - 21 -I t f o l l o w s t h a t w i s u n i t a r y and g —> w i s s t r o n g l y c o n t i n u o u s . The p r o p e r t i e s s t a t e d above f o r w^ a r e s a t i s f i e d . || We now d e f i n e " f l o w under a f u n c t i o n " i n g e n e r a l . D e f i n i t i o n 2.3. A 1) L e t 6 be an automorphism of a W a l g e b r a N . L e t <J> be a p o s i t i v e s e l f - a d j o i n t o p e r a t o r a f f i l i a t e d t o t h e c e n t r e of N . 9 i s c a l l e d a 6 c e i l i n g ( o r j u s t a c e i l i n g i f 6 i s u n d e r s t o o d ) i f f t h e r e i s a p a r t i t i o n o f u n i t y { e ^ : 1 e I ) i n t h e c e n t r e o f N and numbers e. > 0 f o r i e I s u c h t h a t l 9 ( e.) = e. , f o r a l l i e I l x me. > e.e., f o r a l l i e I . x — x x ~ oo 2) I f 9 i s a 0 c e i l i n g , l e t 0 be t h e automorphism of N 8 L (R) ( g i v e n by lemma 2.2) w h i c h s a t i s f i e s : 0(x 8 1) = 0x 8 1, f o r a l l x e N 0(1 8 x ) = 0 ( e 1 S < f l ) 8 x , f o r a l l s e R . s s 0 0 0 Set M = [N 8 L (R) ] ( t h e f i x e d a l g e b r a ) and f o r x e M s e t a f c ( x ) = i d 8 a t ( x ) , f o r a l l t e R . The c o n t i n u o u s a c t i o n a o f R on M i s c a l l e d t h e f l o w b u i l t on t h e automorphism 0 under t h e c e i l i n g 9 . We n e x t show t h a t {N,0} i s d e t e r m i n e d by {M,a} t o g e t h e r w i t h a map - 22,-s , t —>• v ( s , t ) of [0,2TT) x R i n t o t h e u n i t a r i e s i n t h e c e n t r e o f M . F o r t h i s we i n t r o d u c e t h e f o l l o w i n g n o t a t i o n : <5 i s t h e automorphism CO CO of I (Z) d e f i n e d by (6f)(n) = f ( n - l ) f o r f e I (2) . F o r oo i n s 0 < s < 2TT, v i s t h e c h a r a c t e r ( i n £ GZ)) d e f i n e d by v (n) = e — s s Lemma 2.4 ( R e v e r s a l lemma). Suppose {M,a} i s t h e f l o w b u i l t on ~ 00 {N,0} under t h e c e i l i n g 9 . L e t 0 be t h e automorphism of N 8 L (R) as i n d e f i n i t i o n 2.3. There e x i s t s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n s — u s °^ ( t h e group) [0,2TT) i n t h e c e n t r e of CO N 8 L (R) and a f a m i l y {h^: t e R} of s e l f - a d j o i n t o p e r a t o r s a f f i l i a t e d t o t h e c e n t r e of M s u c h t h a t ( i ) 0 ( u g ) = e 1 S u s , f o r a l l s e [0,2TT) ( i i ) spec h f c c z f o r a l l t e R and h t >_ 0 i f t >_ 0 i s h ( i i i ) i d 8 ^ ( u ^ , ) = e u g , f o r a l l s e [0,2TT), t e R . ~ 00 ( i v ) i f a d e n o t e s t h e c o n t i n u o u s a c t i o n of R on M 8 SL (£) g i v e n by lemma 2.2 a p p l i e d t o {M,a} and t h e map i s h t , s —>• e , t h e n t h e r e i s an i s o m o r p h i s m TT o f N 8 L (R) w i t h M 8 l (Z) s u c h t h a t TT(X) = x 8 1, f o r a l l x e M TT(U ) = 1 8 v , f o r a l l s e [0,2ir) s s IT » 9 « IT ^ = i d 8 5 1 o i d 8 o"t « TT ^ = a^, f o r a l l t e R (v) {N,0} i s i s o m o r p h i c t o t h e r e s t r i c t i o n o f i d 8 6 t o t h e f i x e d s u b a l g e b r a under a . - 23 -P r o o f : We f i r s t show t h a t i t s u f f i c e s t o p r o v e t h e e x i s t e n c e o f s —*• u g and t —> h s a t i s f y i n g ( i ) , ( i i ) and ( i i i ) i n cas e t h e c e n t r e o f N i s a - f i n i t e . To see t h i s , n o t e t h a t i f p i s a De-f i n i t e p r o j e c t i o n i n t h e c e n t r e o f N t h e n q = \/ 0 n ( p ) i s a-neZ f i n i t e and 0 i n v a r i a n t . Hence q 8 1 i s i n t h e c e n t r e o f M and i s a i n v a r i a n t . M o r e o v e r , t h e r e s t r i c t i o n o f a t o M i s t h e ' q81 f l o w b u i l t on t h e r e s t r i c t i o n , o f 0 t o under t h e c e i l i n g <j)q • So, by c h o o s i n g a p a r t i t i o n o f u n i t y i n t h e c e n t r e o f N c o n s i s t i n g of a - f i n i t e 0 i n v a r i a n t p r o j e c t i o n s , t h e g e n e r a l c a s e f o l l o w s f r o m t h e a - f i n i t e c a s e . Now suppose t h a t t h e c e n t r e Z of N i s a - f i n i t e . We can f i n d a c o m plete a - f i n i t e measure space (ft , A,y) and a b i m e a s u r a b l e b i j e c t i o n T o f 2 w i t h u o T e q u i v a l e n t t o y so t h a t Z i s i s o m o r p h i c t o 00 L (u) and under t h i s i d e n t i f i c a t i o n 0 i s g i v e n by 0f = f o T - 1 f o r f e L°°(y) . We may a l s o assume t h a t <J> c o r r e s p o n d s t o a m e a s u r a b l e f u n c t i o n CO co —*- <j>(u>) and t h e r e i s a p a r t i t i o n tt = u tt of tt i n t o m e a s u r a b l e n = l s e t s and numbers e > 0 such t h a t <Kco) > e f o r n e tt n — n n CO As i n t h e p r o o f o f lemma 2.1, Z 8 L (R) i s i d e n t i f i e d w i t h 00 — ~ ~ L (y x m ) and 0 i s g i v e n by T : f i x R — > f i x ( R where T ( u , r ) = (To), r - <Kw)) • Now s e t fi^ = {(o),r) e tt x R: 0 <_ r < <))(CJ) } . Then tt x |R = u T n ( f i J neZ * - 24 -i s a p a r t i t i o n o f fi x R i n t o A x L m e a s u r a b l e s e t s . F o r ~k 0 < s < 2TT, k e g and (o),r) e T ( f i x ) s e t - 9 / s i k s u (to.r) = e s ~k ~k+n Fo r t e R, k,n e Z, (w,r) e T ( f i , ) and ( w , r - t ) e T (Qj s e t 9 9 h t ( u , r ) •= n . ~ - l - i s Note t h a t u • T = e u , f o r a l l s e [0,2IT) . A l s o , h i s s s ' t i n t e g e r v a l u e d , non n e g a t i v e i f t >_ 0 and n t ° ^ ^ = n t ^ o r a ^ t £ R . We have, f o r a l l s e [0,2T T), t e R and w,r e fi x (R i s h t ( c o , r ) u (cu,r-t) = e u (w,r) . s s I n terms o f automorphisms we have 0 ( u ) = e " 1 S u , f o r a l l s e [0,2TT) s s i s h ^ i d 8 o ( u g ) = e u s , f o r a l l s e [0,2T T), t e R Hence, p a r t s ( i ) , ( i i ) and ( i i i ) a r e p r o v e n . P a r t ( i v ) now f o l l o w s by CO ~ a p p l y i n g c o r o l l a r y 1.12 t o N 8 L (R) w i t h t h e a c t i o n 6 and t h e r e p r e s e n t a t i o n s —>- u g . P a r t (v) s i m p l y s t a t e s t h a t oo -i A 8("T II [N 0 L ( R ) ] l a B ° = N 8 1 . || Remark: T h i s lemma shows t h a t c e n t r e [N 8 L (R)] e q u a l s oo 9 [ ( c e n t r e N) 8 L (R)] s i n c e b o t h t h e s e s u b a l g e b r a s c o r r e s p o n d t o ( c e n t r e M) 8 1 under TT . We now use lemma 2.4 t o p r o v e a " l i f t i n g lemma". - 25 -Lemma 2.5. L e t {M,a} be a c o v a r i a n t s y s t e m o v e r |R . L e t be ft a W - s u b a l g e b r a o f M such t h a t a^(M^) = M^, f o r a l l t e IR and c e n t r e c c e n t r e M . Set c t ^ ( x ) = a t ( x ) , f o r a l l t e IR, x e Suppose {M^,a~'"} i s i s o m o r p h i c t o t h e f l o w b u i l t on { N ^ , 0 ^ } under cf>^ . Then t h e r e i s an imbedding K of { N ^ , 0 ^ } i n t o a c o v a r i a n t s y s t e m { N , 0 } o v e r Z such t h a t c e n t r e KN^ G c e n t r e N and {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on { N , 0 } under <f> = Kef) . P r o o f : U s i n g lemma 2.4 we o b t a i n a f a m i l y .{h s t e R} of s e l f a d j o i n t o p e r a t o r s a f f i l i a t e d t o t h e c e n t r e o f w i t h spec h t c £ CO CO and an is o m o r p h i s m TT^  o f 8 L (IR) w i t h 8 I (E) such t h a t , i n t h e n o t a t i o n o f lemma 2.4: ^1 ° ^ 1 ° ^l"^ = ^ 8 6 TT^  o i d 8 a f c o -n^~ = a^, f o r a l l t e IR . i s h ^ S i n c e c e n t r e c c e n t r e M and s i n c e t h e map; t , s —>- e s a t i s f i e s ~1 CO t h e c o n d i t i o n s o f lemma 2.2, we can e x t e n d a t o a on M 8 SL (£) s a t i s f y i n g : a (x 8 1) = ci (x) 8 1, f o r a l l x e M t t ' i s h a (1 0 v g ) = e 8 v g , f o r a l l t e IR, s e [0,2TT). L e t N = [M 0 &°°(£)]a and f o r x e N s e t 0 ( x ) = i d 0 < 5(x) . Set KX = TT^(X 0 1 ) " f o r x e . Then K i s an imbedding o f { N ^ , 0 ^ } and ~1 CO 0£ CO QJ c e n t r e KN^ C [ ( c e n t r e M^) 8 I ( Z ) ] c I ( c e n t r e M) 8 % ( Z ) ] c c e n t r e N . To c o n c l u d e t h e p r o o f , s e t v r = TT^(1 8 x r ) f ° r r e IR . Then r —* i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n o f (R i n t h e CO c e n t r e o f M 8 H (Z) and — i r t a (v ) = e v , f o r a l l t , r e IR t r r ' ir<f>1 i d 0 6(v ) = K o e ( e ) v , f o r a l l r e (R . r 1 r 00 By c o r o l l a r y 1.12 t h e r e i s an i s o m o r p h i s m TT of M 0 I (Z) w i t h N 8 L°°(R) such t h a t : TT o ot^ _ ° TT ^ = i d 8 o^, f o r a l l t e (R n • i d 0 { ° n ^ = 0 where i ( x 8 1) = 6x 8 1, f o r a l l x e N 1(1 8 x r ) = 6 ( e i r < 1 > ) 8 xr> f o r a l l r e IR Thus, {M,a} i s i s o m o r p h i c t o t h e r e s t r i c t i o n o f i d 8 a t o t h e 00 ~ II f i x e d s u b a l g e b r a o f N 8 L (IR) under 0 . || The main r e s u l t o f t h i s s e c t i o n i s Theorem 2.6. A c o v a r i a n t system {M,a} o v e r IR i s i s o m o r p h i c t o a f l o w b u i l t under a c e i l i n g i f f t h e r e s t r i c t i o n of a t o t h e c e n t r e o f M i s nowhere t r i v i a l ( i . e . i f e i s a p r o j e c t i o n i n t h e c e n t r e o f M such t h a t a t ( x e ) = xe f o r a l l x i n t h e c e n t r e o f M t h e n e = 0) . - 27 -P r o o f : Assume t h a t {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,0} CO under t h e c e i l i n g 9 . The e x i s t e n c e o f an i s o m o r p h i s m of N 8 L (E) CO w i t h M 8 £ (Z) as i n lemma 2.4 shows t h a t t h e r e s t r i c t i o n o f a t o the c e n t r e o f M i s i s o m o r p h i c t o the f l o w b u i l t on t h e r e s t r i c t i o n o f 0 t o t h e c e n t r e o f N under t h e c e i l i n g 9 . Hence i t s u f f i c e s t o show t h a t when N i s a b e l i a n , t h e f l o w b u i l t on {N,0} under 9 i s CO nowhere t r i v i a l . Now l e t e be a p r o j e c t i o n i n N 8 L (IR) such t h a t 00 0 0(e) = e and i d 8 o^xe) = xe f o r a l l x i n [N 8 L (R) ] . Then e i s o f t h e form f 8 1 f o r f e N and t h e f l o w b u i l t on t h e r e s t r i c t i o n o f 0 t o under <j>f i s t r i v i a l . Hence i t s u f f i c e s t o show t h a t when N i s a b e l i a n t h e f l o w b u i l t on {N,0} under 9 i s n o t t h e t r i v i a l f l o w . S i m i l a r l y , i t s u f f i c e s t o show t h a t when N i s a b e l i a n and a-f i n i t e , t h e f l o w b u i l t on {N,0} under 9 i s n o t t r i v i a l . I n t h i s c a s e , we choose a c o m p l e t e a - f i n i t e measure space ( f i , A , y ) and a b i -m e a s u r a b l e b i j e c t i o n T of fi w i t h y ° T e q u i v a l e n t t o y s u c h 00 -1 t h a t N i s i s o m o r p h i c t o L (y) and 0 i s g i v e n by 0f = f 0 T CO f o r f e L (y) . Lemma 2.1 now shows t h a t t h e f l o w b u i l t on {N,0} X d) under 9 i s i s o m o r p h i c t o t h e f l o w g i v e n by t h e a c t i o n t — W ^ of R on fi^ = { ( C J , S ) e fi x IR: 0 < _ s < 9 ( 0 0 ) } . I t i s easy t o see t h a t t h i s f l o w i s n o n - t r i v i a l . F o r the c o n v e r s e , lemma 2.5 shows t h a t i t s u f f i c e s t o assume t h a t M i s a b e l i a n . We may a l s o assume t h a t M i s a - f i n i t e . To see t h i s , l e t p be a a - f i n i t e p r o j e c t i o n i n M . Then e = V a t . ( p ) (Q i s t h e r a t i o n a l s ) teiq i s a - f i n i t e and a i n v a r i a n t . Now l e t ( e . : i e 1} be a p a r t i t i o n - 28 -of u n i t y i n M c o n s i s t i n g o f a - f i n i t e a i n v a r i a n t p r o j e c t i o n s . We assume t h a t f o r each i e I , t h e r e s t r i c t i o n o f a t o Me. i s i s o m o r p h i c t o t h e f l o w b u i l t on {N.,0.} under d>. . Set N = y N. l e l 0x = T 6.(x.) f o r x = J. x. e N lei 1 1 i € l 1 * = I *± • i e l Then {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,6} under ty . To c o n c l u d e t h e p r o o f we r e f e r t o t h e a p p e n d i x where t h e r e s u l t i s p r o v e n f o r t h e c a s e M a b e l i a n and a - f i n i t e . || We now c h a r a c t e r i z e t h o s e a c t i o n s w h i c h may be r e a l i z e d as a f l o w under a c o n s t a n t c e i l i n g . Theorem 2.7. A c o v a r i a n t s y s t e m {M,a} o v e r R i s i s o m o r p h i c t o a f l o w b u i l t under a c o n s t a n t c e i l i n g o f h e i g h t c i f f t h e r e i s a u n i t a r y u i n t h e c e n t r e o f M such t h a t - i t ( f ) a t ( u ) = e u, f o r a l l t e IR . P r o o f : L e t 0 be an automorphism of N and l e t 0 be t h e automorphism of N 8 L°°(IR) s a t i s f y i n g 0 ( x 8 1) = 0x 8 1, f o r a l l x e N 0(1 8 x ) = e 1 S C 8 xs, f o r a l l s e IR . Then u = 1 8 v„ i s f i x e d by 0 and 2TT J . 2TT - i t -i d 8 a t ( u ) = e ° u, f o r a l l t e (R . ft F o r t h e c o n v e r s e , l e t be t h e W - s u b a l g e b r a o f M g e n e r a t e d by u Then a t(M^) = M^, f o r a l l t e (R and c c e n t r e ' M . Hence, by lemma 2.5 i t s u f f i c e s t o show t h a t t h e r e s t r i c t i o n of a t o i s i s o m o r p h i c t o a f l o w b u i l t under t h e c o n s t a n t c e i l i n g c . S i n c e c s c(u) = u, we g e t a c o n t i n u o u s a c t i o n (3 of t h e group [0,c) (mod c) on by B t ( x ) = a t ( x ) , f o r a l l x e M^, 0 <_ t < c . CO By c o r o l l a r y 1.12 t h e r e i s an i s o m o r p h i s m o f w i t h L [0,c) w h i c h i s 2 TT / c c a r r i e s u t o t h e f u n c t i o n v ( s ) = e and c a r r i e s 3 t o t h e CO 00 u s u a l a c t i o n o f [0,c) on L [0,c) . But L [0,c) i s i s o m o r p h i c t o CO t h e f i x e d s u b a l g e b r a o f L (R) under a . T h i s c o m p o s i t e i s o m o r p h i s m c a r r i e s t o L (R) c and c a r r i e s t h e r e s t r i c t i o n o f a t o t h e r e s t r i c t i o n o f a . Now, t a k e N^ = <G, 0^ = i d e n t i t y . Then t h e r e s t r i c t i o n o f a t o i s i s o m o r p h i c t o t h e f l o w b u i l t on {N^,0^} under t h e c o n s t a n t c e i l i n g c . || - 30 -3. U n i q u e n e s s of F l o w Under a C e i l i n g L e t {M,a} be t h e f l o w b u i l t on {N,0} under t h e c e i l i n g ty . I n t h i s s e c t i o n we i n v e s t i g a t e the e x t e n t t o w h i c h t h e i s o m o r p h i s m c l a s s of {M,oc} d e t e r m i n e s {N,0} and cj> . We f i r s t e x h i b i t two ways of m o d i f y i n g {N,0} and ty so t h a t t h e r e s u l t i n g f l o w s a r e i s o m o r p h i c . Next we s t a t e and p r o v e our u n i q u e n e s s r e s u l t . F i n a l l y , we g i v e a u n i q u e n e s s r e s u l t i n t h e s p e c i a l c a s e of a c o n s t a n t c e i l i n g . Lemma 3.1. L e t 0 be an automorphism of N and ty a 0 c e i l i n g . Suppose f i s a s e l f - a d j o i n t o p e r a t o r a f f i l i a t e d t o t h e c e n t r e o f N such t h a t ty = ty + 0 ( f ) - f i s a l s o a c e i l i n g o p e r a t o r . Then t h e f l o w b u i l t on {N,0} under ty i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,0} under ty . ~ CO P r o o f . L e t 0 be t h e automorphism of N 8 L (IR) as i n d e f i n i t i o n 2.3. i s f S et v = 0(e ) 8 X f o r s £ R . S i n c e s s — i s t i d 8 a (v ) = e v , f o r a l l s , t £ R, t s s co by c o r o l l a r y 1.12, t h e r e i s an i s o m o r p h i s m K o f N 0 L (IR) w i t h CO N 8 L (R) s u c h t h a t K ( X 8 1) = x 8 1, f o r a l l x e N K ( V ) = 1 8 x 5 f ° r a l l s e R . s s I n p a r t i c u l a r K o i d 8 ° K = i d 8 a , f o r a l l t e R K ° 0 ° K _ 1 ( X 8 1) = 0x 8 1, f o r a l l x £ N - 31 -K O 6 ° K X ( 1 8 X ) = eCe""**) 8 x , f o r a l l s e R . s s Hence, K g i v e s an i s o m o r p h i s m between t h e f l o w b u i l t on {N,6} unde r 9 and t h e f l o w b u i l t on {N , 6 } under ^ . || Our second m o d i f i c a t i o n d e a l s w i t h " c u t down" automorphisms. F o r t h i s we i n t r o d u c e t h e n o t i o n o f r e c u r r e n t p r o j e c t i o n . D e f i n i t i o n 3.2. L e t 6 be an automorphism o f N and l e t e be a p r o j e c t i o n i n t h e c e n t r e of N . e i s s a i d t o be r e c u r r e n t i f f e £ V S n(e) and e <_ V 9n(e) . n<0 n>0 CO There i s a c a n o n i c a l way t o p a r t i t i o n e as e = ) e where each e . i n n n = l i s i n the c e n t r e o f N and s a t i s f i e s e(e;L) < e f o r n > 2 0 3 ( e )e = 0 f o r j = 1,2,...,n - 1 and 9 n ( e ) < e . — n n — CO r n We a l s o have e = I 9 (e ) (see [1] d e f i n i t i o n 5.4.1). I t f o l l o w s n = l n f r o m t h e p r o p e r t i e s o f {e : n = 1,2,...} t h a t { 9 3 ( e ) : n = 1,2,... ; n n j = 0 , l , 2 , . . . , n - 1} and {9^(6^): n = 1,2,..., j = l , 2 , . . . , n } a r e o r t h o g o n a l f a m i l i e s and . , 0 0 n - l . °° n V e n ( e ) = I I ehe) = I I ehe) . neZ n = l j=0 n = l j = l We can d e f i n e an automorphism 9 G of N g by 00 0 (x) = 7 9 n ( x e ) f o r x e N . e L , n e n=l - 32 -80 i s c a l l e d t h e c u t down o r r e d u c t i o n o f 8 t o N e e We can a l s o d e f i n e t h e c u t down o r r e d u c t i o n of a 0 c e i l i n g <j> by f e = 1(1 e"M(1>))en . n = l m=l I t i s easy t o see t h a t $ i s a 8 G c e i l i n g . Lemma 3.3. L e t 8 be an automorphism o f N and l e t (j) be a 8 c e i l i n g . L e t e be a r e c u r r e n t p r o j e c t i o n i n t h e c e n t r e o f N w i t h V 6 n ( e ) = 1 . Then t h e f l o w b u i l t on {N,9} under <J> i s i s o m o r p h i c neZ t o t h e f l o w b u i l t on {N , 6 } under i> e e e ~ CO P r o o f : L e t 0 be t h e automorphism of N 8 L (R) as i n d e f i n i t i o n 2.3, Then e 8 1 i s r e c u r r e n t f o r 0 . Mo r e o v e r , i f e = Y e i s t h e n= l CO c a n o n i c a l p a r t i t i o n o f e t h e n e 8 1 = Ye 8 1 i s t h e c a n o n i c a l 1 n n = l p a r t i t i o n o f e 8 1 . F o r x e N we have e ~ 0 0 ~ 0 ft1(x 0 1) = Y 0 n ( x e 8 1) = 8 (x) 8 1 e81 - n e n= l and f o r s e R we have 0 0 ~ i s < j ) 6 e 0 1 ( e 8 *s> = E E X 8 X 8> = G e ( e 6 ) 8 X s • n= l Hence t h e f l o w b u i l t on { N e ' 9 e ^ under <f>e i s i s o m o r p h i c t o t h e CO r e s t r i c t i o n o f i d 8 a t o t h e f i x e d s u b a l g e b r a o f N 8 L (R) under e81 . ~ e Now, i f x c [N 8 L (R)] t h e n x ( e 8 1) e [N 8 L (R) ] F u r t h e r m o r e , i f x ( e 8 1) = 0 the n 6 n ( x e 8 1 ) = x 8 n ( e ) 0 1 = 0 f o r - 33 -a l l n e I . S i n c e V Qn(e) = 1 , i t f o l l o w s t h a t x = 0 . neZ CO ( Hence, t h e map x —> x ( e 8 1) i s an i s o m o r p h i s m of [N 8 L (IR) ] 9 cn oo e o l i n t o [ N £ 8 L (IR) ] . To show t h a t t h i s map i s o n t o , l e t oo ®efli x e [ N £ 0 L (IR)] . S i n c e °° n-1 ekCe ) n = l k=0 we s e t oo n-1^ y = I I e k ( x e 8 l ) . n = l k=0 Then y ( e 8 1) = x and co n - l _ _ e(y) = I I e k ( x e 8 i ) n= l k = l oo _ oo n - l _ , = I 6 n ( x e 8 1) + I I 0 (xe 8 1) n = l n n = l k = l n = 9 e 0 1 ( x ) + I l'«**n 8 1 } n = l k = l = x + I I 0 R ( x e 8 1) n = l k = l Thus, x —> x ( e 8 1) i s o n t o . S i n c e t h i s i s o m o r p h i s m i n t e r t w i n e s t h e r e s t r i c t i o n s of i d 8 a t h e lemma i s p r o v e n . || The main r e s u l t o f t h i s s e c t i o n i s Theorem 3.4. The f l o w b u i l t on {N^,Q^} under o)^ i s i s o m o r p h i c t o t h e f l o w b u i l t on {^^^^2 u n c ^ e r $2 t h e r e e x i s t ( i ) r e c u r r e n t p r o j e c t i o n s e. i n t h e c e n t r e of N. w i t h 3 J V 9?(e.) = 1 for j = 1,2 neZ - 34 -( i i ) an i s o m o r p h i s m K o f ' {(N.) , ( 9 J } w i t h { ( N ) , ( 6 9 ) _ > 1 e l 1 e l 2 2 ( i i i ) a s e l f - a d j o i n t o p e r a t o r g a f f i l i a t e d t o t h e c e n t r e o f ( N , ) s u c h t h a t 1 e± (*,)_ = k(<f> 2) e + C6 ) (g) - g 1 e l 2 1 1 P r o o f : Lemmas 3.1 and 3.3 show t h a t t h e s t a t e d c o n d i t i o n s i m p l y t h a t t h e f l o w s a r e i s o m o r p h i c . F o r t h e c o n v e r s e we b e g i n w i t h Lemma 3.5. The f l o w b u i l t on {N^,8^} under <j>^  i s i s o m o r p h i c t o t h e f l o w b u i l t on {^,62} under $2 i f f t h e r e i s a W - a l g e b r a Q w i t h commuting automorphisms Y - ^ J Y 2 S U C N T N A T : ( i ) t h e r e a r e imbeddings TT^. o f {N_.,6_.} i n t o {Q,Yj} f o r j = 1,2 . Y Y ( i i ) T^OO = Q 2 and T T 2 (N 2 ) = Q 1 ( i i i ) t h e c e n t r e of ir (N_.) i s c o n t a i n e d i n t h e c e n t r e o f Q f o r j = 1,2 . ( i v ) t h e r e i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n r — • v of IR i n t h e c e n t r e o f Q such t h a t r Y - ^ v ^ = T r 1 ( e i e ) v r , f o r a l l r e d -ircf> 2 Y 2 ( v r ) = 7 1 2 ^ 8 2 e ) v r ' f o r a"'"1 r £ R • P r o o f : Assuming t h e c o n d i t i o n s a r e s a t i s f i e d , l e t y ( f o r j = 1,2) be t h e automorphism o f Q 8 L°°((R) s a t i s f y i n g : - 35 -Y (x 8 1) = y.x 8 1, f o r a l l x e Q j 3 ir<f>. Y (1 8 X ) = T,(e.e 3 ) 8 X > f o r a 1 1 r e E . j r 3 J r S e t u = v 8 x f o r r e d . Then r r r Y. (u ) = u , f o r a l l r e IR '1 r ' r Y 2 8 i d ( u r ) = TT 2(9 2(e ) ) u r > f o r a l l r e (R . oo By c o r o l l a r y 1.12 t h e r e i s an automorphism TT of Q 8 L (IR) s u c h t h a t TT(X 8 1) = x 8 1, f o r a l l x e Q TT(1 8 Y ) = u , f o r a l l r e (R . • A r r I n p a r t i c u l a r , T T ^ O i d 8 a • TF = i d 8 a , f o r a l l t e IR t t ' TT ^ o y ° TT = Y^ ® i d TT o 8 i d » it = Y 2 • Y 28id Y L Now, t h e r e s t r i c t i o n o f i d 0 a t o [ (Q 8 L (IR)) ] i s i s o m o r p h i c t o t h e f l o w b u i l t on ^ N j ' e i ^ under <j>^  . S i m i l a r l y , t h e r e s t r i c t i o n Y 8 i d Y 2 o f i d 8 a t o [(Q 8 L (IR)) ] i s i s o m o r p h i c t o t h e f l o w b u i l t on { N 2 , 0 2 } under d>2 . Hence, TT g i v e s an i s o m o r p h i s m between t h e two f l o w s . ~ 00 F o r t h e c o n v e r s e , l e t 6_. be t h e automorphism o f i 8 1 (R) w h i c h s a t i s f i e s : 3.(x 0 1) = 9.x 8 1, for a l l x e N. 3 CI 8 x r ) = 9.(e J) 8 X r> for a l l r e (R - 36 -We a p p l y t h e r e v e r s a l lemma (lemma 2.4) t o {N 2,0 2} a n ( ^ ^2 t C > 6, CO [ CO CO o b t a i n an i s o m o r p h i s m ir o f [N^ 0 L (R) ] 8 I (Z) w i t h N 2 8 L (R) and a f a m i l y {h : t e R} o f s e l f - a d j o i n t o p e r a t o r s a f f i l i a t e d t o t h e 00 * ± c e n t r e o f [N 8 L (R)] w i t h spec h f c c z f o r a l l t e R and h t > 0 f o r t >_ 0 such t h a t O 0 2 0 T T Q = i d 8 6 T T Q 1 O i d 8 c t o T T Q (x 8 1) = ( i d 0 o f c ( x ) ) 8 1, co eL f o r a l l x e [N 8 L (R)] , t e R -1 l s h t 1 = i d 8 ff • T T . ( 1 8 v ) = e 0 v , f o r a l l s e [ 0 , 2 I T ) , t £ R . 0 t 0 s s CO j 0 0 1 S i n c e c e n t r e [N 8 L (R) ] =• [ ( c e n t r e N.^ ) 8 L (R) ] we can use t h e i s h map t , s —> e t o d e f i n e a c o n t i n u o u s a c t i o n 3 of R on 00 00 N 8 L (R) 8 I (Z) (lemma 2.2) s a t i s f y i n g : CO B f c(x 8 1 ) = ( i d 8 a t ( x ) ) 8 1, f o r a l l x e N 8 L ( R ) , t e R i s h 3 (1 0 v g ) = e 0 v g , f o r a l l s e [ 0 , 2 I T ) , t e R . Note t h a t $ commutes w i t h b o t h 6^  0 i d and i d 0 i d 8 6 f o r a l l t £ R . We s e t CO CO ft Q = [N 8 L (R) 0 I ( Z ) ] P . F o r x £ N^, s e t T T ^ ( X ) = x 8 1 8 1 . F o r x e N 2 , s e t n 2 ( x ) = T Q ( X 8 1) L e t be t h e r e s t r i c t i o n o f 0^  8 i d t o Q and l e t be t h e r e s t r i c t i o n o f i d 0 i d 0 6 t o Q . F i n a l l y , f o r r e R s e t v = i r . ( l 0 x*H 8 X 8 1 . r 0 r r - 37 -I t i s easy t o v e r i f y t h a t t h e c o n d i t i o n s o f t h e lemma a r e s a t i s f i e d . Now, d e f i n e TT,Y-^ and Y 2 as i n t h e p r o o f o f lemma 3 . 5 . As p r e v i o u s l y y i oo y2 n o t e d , t h e r e s t r i c t i o n o f i d 8 a t o [Q 8 L (R) ] i s i s o m o r p h i c t o th e f l o w b u i l t ^2,02^ under ty^ • Hence, by t h e r e v e r s a l lemma (lemma 2.4) t h e r e i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n Y 1 oo 1^ oo s —> u g of [0,2TT) i n t h e c e n t r e o f Q 8 L (R) = ( c e n t r e Q) 8 L (R) such t h a t y (u ) = e u f o r a l l s e [0,2TT) . M o r e o v e r , f o r each z. s s i s h t t e R and s e [0,2-n), i d 8 a (u ) = e u , where h i s a s e l f -' t s s t T l a d j o i n t o p e r a t o r a f f i l i a t e d t o t h e c e n t r e o f Q 8 L (R) w i t h spec h^ c z and h > 0 i f t > 0 . We t h e r e f o r e have: ^ t t — — — i s Y2 8 i d ( i r ( u s ) ) = e T r ( u g ) , f o r a l l s e [0,2IT) Y^(TT(U s)) = T r ( u g ) , f o r a l l s e [0,2TT) i s h i d 8 a t ( T r ( u g ) ) = Tr(e ) T T ( u g ) , f o r a l l t e R, s e [0,2TT) . Now, l e t k be t h e s e l f - a d j o i n t o p e r a t o r w i t h spec k c £ a f f i l i a t e d 00 i s l e t o t h e c e n t r e o f Q 8 L (R) such t h a t T r ( u g ) = e , f o r a l l s e [0,2ir) Then Y 2 8 i d ( k ) = k - 1 Y x ( k ) = k i d 8 o t ( k ) >_ k f o r t >_ 0 . We s h a l l show t h a t t h e r e i s a s e l f - a d j o i n t o p e r a t o r k^ a f f i l i a t e d t o the c e n t r e o f Q w i t h spec k^ c E s u c h t h a t - 38 -W ^ k o • CO To show t h i s we l e t a be t h e automorphism o f Q 8 L (R) w h i c h s a t i s f i e s a ( x 8 1) = x 8 1, f o r a l l x e Q -ird> a ( l 0 x ) = T ^ O - ^ ) « x r , f o r a l l r e R . Now, a may be a p p r o x i m a t e d by automorphisms a o f t h e f o r m CO ax = J i d 8 a (TT (e ) 8 l x ) n = l c n 1 n where {e : n = 1,2,...} i s a p a r t i t i o n o f u n i t y i n t h e c e n t r e o f N, n 1 and t > 0 . Hence n — a ( k ) >_ k . S i n c e y^ ® i d = a y^ i t f o l l o w s t h a t y 8 i d ( k ) > k . To show t h e e x i s t e n c e of k^ we may assume as u s u a l t h a t t h e c e n t r e of Q i s a - f i n i t e . I n t h i s c a s e we can f i n d a c o m p l e t e a - f i n i t e measure space ( f i , A , y ) and commuting b i m e a s u r a b l e b i j e c t i o n s T^ and T 2 of Q, w i t h y o T e q u i v a l e n t t o y f o r j = 1 , 2 s u c h t h a t t h e CO c e n t r e o f Q i s i s o m o r p h i c t o L (u) and under t h i s i d e n t i f i c a t i o n — 1 CO CO Y j ( f ) = f ° T f o r f e L ( y ) , j = 1,2 . Hence ( c e n t r e Q) 8 L (R) CO i s i d e n t i f i e d w i t h L (y x m) (where R,L;m) i s Lebesgue.measure on R) . Under t h i s i d e n t i f i c a t i o n y 8 i d i s g i v e n b y . t h e mapping T.(ut,r) = (T^co,r) f o r j = 1,2, and ( u , r ) e fi x R . We choose an 39 -A >< L measurable integer valued function k: (oj,r) — y k(u),r) which represents the operator k . Then kCT^^cjjr) = k(w,r) - 1 for a.e.(u),r) k(T^ "*"co,r) >_ k(w,r) for a.e.(w,r) . We choose an e R so that for a.e. oi k(T2 1a),r ( )) = k ( c j,r 0) - 1 \aCl~^,rQ) >_ k(a),r0) . Then k^Cw) = k C^r^) satisfies the conditions Y 2 ( k Q ) = k Q - 1 2 "^^ 0^ 2 is 2 Now set u g = e for s e [0,2T0 . Since ^ ( U g ) = e u s ' there is a positive, self-adjoint operator h^ affiliated to the centre of N^  with spec h^ c z such that 2 l s h l 2 v, (u ) = Tr_ (e )u , for a l l s e [0,2Tr) . I s 1 s Since the roles of and Y 2 a r e interchangable we have Lemma 3.6. In the situation of lemma 3.5 there are, for j = 1,2, strongly continuous unitary representations u 3 of [0,2TT) in the centre of Q and positive self-adjoint operators h^ affiliated to the centre of N. with spec h. c <E such that 3 3 - 40 -i i 1 i s n 2 1 Y 1 ( V = 6 Y 2 ( V = 7 T 2 ( e ) u s f o r S £ [ ° ' 2 7 T ) „ - 9 ? l s h l ? Y o ( u ) = e u , yAu) = ir.Ce V , for s e [ 0 , 2 i r ) . Z. S S X s X s U s i n g t h i s lemma, we s h a l l show t h a t i t s u f f i c e s t o assume t h a t 0^ and 0£ a r e c o n s e r v a t i v e . R e c a l l t h a t an automorphism 0 on N i s c a l l e d c o n s e r v a t i v e i f f t h e r e a r e no no n - z e r o c e n t r a l p r o j e c t i o n s p i n N such t h a t {0 ( p ) : n e Z} i s an o r t h o g o n a l f a m i l y . By a m a x i m a l i t y argument we can f i n d a c e n t r a l p r o j e c t i o n p i n N^ such t h a t {0 n(p): n e Z] v n i s an o r t h o g o n a l f a m i l y and i f q d e n o t e s £ 0 (p) t h e n 0.. neZ r e s t r i c t e d t o ( N n ) 1 i s c o n s e r v a t i v e . I t f o l l o w s t h a t 0_ 1 l - q 7 1 1 0 0 r e s t r i c t e d t o (1SL) i s i s o m o r p h i c t o i d 8 6 on (N.,) 8 I (Z) . 1 q-L q l Hence, t h e r e i s a s e l f - a d j o i n t o p e r a t o r h a f f i l i a t e d t o t h e c e n t r e o f .(N.) w i t h spec h c z such t h a t 1 q l V l = h " 9 l ( h ) * S i n c e 1 1 3 ^ " " " S ^ o t n ^1 a n c* ^2 i n v a r i a n t i n t n e c e n t r e o f Q, t h e r e e x i s t s a © 2 i n v a r i a n t p r o j e c t i o n q 2 i n t h e c e n t r e o f N 2 such t h a t IT (q ) = i v 9 ( q 9 ) . Now s e t u 2 = IT ( e ^ S ' 1 ) u 2 f o r s e [0,2ir) J. X ^ Z S X s Then -2 -2 Y, (u ) = u , f o r a l l s e [0,2IT) I s s -2 - i s - 2 Y 0 ( u ) = e u , f o r a l l s e [0,2ir) . 2 s s Hence, t h e r e e x i s t s s —*• u i n t h e c e n t r e o f (N„) such t h a t s 2 q 2 _2 Tr„(u ) = u-., f o r a l l s e [0,2TT) . I s s - 41 -— X S I n p a r t i c u l a r 0 9 ( u ) = e u.., f o r a l l s e [0,2TT) . So 8„ z s s fl / 9 2 r e s t r i c t e d t o (N„) i s i s o m o r p h i c t o i d 8 6 on (N„ )••; 8 I (L) . 2 q, 2 q -1 6 1 9 2 But i r ^ o TT^  i s an i s o m o r p h i s m of w i t h N 2 • Hence, t h e r e i s an i s o m o r p h i s m K of t h e r e s t r i c t i o n of 0^ t o ( ^ ^ ) ^ w i t h t h e r e s t r i c t i o n of t 0 ( N2^q ' T ° s n o w t n a t t n e c e i l i n g s and $2^2 c o r r e s p o n d i n t h e r i g h t way we n o t e t h a t i f 9 i s a s e l f - a d j o i n t CO A o p e r a t o r a f f i l i a t e d t o t h e c e n t r e of A 8 JI (Z) (A i s any W - a l g e b r a ) t h e n t h e r e i s a s e l f - a d j o i n t o p e r a t o r g a f f i l i a t e d t o t h e c e n t r e of CO A 8 % (E) s u c h t h a t 9 = g - i d 8 6(g) . The same r e a s o n i n g a p p l i e d t o t h e r e s t r i c t i o n of 0^  t o (^2^1 ^ shows t h a t t h i s r e s t r i c t i o n i s c o n s e r v a t i v e . Hence, f o r t h e r e s t of t h e p r o o f we assume t h a t 0^ and 0^  a r e c o n s e r v a t i v e . The p r o p e r t y of c o n s e r v a t i v e automorphisms w h i c h we w i l l use i s t h a t e v e r y n o n - z e r o c e n t r a l p r o j e c t i o n i s r e c u r r e n t . We n e x t r e d u c e t o t h e c a s e where t h e o p e r a t o r h^ i s one t o one. L e t e^ be t h e s u p p o r t of h^ and s e t e^ = 0^(e^) . We f i r s t show t h a t e = 1 . There i s a 0 2 i n v a r i a n t p r o j e c t i o n f i n t h e c e n t r e o f N 2 s u c h t h a t v^el^ = ^ 2 ^ ^ " Moreover Y, (uV (1-e )) = U2TT (1-e ) , f o r a l l s e [0,2TT) X S X X S X X Y 9 ( U 2 T T ( 1 - e . ) ) = e i S u 2 T r (1-e ) , f o r a l l s e [0,2TT) . Z S X X S X X Hence, t h e r e e x i s t s s —> u g i n t h e c e n t r e of ( N 2 ) 1 _ f s u c h t h a t TT 0(U ) = U 2TT, (1-e,) f o r a l l s e [0,2TT) . I n p a r t i c u l a r 2 s s i 1 )_(u ) = e 1 S u , f o r a l l s e [0,2TT) z s s - 42 -T h i s c o n t r a d i c t s t h e a s s u m p t i o n t h a t 0 2 i - s c o n s e r v a t i v e u n l e s s e = 1 . We now c u t down t o • ^ n e m a P x —* 7r^Cx) i s a m imbedding °f "Ve/Ve/ i n t ° { Q T T 1 ( e 1 ) ' ( V T r 1 ( e 1 ) } a * d s i n -Y^(Tr^(e^)) = 1, t h e map x —> Tr 2(x)Tr^(e^) i s am imbedding of neZ {N o,0~} i n t o t h e r e s t r i c t i o n o f Yo t o Q (e.) . We a l s o have: l I z TT^  1 ir((() ) e 1 ( e ) ( v r 7 r i ( e i ) ) = ^ l ^ V e 6 > v r 7 r 1 ( e 1 ) , f o r a l l r € R -ir<f> 2 Y 2 ( v r T T 1 ( e 1 ) ) = [ 1 T 2 ( 0 2 e ) ^ 1 ( e 1 ) ] v r i r (e ) , f o r a l l r e R ("Vi) f a S(U 2TT (e ) ) = TT ( e l s h ) u 2 T r ) , f o r a l l s e [0,2ir) 1 ± s 1 1 1 s 1 where h i s p o s i t i v e s e l f - a d j o i n t and 1 - 1 . Hence i t s u f f i c e s t o p r o v e t h e theorem under t h e a s s u m p t i o n t h a t i n t h e s i t u a t i o n o f lemma 3.6 h^ i s 1 - 1 . We n e x t r e d u c e t o t h e c a s e where h^ = 1 . L e t p be t h e p r o -j e c t i o n i n t h e c e n t r e of Q s u c h t h a t u 2 = 1 e i n S Y 9 ( p ) , f o r a l l s e [0,2TT) . S neZ 1 -1 2 -1 " i s n i 2 We have y. (u ) = TT. (6_ e ) u , f o r a l l s e [0,2TT) . L e t I s 1 1 s 0 ^ ( h O = £ n e n be t h e s p e c t r a l r e s o l u t i o n o f 9 i ^ ( h ] ) • I l : f o l l o w s n > l n t h a t -1 n > l -m 2 -m ~ i s n i -1 " i s h ^ 2 S i m i l a r l y , f o r m >^  1, y ( u g ) = 1 T^(0-^ ( e ) . . . 6^ (e ^ u s ^ o r s e [0,2ir) . We l e t 8^0^) + ... + ^ O ^ ) = I n e ^ b e t h e s p e c t r a l n > l r e s o l u t i o n . Then Y-^Cp) = I V e n ) Y 2 ( p ) - 43 -Y " m ( P ) = I M e ^ Y ^ C p ) . n > l The p r o j e c t i o n s e ^ m ^ have t h e p r o p e r t i e s ( i ) e ^ = 0 f o r n < m ( s i n c e h.. > 1) n i — ( i i ) e ^ m ^ e ^ = 0 f o r m < j and n > k ( s i n c e h.. > 1) . n k — - 1 — S i n c e Y - ^ C P ) P = 0, f o r a l l m >_ 1, i t f o l l o w s t h a t { Y ^ ( P ) ' k e Z} i s an o r t h o g o n a l f a m i l y . L e t f be t h e p r o j e c t i o n i n t h e c e n t r e o f N 2 s u c h t h a t *2<f) = I Y l ( p ) • neZ Note t h a t 6 n ( f ) = 1 ( s i n c e TT ( f ) >_ p) . We s h a l l show t h a t t h e neZ c u t down ( Y o ) / J T \ s a t i s f i e s 2 7 r 2 ( f ) ( Y 2 ) i r 2 ( f ) ( P ) = Y 1 1 ( P ) • F o r t h i s , d e f i n e p r o j e c t i o n s -f <_ f i n t h e c e n t r e o f N 2 by W = ^ Y 1 U l ( £ n 1 ) ) p ) • keZ x v (1) v S i n c e ) e = 1 i t f o l l o w s t h a t f = > f . We c l a i m t h a t t h i s i s Zt n n n > l n > l t h e c a n o n i c a l p a r t i t i o n o f f . F i r s t , Y 2 ( n 2 ( f 1 ) ) = I Y k ( T r 1 ( e ^ 1 ) ) Y 2 ( p ) ) . keZ S i n c e ^ ( e ^ ^ Y ^ p ) ± TT 2(f) we have e^f.^) <_ f . N e x t , f o r n > 1 and 1 <_ j <_ n - 1 we have * 2 < * 2< fn» - j A ^ n ^ * " ' keZ - 4 4 -But T r 9 ( f ) y J ( p ) = I TT.(e! A )) YJ(p) and e ^ e f 5 = 0 f o r £ = l , . . . , j z z £=1 x J J and 1 <_ j <_ n - 1 . Hence Q^(f )f = 0 f o r 1 <_ j <_ n - 1 . F i n a l l y Y 2 ( u 2 ( f n ) } = ^ ^ i ^ f ^ ^ P ) ) a n d i r i ( e n 1 M ( P ) - ^ 2 ( f ) ' H e n c e keZ 3 o ( f ) < f . T h i s shows t h a t f = 7 f i s t h e c a n o n i c a l p a r t i t i o n o f 2 n — n n > l We now compute ( V . C f ) ^ = ^ 2 ^ 2 ( f n ) p ) 2 n >1 n > l = y f ( p ) • v - i n s n , s Now s e t u g = I e Y ^ P ) • Then neZ xs y ^ ( u g ) = e u g , f o r a l l s e [0,2TT) (Y,) res (u ) = e 1 S u , f o r a l l s e [0,2TT) . z '"'2^  ' s s Hence, as b e f o r e , i t s u f f i c e s t o p r o v e t h e theorem under t h e a s s u m p t i o n t h a t i n lemma 3.6 h^ = 1 . CO I n t h i s c a s e , by c o r o l l a r y 1.12, we may assume t h a t Q = N^ 8 £ (Z), Tf^Cx) = x 8 1 f o r a l l x e N^, y^ = 9^  8 <5^  and y 2 = i d 8 6 . CO I f we r e g a r d N^ 8 £ (Z) as bounded f u n c t i o n s x: n —>- f r o m Y l E t o N^ we see t h a t Q c o n s i s t s o f t h o s e o p e r a t o r s x s a t i s f y i n g x = 9,nxrt . Hence K: X —> Tr„(x) n i s an i s o m o r p h i s m of N„ w i t h N n 1 (J z u z ± ^ —n *~n suc h t h a t K o 9„ ° K = Q. . So f o r x e N„, ir '(x) = 0, ( K X ) = K(9„ ) X z 1 z z n 1 z f o r a l l n e Z . Now, choose a s e l f - a d j o i n t o p e r a t o r h a f f i l i a t e d t o t h e c e n t r e 00 of N 8 £ (£) such t h a t - 45 -v = e"""'"11, f o r a l l r e IR . r ' Then Y l ( h ) = h + Tr1(61<).1) Y 2 ( h ) = h - Tr2(e2<f>2) • I f n — • h r e p r e s e n t s h we have n 8,(h ,,) =h + B.ty., f o r a l l n e Z 1 n+1 n 11 h = h - e^CKe-t))-), f o r a l l n e Z •. n - 1 n 1 z L Choose n = 0 i n t h e f i r s t e q u a t i o n and n = 1 i n t h e second t o o b t a i n W = ho + 91*1 h Q = h - K $ 2 . That i s h Q ;= S'VQ) + <j>1 - K $ 2 . Now t a k e g = e^Ch^), t h e n *1 = K ^ 2 + 6 1 S ~ g * T h i s c o n c l u d e s t h e p r o o f o f theorem 3.4. || F o r t h e c o n s t a n t c e i l i n g c a s e we have Theorem 3.7. The f l o w b u i l t on {N^,6^} under t h e c o n s t a n t c e i l i n g <f>^  = c i s i s o m o r p h i c t o t h e f l o w b u i l t on {^,82} under t h e c o n s t a n t c e i l i n g § 2 = c i f f {N^,6^} i s i s o m o r p h i c t o {^,62) • P r o o f : I f {N^,0^} i s i s o m o r p h i c t o {^,62} th e n t h e f l o w s a r e i s o m o r p h i c . - 46 -F o r t h e c o n v e r s e , we a p p l y lemma 3.5 t o o b t a i n Q, y^, y^, TT^, TT and r —*• v . We have r Yi (v ) = e v , ' l v r r ' i r c f o r a l l r e (R — i r e y„(v ) = e v , f o r a l l r e 2 r r Hence, v 2 i s f i x e d by b o t h y^ and y^ • Choose a s e l f - a d j o i n t c o p e r a t o r k a f f i l i a t e d t o t h e c e n t r e o f Q f i x e d by b o t h y and Y 2 s u c h t h a t .2TT. x — k c e = v„ . 2TT c . k - i s — Set w = e v , f o r s e (R . Then w„ = 1 so s —> u = w s s/c 2TT s S f o r s e [0,2TT) i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n o f [0,2TT) i n t h e c e n t r e o f Q such t h a t i s Y ^ ( u 2 ) = e u g , f o r a l l s e [0,2IT) " ~ i s Y 2 ( u s ) = e U g , f o r a l l s e [0,2TT) . As i n t h e p r o o f o f theorem 3.5, {N^,0^} i s i s o m o r p h i c t o { N 2 , 6 2 } . - 47 -4. Fl o w under a c e i l i n g and weak e q u i v a l e n c e I n t h i s s e c t i o n we show t h a t t h e u n i q u e n e s s r e s u l t s o f s e c t i o n 3 h o l d w i t h weak e q u i v a l e n c e r e p l a c i n g i s o m o r p h i s m . We s h a l l need P r o p o s i t i o n 4.1. L e t {M,a} be a c o v a r i a n t s y s t e m o v e r a l o c a l l y compact a b e l i a n group G . L e t t — u t be an a c o c y c l e i n M (see d e f i n i t i o n 1.5). I f t h e r e i s a s t r o n g l y c o n t i n u o u s u n i t a r y r e p r e s e n t a t i o n p —>• v ^ of G i n t h e c e n t r e o f M such t h a t a (v ) = <t,p> v , f o r a l l p e G, t e G t p p t h e n t h e r e i s a u n i t a r y u e M such t h a t ft u t ~ u a t ^ u )» f ° r a ^ t e G P r o o f : L e t F 2 be t h e 2 x 2 m a t r i c e s o v e r <C w i t h m a t r i x u n i t s {e : i , j = 1,2} . D e f i n e a c o n t i n u o u s a c t i o n 3 o f G on M 8 by: P t ( . I A j 9 e i j } = a t ( x l l } 8 e l l + 8 6 2 1 x , j = l , 2 + a t ( x 2 1 ) u * 0 e 1 2 + u t a t ( x 2 2 ) u j 0 e 2 2 where x = 7 x.. 8 e.. i s i n M 8 F. and t e G . Note t h a t 1,3=1,2 J g t h e p r o j e c t i o n s e = 1 0 e ^ and f = 1 0 e 2 2 a r e i n (M 0 F 2 ) and t h e y a r e e q u i v a l e n t r e l a t i v e t o M 0 F 2 . We c l a i m t h a t i t s u f f i c e s t o show t h a t e and f a r e e q u i v a l e n t 3 I r e l a t i v e t o (M 0 F 2 ) . F o r i f w i s a p a r t i a l i s o m e t r y i n (M 0 F 2 ) such t h a t - 48 -ww = f and w w = e t h e n t h e r e i s a u n i t a r y u i n M such t h a t w = u 8 e 2 l * S i n c e B t(w) = w f o r a l l t e- G we have u 8 ~ u t a t ( u ) 8 e^^ f o r a l l t e G . Hence * u t = u a t ^ u ^ ^ o r a ^ t e G . p We now show t h a t e and f a r e e q u i v a l e n t r e l a t i v e t o (M 8 T^) U s i n g t h e map p —>- v , we a p p l y c o r o l l a r y 1.12 t o deduce t h a t 8 0 0 M 8 i s i s o m o r p h i c t o (M 8 F^) 8 L (G) . We c o n c l u d e t h e p r o o f w i t h t h e f o l l o w i n g lemma. Lemma 4.2. L e t e and f be p r o j e c t i o n s i n a W - a l g e b r a M . L e t A be a f i n i t e W - a l g e b r a and suppose t h a t e 8 1 and f 8 1 a r e e q u i v a l e n t p r o j e c t i o n s i n M 8 A . Then e and f a r e e q u i v a l e n t i n M . P r o o f : We may assume t h a t A i s a - f i n i t e s i n c e t h e s u p p o r t o f any n o r m a l f i n i t e t r a c e i s a a - f i n i t e c e n t r a l p r o j e c t i o n . S i n c e M 8 A i s i s o m o r p h i c t o M, 8 A we c a n r e d u c e t o t h e e f c a s e where e and f a r e f i n i t e o r p r o p e r l y i n f i n i t e . By t h e c o m p a r i s o n theorem ( [ 7 ] p. 218 theoreme 1) we may assume t h a t e <_ f . I n case b o t h e and f a r e f i n i t e , l e t T be a n o r m a l t r a c e on and l e t d> be a n o r m a l f i n i t e t r a c e on A w i t h <j)(l) i 0 . Then x 0 d>(e 8 1) = x 8 <|>(f ® 1) s i n c e e 8 1 ^ f 8 1 . Hence x ( e ) = x ( f ) . S i n c e e <_ f we c o n c l u d e t h a t e ^ f . We now assume t h a t b o t h e and f a r e p r o p e r l y i n f i n i t e . We - 49 -show t h a t t h e r e i s a c e n t r a l p r o j e c t i o n g such t h a t 0 4- ge ^ gf . S i n c e e i s i n f i n i t e we can f i n d an i n f i n i t e m u t u a l l y o r t h o g o n a l f a m i l y o f e q u i v a l e n t a - f i n i t e p r o j e c t i o n s i n M g . By [7] p. 218 c o r o l l a i r e 2, t h e r e i s a c e n t r a l p r o j e c t i o n g^ and a m u t u a l l y o r t h o g o n a l i n f i n i t e f a m i l y { e ^ = i e 1} of e q u i v a l e n t a - f i n i t e p r o j e c t i o n s s u c h t h a t 0 * g e = I e . i e l By t h e same r e s u l t , t h e r e i s a c e n t r a l p r o j e c t i o n g 2 and a m u t u a l l y o r t h o g o n a l f a m i l y { f : j e J } of e q u i v a l e n t p r o j e c t i o n s s u c h t h a t I c J f o r each i e l , f . ^ e.g„ and l I 2 0 * g 2 f = I f . j e J S i n c e e. < g„ f o r a l l i e l and s i n c e f . ^ e.g„ f o r a l l i e l i — l l l 2 we have f <_ g^ f o r a l l j e J . Hence g = g ^ g 2 ^ 0 a n d 0 + ge = I ge-i e l 0 * gf = I f . j e J 3 Now, u s i n g t h e e q u i v a l e n c e e 8 1 ^ f 8 1 we o b t a i n a m u t u a l l y o r t h o g o n a l f a m i l y {e^: i e 1} o f e q u i v a l e n t a - f i n i t e p r o j e c t i o n s i n 8 A such t h a t I f 8 1 = ( g f ) 8 1 = I e. . j e J J i e l S i n c e each g f . i s a - f i n i t e and each e. i s a - f i n i t e , t h e c a r d i n a l i t y J i o f I i s t h e same as t h e c a r d i n a l i t y o f J ( [ 7 ] p. 224 lemma 6 ) . Hence ge ^ g f . - 50 -Now l e t {g, : k e K} be a maximal f a m i l y o f m u t u a l l y o r t h o g o n a l c e n t r a l p r o j e c t i o n s i n t h e c e n t r e o f M such t h a t g^e ^ g f 4 0 . Set g Q = I g k . I f ( l - g Q ) f 4 0 t h e n ( l - g 0 ) e 4 0 s i n c e e and keK f have t h e same c e n t r a l s u p p o r t (due t o e 8 1 ^ f 8 1) . I n t h i s c a s e we can r e p e a t t h e above argument t o c o n t r a d i c t t h e m a x i m a l i t y of { g ^ 1 k e K } . Hence g^f = f and g^e = e so t h a t f ^ e . T h i s c o n c l u d e s t h e p r o o f o f p r o p o s i t i o n 4.1. || We a p p l y p r o p o s i t i o n 4.1 t o f l o w under a c e i l i n g : Lemma 4.3. L e t {M,a} be t h e f l o w b u i l t on {N,0} under <f>0 . L e t w be a u n i t a r y i n N and s e t 6^(x) = w6(x)w f o r x e N . Then {M,a} i s w e a k l y e q u i v a l e n t t o t h e f l o w b u i l t on {N,0^} under <f> . P r o o f : L e t 0 and 0^ be t h e automorphisms o f N 8 L (R) w h i c h s a t i s f y 0 ( x 8 1) = 0x 8 1, 0 x ( x 8 1) = Q x 8 1, f o r a l l x e N 0(1 8 x ) = 6 e 1 S < ) ) 8 x > 0 , (1 8 x ) = e.e 1 8* 8 X , f o r a l l s e R. S S X S X s Then Q^x) = w 0 1 0 ( x ) ( w 8 1)*, f o r a l l x e N 8 L°°((R) . By t h e CO r e v e r s a l lemma (lemma 2.4) we know t h a t N 8 L (IR) i s i s o m o r p h i c t o CO ~ M 8 Z (Z) i n such a way t h a t 0 c o r r e s p o n d s t o i d 8 6 . Hence, by 00 p r o p o s i t i o n 4.1, t h e r e i s a u n i t a r y u i n N 8 L (IR) s u c h t h a t w 8 1 = u 0 ( u ) . Set u f c = u i d 8 a f c(u) f o r t e IR . Then u f c i s f i x e d by 0, so u t e M, f o r a l l t e R . Moreover, t — u t i s an a c o c y c l e . - 51 -L e t TT be t h e automorphism o f N 8 L (R) g i v e n by: * 0 0 , » TT(X) = uxu , f o r a l l x e N 8 L (R) Then IT - 0 » TT TT = ( a d u t o i d 8 a ) ° ir = i d 8 a^, f o r a l l t e R . * 00 (Here adu (x) = u xu , f o r a l l x e N 8 L (R)) . Hence, t h e f l o w t t t b u i l t on {N,0^} under d> i s i s o m o r p h i c (by ir) t o t h e f l o w t —>• a d u t ° i d 8 a on M . That i s , t h e f l o w s a r e w e a k l y e q u i v a l e n t . A " c o n v e r s e " o f lemma 4.3 i s : Lemma 4.4. Suppose {ML^a"'"} i s w e a k l y e q u i v a l e n t t o t h e f l o w b u i l t on {N,0} under ty . Then t h e r e i s a u n i t a r y w i n N s u c h t h a t {M^ ,a"*"} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N, adw° 0} under ty . ~ 00 P r o o f : L e t 0 be t h e automorphism o f N 8 L (R) as i n d e f i n i t i o n 2.3. There i s an i d 8 a c o c y c l e t —*• u i n [N 8 L (R) ] ^  s u c h t h a t a"*" oo e i s i s o m o r p h i c t o t h e r e s t r i c t i o n o f t —> adut«> i d 8 a t o [N 8 L (R) ] 00 By p r o p o s i t i o n 4.1, t h e r e i s a u n i t a r y u i n N 8 L (R) such t h a t u = u i d 8 a t ( u ) , f o r a l l t e R . Hence u 6 ( u ) i s f i x e d by i d 8 a, so t h e r e i s a u n i t a r y w e N w i t h w 8 1 = u 0 ( u ) . L e t TT = adu, t h e n T7O0«,TT ^ = a d ( w 8 1) ° 0 TT o a d u ^ i d 8 a ° TT ^  = i d 8 a , f o r a l l t e R . - 52 -Hence {H^,a^} i s i s o m o r p h i c t o t h e f l o w b u i l t on {-N, adw=> 6} under • . II The main r e s u l t o f t h i s s e c t i o n i s : Theorem 4.5. The f l o w b u i l t on {N ,0 } under <J>^  i s w e a k l y e q u i v a l e n t t o t h e f l o w b u i l t on {N 2,0 2> under cf>2 i f f t h e r e e x i s t ( i ) r e c u r r e n t p r o j e c t i o n s e^ i n t h e c e n t r e o f ISL w i t h e 2 V 0 n ( e . ) = 1 f o r j = 1,2 neZ 2 2 ( i i ) an i s o m o r p h i s m K o f ( N i ^ e w i t h ( N 2 ) ( i i i ) a u n i t a r y u i n ( N 9 ) such t h a t e 2 K - (0 ) ° K - 1 = adu o (0„) 1 e 1 e 2 ( i v ) a s e l f - a d j o i n t o p e r a t o r g a f f i l i a t e d t o t h e c e n t r e o f (N„) such t h a t 2 e 2 P r o o f : I f t h e c o n d i t i o n s a r e s a t i s f i e d t h e n theorem 3.4 and lemma 4.3 show t h a t t h e f l o w s a r e w e a k l y e q u i v a l e n t . C o n v e r s e l y , i f t h e f l o w s a r e w e a k l y e q u i v a l e n t t h e n by lemma 4.4 we can f i n d a u n i t a r y w i n N 2 s u c h t h a t t h e f l o w b u i l t on { N 2 , adw o 0 2> under (J>2 i s i s o m o r p h i c t o t h e f l o w b u i l t on {N^,0^} under d>^  . Now, a p p l y theorem 3.4 t o o b t a i n r e c u r r e n t p r o j e c t i o n s e. i n t h e c e n t r e o f N. w i t h J J \ / 0 ? ( e . ) = 1 f o r j = 1,2, an i s o m o r p h i s m K o f {(N,) ,(0,) } neZ J J 1 1 w i t h {(N„) ,(adw» 0„) } and a s e l f - a d j o i n t o p e r a t o r g a f f i l i a t e d 2 e 2 2 e 2 - 53 -to t h e c e n t r e o f (N„) such t h a t 2 e 2 K(<J) ) = (<|> ) + (adw 8 ) (g) - g . X 1 2 2 S i n c e (adw » 9 2^ e2 = a d ( w e 2 ^ ° ^ 2 ^ e 2 W e "'"et U = W 6 2 ' T ^ e c o n d l t i o n s of t h e theorem a r e s a t i s f i e d . || Fo r t h e case o f a c o n s t a n t c e i l i n g we have Theorem 4.6. The f l o w b u i l t on {1^,8^} under t h e c o n s t a n t c e i l i n g c i s w e a k l y e q u i v a l e n t t o t h e f l o w b u i l t on {N,^^} under c i f f {N^,6^} i s w e a k l y e q u i v a l e n t t o {^,82} . P r o o f : Lemma 4.3 shows t h a t i f {N,8^} i s w e a k l y e q u i v a l e n t t o {^,82} t h e n t h e r e s u l t i n g f l o w s a r e w e a k l y e q u i v a l e n t . F o r t h e c o n v e r s e , lemma 4.5 shows t h a t t h e r e i s a u n i t a r y w i n N such t h a t t h e f l o w b u i l t on {N^, adw ° B^} under c i s i s o m o r p h i c t o th e f l o w b u i l t on {N^,6^} under c . Theorem 3.7 shows t h a t {N2, adw ° Q^} i s i s o m o r p h i c t o {N^,8^} . That i s , {^,62) i s we a k l y e q u i v a l e n t t o {N^,8^} . || We c o n c l u d e t h i s s e c t i o n w i t h a r e s u l t c o n n e c t i n g W {N,8} t o W {M,a}. P r o p o s i t i o n 4.7. L e t {M,a} be t h e f l o w b u i l t on {N,8} under d) . Then M i s p r o p e r l y i n f i n i t e i f f N i s p r o p e r l y i n f i n i t e . I n t h i s * * case W {N,8} i s i s o m o r p h i c t o W {M,a} . P r o o f : I n t h e n o t a t i o n o f t h e r e v e r s a l lemma (lemma 2 . 4 ) , t h e r e i s an CO 00 i s o m o r p h i s m ir of N 8 L (IR) w i t h M 8 I (£) such t h a t - 54 -T r o Q o 7 r " L = i d 8 S TT ° i d 8 a ° TT ^  = a f c , f o r a l l t e IR . Hence M i s p r o p e r l y i n f i n i t e i f f N i s . F o r the r e s t o f t h e p r o o f we assume t h a t M and N a r e p r o p e r l y i n f i n i t e . C o n s i d e r t h e 00 c o n t i n u o u s a c t i o n 3 o f Z x | R on N 8 L ((R) d e f i n e d by 6, - = 0 n» i d 8 a , f o r a l l ( n , t ) e Z x (R . vn, L y r. A oo A We s h a l l show t h a t W {N 8 L (IR),8} i s i s o m o r p h i c t o W {N,0} . Now, A co W {N 8 L (!R),B} i s g e n e r a t e d by t h e o p e r a t o r s CO { T r g ( x ) : x e N 8 L (IR)}, {X (1,0)} and {X ( 0 , t ) : t e R} . M o r e o v e r , t h e r e a r e commuting a c t i o n s s —*• y g of [0,2TT) and 2 A oo P —*" Y p of R on W (N 8 L (R) ,8} s u c h t h a t 1 2 3, \ = Y ° Y > f o r a l l ( s , p ) e [0,2TT) X (R . ( s , p ) s p A oo L e t P be t h e W a l g e b r a g e n e r a t e d by { T T 0 ( X ) : X e N 8 L (R)} and 3 { A D ( 0 , t ) : t € R} . S i n c e p 2 oo Y (TT„(X)) = T T 0 ( X ) , f o r a l l x e N 8 L ( R ) , p e R P P 3 Yp(Ag ( 0,t)) = e " i p t A g ( 0 , f ) , f o r a l l t e R, p e R i t f o l l o w s , by theorem 1.11 ( t h e c h a r a c t e r i z a t i o n t h e o r e m ) , t h a t P CO i s i s o m o r p h i c t o t h e c r o s s e d p r o d u c t o f N 8 L (R) by i d 8 a . A n o t h e r a p p l i c a t i o n o f theorem 1.11 shows t h a t t h i s c r o s s e d p r o d u c t 2 i s i s o m o r p h i c t o N 8 B ( L (R)) . The c o m p o s i t e i s o m o r p h i s m c a r r i e s - 55 -TT.(X) t o x f o r x e N 8 L~(IR) FS A _ ( 0 , t ) t o 1 8 A f o r t e R . p t (Here (A £)(s) = £(s-t) f o r ? e i- 2(!R)) • S i n c e Y g ( y ) = y, f o r a l l y e P, s e [0,2TT) YgCXgd.O).) = e ~ 1 S A g ( l , 0 ) , , f o r a l l s e [0,2*) ft co i t f o l l o w s by theorem 1.11, t h a t W {N 8 L ((R),3) i s i s o m o r p h i c t o t h e c r o s s e d p r o d u c t o f P by a d A D ( l , 0 ) . P I n summary, W {N 8 L (iR),3} i s i s o m o r p h i c t o t h e c r o s s e d p r o d u c t 2 of N 8 B ( L (IR)) by t h e automorphism 6^ w h i c h s a t i s f i e s e i(x 8 1) = 6x 0 1, f o r a l l x e N 8 (1 8 x ) = Q e " * 8 xr> f o r a l l r e (R 9 ( 1 8 A ) = 1 8 A , f o r a l l t e (R . A c c o r d i n g t o lemma 2.2, 9^ i s w e a k l y e q u i v a l e n t t o 6 8 i d . S i n c e 2 2 N i s p r o p e r l y i n f i n i t e and s i n c e B ( L ( E ) ) i s i s o m o r p h i c t o B(£ (£)) i t f o l l o w s t h a t 9^ i s w e a k l y e q u i v a l e n t t o 9 (see t h e remark ft 00 f o l l o w i n g theorem 1.8). Hence, by p r o p o s i t i o n 1.6, W {N 8 L (IR),fB} ft CO i s i s o m o r p h i c t o W {N,0} . A s i m i l a r a n a l y s i s o f M 8 I (E) w i t h t h e a c t i o n ( n , t ) —> i d 8 6 n° a , f o r a l l ( n , t ) e E x (R ft 00 ft II shows t h a t W {N 8 L (IR),3) i s a l s o i s o m o r p h i c t o W {M,a} . || - 56 -5. A p p l i c a t i o n t o P r o p e r l y I n f i n i t e W - a l g e b r a s We s h a l l a p p l y our r e s u l t s on f l o w under a c e i l i n g t o t h e s i t u a t i o n g i v e n by t h e f o l l o w i n g theorem o f T a k e s a k i . Theorem 5.1 ( [ 2 ] Theorem 8.1, lemma 8.2 and c o r o l l a r y 8.4). L e t P * be a p r o p e r l y i n f i n i t e W a l g e b r a . Then t h e r e i s a c o v a r i a n t s y s t e m {M,a} o v e r IR w i t h t h e p r o p e r t i e s : ( i ) M i s p r o p e r l y i n f i n i t e and s e m i - f i n i t e ( i i ) t h e r e i s a f a i t h f u l , n o r m a l , s e m i - f i n i t e ( a b b r e v i a t e d f . n . s - f ) t r a c e T on M such t h a t T o c(j_ = e ^"T, f o r a l l t e IR ( i i i ) W {M,a} i s i s o m o r p h i c t o P . Mor e o v e r , {M,a} i s u n i q u e up t o weak e q u i v a l e n c e . || We s h a l l r e f e r t o {M,a} as a c o n t i n u o u s d e c o m p o s i t i o n o f P . Note t h a t t h e c o v a r i a n t s y s t e m c o n s i s t i n g o f t h e r e s t r i c t i o n o f a t o t h e c e n t r e o f M i s u n i q u e up t o i s o m o r p h i s m . We f i r s t examine t h e i m p l i c a t i o n s of p r o p e r t y ( i i ) i n cas e {M,a} i s t h e f l o w b u i l t on {N,0} under <j) . P r o p o s i t i o n 5.2. L e t {M,a} be t h e f l o w b u i l t on {N,6} under . Then (a) M i s s e m i f i n i t e i f f N i s s e m i f i n i t e . (b) There i s a f . n . s - f t r a c e on M such t h a t T, ° a = e ^ , f o r a l l t e IR 1 1 t 1 i f f t h e r e i s a f . n . s - f t r a c e on N such t h a t x ^ 0 9 = X2(e^*) (see [8] f o r an e x p l a n a t i o n o f t h e n o t a t i o n ) . - 57 -P r o o f : I n t h e n o t a t i o n o f t h e r e v e r s a l lemma (lemma 2 . 4 ) , t h e r e i s 00 00 an i s o m o r p h i s m TT o f N 8 L (R) w i t h M 8 SL (£) such t h a t ir« i d 8 a » ir ^  = a , f o r a l l t e R . t t oo 6 H e r e , M = [N 8 L ( R ) ] and f o r x e M, t e R a t ( x ) = i d 8 a f c ( x ) TT(X) = x 8 1 . Thus, p a r t (a) i s p r o v e n . L e t and be f . n . s - f t r a c e s on N and M r e s p e c t i v e l y . CO CO L e t m and n be t h e u s u a l t r a c e s on L (R) and SL (Z) r e s p e c t i v e l y . L e t p ( r e s p e c t i v e l y p t , t e R) be t h e p o s i t i v e s e l f - a d j o i n t 1 - 1 o p e r a t o r a f f i l i a t e d t o t h e c e n t r e o f N ( r e s p e c t i v e l y M) such t h a t 0 =T^p') ( r e s p e c t i v e l y O a f c = T ^ ( P t " ) f ° r t e R) . We have 8 m o i d 8 = 8 m, f o r a l l t e R T, 8 n » i d 8 { = T, 8 n . 4 4 By a p p r o x i m a t i n g 0 by automorphisms 9 o f t h e f o r m CO 0 ( x ) = I 0 8 a (e 0 l x ) f o r x e N 8 L°°(R) n = l n where {e : n = 1,2,..} i s a p a r t i t i o n o f u n i t y i n t h e c e n t r e o f N n we see t h a t T 3 8 m » 0 = T 3 ° 0 8 m = x 3 8 m(p 8 1«) . S i m i l a r l y - 58 -8 n o a = (T^ O a t ) 0 n = 8 n ( p ^ 8 ]_•) , f o r a l l t e IR. Now, s e t T R = x_ 8 m and t , = T . 8 n ° I T , t h e n 5 3 6 4 (1) T C O 6 = T (p 0 1«), i c » i d 8 ff, = x c , f o r a l l t e R 5 5 5 t 5 (2) x , ° 6 = x,, T<= i d 0 o = T . ( p • ) , f o r a l l t e R . D D O t o t L e t k be t h e p o s i t i v e s e l f - a d j o i n t 1 - 1 o p e r a t o r a f f i l i a t e d t o t h e CO c e n t r e of N 8 L (R) such t h a t T 5 = T 6 ^ k " - ) ' A c o m p u t a t i o n u s i n g (1) and (2) shows e _ 1 ( k ) = k ( p 8 1) i d 8 o _ t ( k _ 1 ) = k - 1 p t > f o r a l l t e R . Now, l e t h be t h e s e l f - a d j o i n t o p e r a t o r a f f i l i a t e d t o t h e c e n t r e o f oo i r h N 8 L (R) such t h a t e = 1 8 xr> f o r a l l r e R . Then i d 8 o _ t ( e h ) = e t e h , f o r a l l t e R Q - 1 ^ ) = ( e * 8 l ) i h . Set k x = ke . Then 9 _ 1 ( k 1 ) = k ^ p e * 8 1) i d 8 a _ t ( k ~ 1 ) = k ~ 1 ( p t e t ) , f o r a l l t e R . Now suppose T4 ° a t = e^Tl^, ^ o r a ^ fc 6 ' That i s p = e C f o r a l l t e R . Then i d 0 a _ t . ( k 1 ) = k^. f o r a l l t e R . Hence, t h e r e i s a p o s i t i v e s e l f - a d j o i n t 1 - 1 o p e r a t o r a f f i l i a t e d t o t h e - 59 -c e n t r e o f N such t h a t k = k„ 8 1 . S i n c e 6 1 ( k _ ) = k -(pe* 8 1) we see t h a t 3 1(k2) = k 2 p e * Set x 2 = x2^2^'^ ' A c o m p u t a t i o n shows T 2 » 0 = x 2 ( e v-) C o n v e r s e l y , i f e = ^ ( e ^ ) t h e n 3 1 ( k 1 ) = k x . Thus, k^ i s a f f i l i a t e d t o t h e c e n t r e o f M . S i n c e i d 8 a_t^j^") = (p-^et), f o r a l l t e IR we have a _ t ( k 1 ^ ) = k 1 " ' " ( p t e t ) , f o r a l l t e Set X, = x , ( k •) . A c o m p u t a t i o n shows 1 4 1 - t x ^ o a t = e x^ Due t o p r o p o s i t i o n 5.2 we make t h e f o l l o w i n g d e f i n i t i o n : D e f i n i t i o n 5.3. A d i s c r e t e d e c o m p o s i t i o n of a p r o p e r l y i n f i n i t e ft W - a l g e b r a P i s a c o v a r i a n t system {N,0} o v e r Z w i t h t h e p r o p e r t i e s ( i ) N i s p r o p e r l y i n f i n i t e and s e m i - f i n i t e ( i i ) t h e r e i s a f . n . s - f t r a c e x on N and a G c e i l i n g o p e r a t o r <j> such t h a t x •>0 = j(e~^') . ft ( i i i ) W {N,G} i s i s o m o r p h i c t o P . - 60 -P r o p o s i t i o n 5.2 and theorem 5.1 y i e l d t h e f o l l o w i n g c o n n e c t i o n between d i s c r e t e and c o n t i n u o u s d e c o m p o s i t i o n s : ft Theorem 5.4. L e t P be a p r o p e r l y i n f i n i t e W - a l g e b r a , {M,a} a c o v a r i a n t system o v e r R, {N,6} a c o v a r i a n t system o v e r L and ty a 9 c e i l i n g o p e r a t o r . Then any two of t h e f o l l o w i n g i m p l y t h e t h i r d : ( i ) {M,a} i s a c o n t i n u o u s d e c o m p o s i t i o n of P ( i i ) {N,9} i s a d i s c r e t e d e c o m p o s i t i o n of P w i t h x ° 9 = T ( e ^•) f o r some f . n . s - f t r a c e x on N ( i i i ) {M,a} i s w e a k l y - e q u i v a l e n t t o t h e f l o w b u i l t on {N,9} under ty . P r o o f . We f i r s t p r o v e t h a t ( i ) and ( i i ) i m p l y ( i i i ) . L e t {M^ ,a"'"} be t h e f l o w b u i l t on {N,9} under ty . P r o p o s i t i o n 5.2 and p r o p o s i t i o n 4.7 show t h a t {M^ ,©;"'"} i s a c o n t i n u o u s d e c o m p o s i t i o n of P . Theorem 5.1 shows t h a t {M,a} and {M^a"'"} a r e w e a k l y e q u i v a l e n t . Now assume t h a t {M,a} i s w e a k l y e q u i v a l e n t t o {M^jOi"*"} and {M,a} i s a c o n t i n u o u s d e c o m p o s i t i o n o f P . Then, t h e r e i s a f . n . s - f t r a c e x^ on such t h a t x^ o ot^ = e t x ^ , f o r a l l t e R . P r o p o s i t i o n 5.2 shows t h a t t h e r e i s a f . n . s - f t r a c e x on N such -ty * t h a t T° 9 = x ( e •) . P r o p o s i t i o n 4.7 shows t h a t W {N,9} i s i s o m o r p h i c ft 1 ft t o W {M^,a } w h i c h i s i s o m o r p h i c t o W {M,a} (by weak e q u i v a l e n c e ) and so W {N,9} i s i s o m o r p h i c t o P . Thus ( i i ) h o l d s . F i n a l l y we show t h a t ( i i ) and ( i i i ) i m p l y ( i ) . P r o p o s i t i o n 5.2 shows t h a t t h e r e i s a f . n . s - f t r a c e x^ on such t h a t x n o = e t x , f o r a l l t e R . By weak e q u i v a l e n c e , t h e same i s t r u e - 61 -of {M,a} . P r o p o s i t i o n 4.7 shows t h a t W {M^,a x} i s i s o m o r p h i c t o A A W {N,6} w h i c h i s i s o m o r p h i c t o P . Hence W {M,a} i s i s o m o r p h i c t o P . Thus ( i ) h o l d s . || Theorems 5.1, 2.6 and 5.4 g i v e a g e n e r a l i z a t i o n o f [1] theoreme 5.3.1. A C o r o l l a r y 5.5. A p r o p e r l y i n f i n i t e W a l g e b r a P has a d i s c r e t e d e c o m p o s i t i o n i f f f o r any (and hence a l l ) c o n t i n u o u s d e c o m p o s i t i o n s {M,a}, t h e r e s t r i c t i o n o f a t o t h e c e n t r e of M i s nowhere t r i v i a l . P r o o f : L e t {M,a} be a c o n t i n u o u s d e c o m p o s i t i o n of P such t h a t t h e r e s t r i c t i o n of a t o t h e c e n t r e of M i s nowhere t r i v i a l . Theorem 2.6 shows t h a t t h e r e e x i s t s {N,6} and 9 s u c h t h a t {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,6} under 9 . By theorem 5.4, {N,8} i s a d i s c r e t e d e c o m p o s i t i o n of P . C o n v e r s e l y , l e t {N,6} be a d i s c r e t e d e c o m p o s i t i o n of P . L e t {M,a} be t h e f l o w b u i l t on {N,6} under 9 . Theorem 2.6 shows t h a t t h e r e s t r i c t i o n o f a t o t h e c e n t r e of M i s nowhere t r i v i a l and theorem 5.4 shows t h a t {M,a} i s a c o n t i n u o u s d e c o m p o s i t i o n of P . || Theorems 4.5, 5.1 and 5.4 g i v e a g e n e r a l i z a t i o n o f [1] theoreme 5.4.2. C o r o l l a r y 5.6. F o r j = 1,2, l e t {N ,9_.} be a d i s c r e t e d e c o m p o s i t i o n of P . Then P^ i s i s o m o r p h i c t o i f f f ° r 3 = 1>2 t h e r e a r e r e c u r r e n t p r o j e c t i o n s e. i n t h e c e n t r e of N. w i t h \J 9 ? ( e . ) = 1 3 J n e Z 3 3 such t h a t t h e r e d u c t i o n s {(ISL) ,(9.,) } and {(N„) ,(9„) } a r e 1 e l 1 e l 2 e 2 2 6 2 w e a k l y e q u i v a l e n t . P r o o f : Assume t h a t P i s i s o m o r p h i c t o P 2 . Theorems 5.1 and 5.4 show t h a t t h e f l o w b u i l t on {N ,9 } under <j>1 i s w e a k l y e q u i v a l e n t t o t h e f l o w b u i l t on { ^ , 9 2 ^ under ^ • By theorem 4.5 t h e c o n -c l u s i o n of t h e theorem h o l d s . C o n v e r s e l y , f o r j = 1,2, l e t T. be the f . n . s - f t r a c e on N. su c h t h a t x.° 6. = x. ( e • ) . Then f o r 3 3 3 3 3 = 1,2 x. • (9.) = x. ( e 3 3«) 3 3 e. 3 (where x. i s t h e r e s t r i c t i o n o f x. t o (N.)e.) . L e t K be an 3 3 - 3 3 iso m o r p h i s m of (N ) w i t h (N„) and u a u n i t a r y i n (N ) ^1 ^2 ^2 such t h a t K » (6,) ° K = adu o (9„) 1 e l 2 6 2 L e t f be t h e s e l f - a d j o i n t o p e r a t o r a f f i l i a t e d t o t h e c e n t r e o f (N„) such t h a t " ^2 - i - . - f . T l ° K = T 2 *^  ' A c o m p u t a t i o n shows ^l)e1 = <*2>e2 + ( 92>; 2 ( f> " f ' Thus, by theorem 4.5, t h e f l o w b u i l t on {N^,9^} under <f>^  and t h e f l o w b u i l t on {^,92} under ty^ a r e w e a k l y e q u i v a l e n t . These c o v a r i a n t systems g i v e c o n t i n u o u s d e c o m p o s i t i o n s o f P^ and (theorem 5.4). Hence P^ i s i s o m o r p h i c t o P2 • || The c o n s t a n t c e i l i n g c a s e g i v e s r i s e t o a g e n e r a l i z a t i o n o f [1] theoreme 4.4.1. C o r o l l a r y 5.7. L e t P be a p r o p e r l y i n f i n i t e W a l g e b r a and c a - 63 -p o s i t i v e r e a l number. P has a d i s c r e t e d e c o m p o s i t i o n {N,0} such t h a t —c N has a f . n . s - f t r a c e T w i t h TO 0 = e x i f f f o r any (and hence a l l ) c o n t i n u o u s d e c o m p o s i t i o n s {M,a} of P, t h e r e i s a u n i t a r y u i n - J t 2 l T t h e c e n t r e o f M w i t h a^(u) = e c u , f o r a l l t e R. - i t ( 2 j 0 P r o o f : Suppose we have {M,a} and u w i t h a t ( u ) = e c u, f o r a l l t e IR . Theorem 2.7 shows t h a t t h e r e e x i s t s {N,6} s u c h t h a t {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,6} under c . By theorem 5.4, {N,0} i s a d i s c r e t e d e c o m p o s i t i o n o f P and N has a —c f . n . s - f t r a c e x s u c h t h a t x ° 0 = e x . C o n v e r s e l y , assuming we have {N,0} and x such t h a t x «> 0 = e x, we l e t {M,a} be t h e f l o w b u i l t on {N,0} under c . Theorem 2.7 shows t h a t t h e r e i s a u n i t a r y - i t 2 T T u i n t h e c e n t r e o f M such t h a t c t t ( u ) = e c u, f o r a l l t e [R . Theorem 5.4 shows t h a t {M,a} i s a d i s c r e t e d e c o m p o s i t i o n o f P . The c o r r e s p o n d i n g u n i q u e n e s s r e s u l t i s : C o r o l l a r y 5.8. F o r c > 0, j = 1,2, l e t ^ N j ' 6 j ^ b e a d i s c r e t e d e c o m p o s i t i o n o f P. such t h a t N. has a f . n . s - f t r a c e x. w i t h 1 3 3 x . o 0. = e °x. . Then P, i s i s o m o r p h i c t o P„ i f {N-,9-} i s 3 3 3 1 2 1 1 we a k l y e q u i v a l e n t t o {^,82} • P r o o f : Assume t h a t P^ i s i s o m o r p h i c t o P2 • Theorems 5.1 and 5.4 show t h a t t h e f l o w s b u i l t on {N.,0,} under c f o r j = 1 , 2 a r e 3 3 w e a k l y e q u i v a l e n t . By theorem 4.6, t h e c o v a r i a n t systems {N^,8^} and {^,62} a r e w e a k l y e q u i v a l e n t . C o n v e r s e l y , i f {N^,6^} and {^,82} a r e w e a k l y e q u i v a l e n t t h e n t h e c o r r e s p o n d i n g f l o w s a r e w e a k l y e q u i v a l e n t (theorem 4.6). These c o v a r i a n t systems g i v e c o n t i n u o u s d e c o m p o s i t i o n s of P n and P 0 (Theorem 5.4). Hence P 1 i s i s o m o r p h i c "1 t o P 2 - 64 -R e f e r e n c e s [1] Connes, A. "Une c l a s s i f i c a t i o n des f a c t e u r s de t y p e I I I " . Ann. S c i e n t . E c. Norm. Sup., 4 e s e r i e , t . 6 , 1973, p. 133-252. [2] T a k e s a k i , M. " D u a l i t y f o r C r o s s e d P r o d u c t s and the S t r u c t u r e o f von Neumann A l g e b r a s o f Type I I I " . A c t a Math. 131, 1973, 249-310. [3] Ambrose, W. " R e p r e s e n t a t i o n o f E r g o d i c f l o w s " . A n n a l s o f Math. V. 42, No. 3, 1941. [4] Kubo, I . " Q u a s i - F l o w s " . Nagoga Math. J . V. 35, 1969, 1-30. [5] L a n d s t a d , M. " D u a l i t y t h e o r y f o r C o v a r i a n t Systems". T h e s i s , U n i v e r s i t y of P e n n s y l v a n i a . [6] L a n d s t a d , M. " D u a l i t y f o r C o v a r i a n c e A l g e b r a s " . P r e p r i n t no. 1/76. [7] D i x m i e r , J . "Les A l g e b r e s D ' o p e r a t o r dans L'espace H i l b e r t i e n " , 2nd e d i t i o n , P a r i s , G a u t h i e r - V i l l a r s , ( 1 9 6 9 ) . [8] P e d e r s o n , G.K. and T a k e s a k i , M. "The Radon-Nikodym Theorem f o r von Neumann A l g e b r a s " . A c t a . Math. 130 ( 1 9 7 3 ) , 53-87. A p p e n d i x I n t h i s a p p e n d i x we g i v e a p r o o f o f : Theorem 1. L e t {M,a} be a c o v a r i a n t system o v e r R w i t h M a b e l i a n and 0 - f i n i t e . I f a i s nowhere t r i v i a l t h e n t h e r e i s an a b e l i a n W - a l g e b r a N w i t h an automorphism 0 and a 8 c e i l i n g 9 s u c h t h a t {M,a} i s i s o m o r p h i c t o t h e f l o w b u i l t on {N,0} under 9 . || The p r o o f c o n s i s t s of showing t h a t t h e arguments of [3] and [4] a r e v a l i d under weaker c o n d i t i o n s . We s h a l l need the f o l l o w i n g measure t h e o r e t i c a l n o t i o n s . D e f i n i t i o n 2. L e t ( f t , A , y ) be a c o m p l e t e a - f i n i t e measure space. ( i ) An automorphism T of ( f t , A , y ) i s a b i m e a s u r a b l e b i s e c t i o n o f ft such t h a t y ° T i s e q u i v a l e n t t o y . ( i i ) I f T i s an automorphism of ( f t , A , y ) , a T c e i l i n g i s a m e a s u r a b l e f u n c t i o n 9: ft —>• [O, 0 0) such t h a t t h e r e i s a m e a s u r a b l e CO p a r t i t i o n ft = u ft o f ft i n t o T i n v a r i a n t s e t s and numbers n= l e > 0 f o r each n such t h a t 9 (to) > e f o r to e ft and n = 1,2,... n — n n ( i i i ) A m e a s u r a b l e a c t i o n of a l o c a l l y compact a-compact ( f o r c o n v e n i e n c e ) a b e l i a n group G on ( f t , A , y ) i s a f a m i l y {W^ .: t e G} o f automorphisms of (ft , A,y) w h i c h s a t i s f i e s (a) W , = W o W , f o r a l l s , t e G t+s t s' (b) i f (G,L,m) de n o t e s Haar measure on G and we e q u i p ft x G w i t h t h e c o m p l e t i o n A x L o f A x L w i t h r e s p e c t t o y x m t h e n t h e map (o),t) —> Wt(o>) i s m e a s u r a b l e . - 66 -( i v ) M e a s u r a b l e a c t i o n s {W : t e G} and {W^: t e G} o f G on (f i , A , y ) and ( f i , A , y ) r e s p e c t i v e l y a r e s a i d t o be i s o m o r p h i c i f f t h e r e a r e i n v a r i a n t m e a s u r a b l e s e t s ft^ c ft and ft^ c fi w i t h y(ft\ftQ) = 0, y ( f i \ f i g ) = 0 and a b i m e a s u r a b l e b i j e c t i o n S: fig —»• fi^ such t h a t S o W = W o S and such t h a t y » S i s e q u i v a l e n t t o y (v) I f T i s an automorphism of ( f i , A , y ) and (j) i s a T c e i l i n g X (f) we r e f e r t o t h e m e a s u r a b l e a c t i o n t —>• W of R on ( f i . , A , , y . ) c o n s t r u c t e d i n s e c t i o n 2 as t h e f l o w b u i l t on t h e automorphism T under t h e f u n c t i o n <j> . Note t h a t i f T i s a b i j e c t i o n o f a s e t fi and <f>: fi —>• [e,°°) X (b f o r some e > 0 we can c o n s i d e r t h e a c t i o n t —> W £ of (R on t h e s e t fi^ = {(w,s) e f i x R : 0 <_ s < <j>(u>)} . Our f i r s t r e s u l t i s a g e n e r a l i z a t i o n df theorem 1 o f [ 3 ] ; t h e p r o o f i s a l m o s t t h e same. Lemma 3. L e t T be a b i j e c t i o n o f a s e t fi and l e t d>: fi —[e,°°) f o r some e > 0 . Suppose t h a t t h e r e i s a a - a l g e b r a B on fii = { ( w > s ) : 0 < s < <Kw)} and a measure v on 8 s u c h t h a t <P ~ ( i ) (ft,,B,v) i s a c o m p l e t e a - f i n i t e measure space 9 X d) ( i i ) t —>• ¥ ^ i s a m e a s u r a b l e a c t i o n ( i i i ) t h e f u n c t i o n s F and G d e f i n e d b e l o w a r e 8 m e a s u r a b l e , F(w,s) = d>((jj) f o r (u,s) e fi^ G(u,s) = s f o r (w,s) e fi. . <P 67 -Then, t h e r e i s a complete a - f i n i t e measure u on a a - a l g e b r a A o f s u b s e t s of fi such t h a t T i s an automorphism of (fi , A,u), cb i s A m e a s u r a b l e , 8 = A, and v i s e q u i v a l e n t t o u, . P r o o f : As i n t h e p r o o f o f theorem 1 of [3] we s e t A = {E c fi; E x [0,e) e 8} . S i n c e G i s 8 m e a s u r a b l e , fi x [0,e) e 8 so t h a t fi e A and A i s a a - a l g e b r a . Note t h a t <j> i s A m e a s u r a b l e because F i s 8 m e a s u r a b l e . F o r 0 <_ s < e, d e f i n e b i j e c t i o n s V g of fix [0,e) by: V g ( o ) , t ) = i Then V g ( a ) , t ) = f W T , < t > ( t o , t ) , i f 0 <_ G •W A' H'(u),t) < e (u»,t+s) , i f 0 £ t + s < e (oi,t+s-e) , i f t + s >_ e . [ W ^ C u . t ) , i f G o w T , < 1 )(a),t) > e . S i n c e s -> W1'* i s a m e a s u r a b l e a c t i o n o f R i t f o l l o w s t h a t s —> V c s i s a m e a s u r a b l e a c t i o n o f t h e group [0,e) on t h e r e d u c t i o n (fi x [ 0 , e ) , 8 fix[0,£)' V fix[0,e)) We adopt t h e n o t a t i o n (fi x [ 0 , e ) , B , V £ ) f o r t h i s measure s p a c e . S i m i l a r l y ( [ 0 , e ) , L £ , m ) d e n o t e s t h e r e d u c t i o n o f Lebesgue measure (R,L,m) t o [0,e) . Now, f o r f a bounded 8 m e a s u r a b l e f u n c t i o n on fix [ 0 , e ) , t h e f u n c t i o n - 68 -(u>,t,s) f V_(u,t) on ft x [0,e) x [0,e) i s B& x L £ measurable. So for v a.e. (o),t) e fi x [0,e) the function s —> f ° V (o),t) on [0,e) i s L measurable. In f a c t the set of a l l (w,t) e fi x [0,e) for which t h i s i s L measurable i s a V invariant set of the form £ s fi. x [0,e) where fif e A ' and (fi\fif) x [0,e) i s n u l l . Set f(o),t) = f£ f V ( t o , t ) d s , i f 0) e fif 0 [ 0 , otherwise. Then f o V = f for a l l 0 < s < e . Hence, there i s an A measurable s {uwc t. i oil f such that f(o)) = f(w,t), for a l l (w,t) e fi x [0,e) Define a measure v' on B £ by: fdv' = fdv for f bounded and B measurable Then v' i s equivalent to V £ (by Fubini's theorem) and v' o V g = v', for a l l s e [0,e) Define a measure y on A by y(E) = v'(E x [0,e))( = v(E x [0,e))) for E e A . Then (fi ,A,y) i s a complete a - f i n i t e measure space. - 6 9 -Note that since G i s 8 measurable, A x lP c 8 where i s the a-algebra on [0,e) generated by the i n t e r v a l s . Since fdv' = fdv' for f bounded and 8 measurable, i t follows that J e ,0 for E e A x JL e v' (E) = u x m (E) . Hence A x / , c 8 and v'(E) = u x m (E) f o r E e A x L . We now e e e e show that T i s an automorphism. For t h i s we consider the sets E = {to e fi: k2 _ n < <j>(w) < (k+l)2~ n} n,k — oo for n = 1,2,... and k = 1,2,... . Then fi = u E i s a p a r t i t i o n k=l n ' k of fi into d i s j o i n t sets and since F i s 8 measurable we have E . £ A for a l l n and k . For E c Q we have, for s u f f i c i e n t l y n,k large n, Hence {(To), s) : iii e E ii E , , 0 <_ s < e - 2 n} c w ( ( E n E , ) x [0 ,e ) ) n fi x [0,e) k2"n n ' k {(Tco,s) : c o e E , 0 < _ s < e - 2 n } u W ((E n E . ) x [0,e)) n fi x [0,e) k=l k2"n n ' k So CO 00 TE x [0,e) = u u [W ((E n E . ) x [0,e)) n fi x [0,e)] n=l k=l k2"n n ' k This shows that f o r E e A, TE e A and u(E) = 0 implies V(TE) = 0 Si m i l a r l y we can show that f o r E e A T _ 1 E e A . Hence T i s an automorphism of (fi,A,i_i) . - 70 -We n e x t show t h a t A x L = 8 . F o r t h i s we c o n s i d e r t h e a-e E a l g e b r a 8^  x L . That i s , the c o m p l e t i o n o f 8^  x w i t h r e s p e c t t o v' x . We a l s o c o n s i d e r t h e maps R(co,t,s) = (V ( o ) , t ) , s ) and S ( i o , t , s ) = ( u . t , G o V g ( w , t ) ) f o r ( w , t , s ) e fi x [ 0 , E ) X [ 0 , E ) . U s i n g a d d i t i o n mod E we have R(w,t,s) = (w,t+s,s) f o r (w,t,s) e fi x [0,e) x [ 0 , E ) S ( w , t , s ) = (w,t,s+t) f o r ( w , t , s ) £ fi x f0,e) x [0,e) . S i n c e s —*• V i s a m e a s u r a b l e a c t i o n and s i n c e G i s 8 m e a s u r a b l e s i t f o l l o w s t h a t R and S a r e automorphisms of fi[0,e)x[0,e). I n f a c t R and S p r e s e r v e v' x m £ . W r i t i n g - t f o r t h e a d d i t i v e i n v e r s e of t mod £ we have: R o S _ 1 ° R(o3,t,s) = ( w , s , - t ) S o R - 1 o S ( w , t , s ) = ( w , - s , t ) f o r ( u , t , s ) e fi x [ 0 , E ) x [ 0 , E ) . Now c o n s i d e r C = {E e 8_ x : { ( o ) , s ) : ( o o , t , s ) e E} e A x /. f o r a.e. t e [ 0 , e ) } . £ £ We s h a l l show t h a t 8 x L = C . E E Note t h a t i f E e 8 x L t h e n f o r a.e. s e [ 0 , E ) £ £ { ( u ) , t ) : (o),t,s) e E} e B £ . -1 -1 F o r E e 8 ,^ we a p p l y t h i s t o R a S o R E x [0,e) . We have: - 71 -{ ( u ) , t ) : (a>,t,s) e R ^ o s o R - 1 E x [0,e } = {(w,t) : (u),s,-t) £ E x [ 0 , e ) } = {to: (to,s) £ E} x [0,e) . Hence, f o r E e 8 £ and f o r a.e. t e [ 0 , e ) {w: (to,t) £ E} i s A m e a s u r a b l e . Thus, f o r E £ B and F e L and f o r a.e. t {(co,s): (to,t,s) £ E x F} = {to: (w,t) £ E} x F e A x L £ . S i n c e C i s c l e a r l y a a - a l g e b r a we have B £ x L £ c C . To show t h a t B x i = Q we need o n l y show t h a t C i s c o m p l e t e w i t h r e s p e c t t o v' x m . F o r t h i s we assume t h a t E c F £ C and v' x m (F) = 0 . E £ Hence v' x m ( S - 1 c R o S _ 1 ( F ) ) = 0 . So f o r a.e. s £ { ( w , t ) : (co,t,s) £ S-1.. R o - S _ 1 ( F ) } i s v' n u l l . That i s , f o r a.e. s { ( u , t ) : ( u , - s , t ) e F} i s v 1 n u l l . That i s , f o r a.e. t { ( t o , s ) : ( t o , t , s ) £ F} i s v 1 n u l l . S i n c e { ( u , s ) : (w,t,s) £ E} c { ( a ) , s ) : ( u , t , s ) e F } , t h e f o r m e r s e t i s A x i m e a s u r a b l e . £ T h i s shows t h a t C = B x L . Now f o r E £ B we have £ £ E R - 1 o S o R - 1 E x [0,E) £ C . That i s , f o r a.e. t { ( t o , s ) : ( w , t , s ) £ R _ 1 o s 9 R _ 1 E x [ 0 , e ) } i s A x L m e a s u r a b l e . That i s , f o r a.e. t £ { ( u , s ) : (co,s,-t) £ E x [0,£)} = E - 72 -A x L m e a s u r a b l e . T h i s shows t h a t B = A x L To c o n c l u d e e t h e p r o o f we f i r s t n o t e t h a t W 1^ i s a m e a s u r a b l e f l o w r e l a t i v e t o ( f t , , A x L 9 ft ' ^ X m 9 ) . F o r n = 1,2,... s e t 9 E = { ( i o , t ) ( n - l ) e <_ G(w,t) < n e } Then f o r any s e t E, E = u E n E n= i co u T/'W'V n E ) ] ., ne -ne n=l But W T' E(E n E ) c ft x [0,e) . Thus E i s B-measurable i f f E i s -ne n A x L m e a s u r a b l e . I n t h e same way v ( E ) = 0 i f f y * m(E) - 0 ft, 9 Our n e x t r e s u l t i s a g e n e r a l i z a t i o n o f theorem 2 o f [ 3 ] . The p r o o f was g l e a n e d from the p r o o f s o f theorem 2 o f [3] and theorem 3.1 o f [ 4 ] . Theorem 4. A nowhere t r i v i a l m e a s u r a b l e a c t i o n o f !R i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . ( H e r e , nowhere t r i v i a l means t h e c o n -t i n u o u s a c t i o n o f iR on L i s nowhere t r i v i a l ) . P r o o f : L e t (X,B,v) be a c o m p l e t e a - f i n i t e measure space and t — • W a nowhere t r i v i a l m e a s u r a b l e a c t i o n o f R on (X,B,v) . S i n c e t h e a c t i o n i s n o n - t r i v i a l we can f i n d a m e a s u r a b l e s e t E and a p o s i t i v e number t such t h a t v((W E)\E) 4 0 . S i n c e t h e a c t i o n i s m e a s u r a b l e , U 0 f o r v a.e. x t h e f u n c t i o n t -> 1 E o W t(x) - 73 -i s Lebesgue m e a s u r a b l e . I n f a c t t h e s e t Y of a l l x such t h a t t —> 1 ° W (x) i s Lebesgue m e a s u r a b l e i s W i n v a r i a n t and E t t v(X\Y) = 0 . F o r each number a > 0 we d e f i n e * a ( x ) = 1 a 0 0 K " W f x ) d t , i f x e Y E t i f x 4 Y . ty i s m e a s u r a b l e by F u b i n i ' s theorem. S i n c e t h e a c t i o n i s m e a s u r a b l e a 00 f —> f o w d e f i n e s a c o n t i n u o u s a c t i o n o f R on L (v) and so ty 1 a-weakly. Hence, we may choose a s m a l l enough so t h a t a E v(E± n wt E 2 ) 4 0 o where E, = .{x e X: ty (x) < 1/4} 1 3. E„ = {x e X: ty (x) > 3/4} Note t h a t s —> ty ° W (x) i s c o n t i n u o u s f o r each x . I n f a c t f o r r a s t , s e R and x e X, ty (W. ( x ) ) - ^ (W ( x ) ) | < Ht-s| a L a s a D e f i n e extended r e a l v a l u e f u n c t i o n s x a n d X o n x b Y : X ( x ) = sup{u: W x e E_ n Wr (E )} u 1 t Q 2 - o o , i f t h e s e t i s empty X(x) f i n f { u : W x e E n W ( E , ) } 1 t Q 2 [ -H», i f t h e s e t i s empty - 74 -Then x and x a r e m e a s u r a b l e f o r by t h e c o n t i n u i t y o f ty&, X and x a r e t h e sup and i n f r e s p e c t i v e l y o f the m e a s u r a b l e e x t e n d e d r e a l v a l u e d f u n c t i o n s x u and Xu f o r u r a t i o n a l where X • (x) = u f u, i f W (x) e E_ n (E ) 1 u 1 t Q 2 -co, o t h e r w i s e u, i f W (x) e E- n W (E ) 1 t Q 2 [ +°°, o t h e r w i s e . D e f i n e m e a s u r a b l e s e t s x l = {x e X: X ( x ) = CO 9 X ( x ) = _cx>} X 2 - {x £ X: X ( x ) = CO 5 X ( x ) > _M} x 3 = {x e X: -co < X ( x ) < oo} X 4 = {x e X: X ( x ) = —CO } . These s e t s a r e m e a s u r a b l e , d i s j o i n t and t h e y c o v e r X . S i n c e E l ° W t ^ E 2 ^ C X l u X 2 u X 3 ' o n e o f X l ' X 2 ° r X 3 i s n o n - n u l ± -S i n c e each X^ i = 1,...,4 i s W i n v a r i a n t we s h a l l show t h a t t h e r e d u c t i o n o f t —> Wfc t o each X_^  i = 1,2,3 i s i s o m o r p h i c t o a f l o w b u i l d under a f u n c t i o n . F o r x e X,, t h e s e t {u: W x e E n n W (E„)} c o n t a i n s a r b i t r a r i l y 1 1 t Q 2 l a r g e p o s i t i v e and n e g a t i v e numbers u . Set tt = {x £ X, : ty (x) = \ and ty W. (x) > \ f o r 0 < t < f } . i a Z a t Z o As i n t h e p r o o f of theorem 2 of [ 3 ] , f o r each x e X± t h e t r a j e c t o r y - 75 -{W^x: t e R} i n t e r s e c t s ft f o r a r b i t r a r i l y l a r g e p o s i t i v e and n e g a t i v e numbers t . F o r w e ft s e t <j>(w) = i n f { t > 0: W t(w) e ft} . Then <j> >_ a/8 . F o r w e f t , s e t Tw = W g(w) where s = i n f { t > 0: Wt(co) eft}. We map ft^ onto by: S(w,t) = W^Cu), f o r (w,t) e ft^ . t d> Note t h a t S o W T^ = Wfc ° S, f o r a l l t e IR . U s i n g S we o b t a i n a T (b complete a - f i n i t e measure on ft, f o r w h i c h W i s a m e a s u r a b l e 9 t a c t i o n . As i n t h e p r o o f o f theorem 2 of [ 3 ] , t h e f u n c t i o n s F and G of lemma 3 a r e m e a s u r a b l e . Hence, by lemma 3, t h e r e d u c t i o n t o X^ i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . We n e x t show t h a t t h e r e d u c t i o n t o X^ i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . F o r x e X 2 , -°° < x(x) < °° and x(W tx) = x(x) - t f o r a l l t e R . Set CO ft = u {x e X 2 : x(x) = n ) n=-°° <j>(w) = 1 f o r w e f t Tw = W 1(w) f o r w e f t . D e f i n e a b i j e c t i o n S o f ft^ onto X 2 by S(w,t) = W t(w) f o r (w,t) e ft^ . Then S ° W T^ = Wfc ° S, f o r a l l t e (R . U s i n g S we o b t a i n a complete a - f i n i t e measure on Q,A s u c h t h a t t —>• W i s a m e a s u r a b l e V 9 t a c t i o n . The f u n c t i o n s F and G of lemma 3 a r e m e a s u r a b l e so by - 76 -lemma 3 t h e r e d u c t i o n of t —>- W t o i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . F o r X^  we p r o c e e d as f o r X^  u s i n g x • Now s e l e c t a maximal d i s j o i n t f a m i l y of n o n - n u l l , m e a s u r a b l e , W i n v a r i a n t s e t s s u c h t h a t t h e r e d u c t i o n t o e ach i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . S i n c e t h e s e t s a r e d i s j o i n t and non-n u l l t h e f a m i l y i s c o u n t a b l e . S i n c e t h e f l o w i s nowhere t r i v i a l t h e f i r s t p a r t of the p r o o f shows t h a t t h e complement of t h e u n i o n of t h i s f a m i l y i s n u l l . By t a k i n g t h e " d i r e c t sum" of t h e f l o w s b u i l t under f u n c t i o n s we o b t a i n an i s o m o r p h i s m of W w i t h a f l o w b u i l t under a f u n c t i o n . || The f o l l o w i n g r e s u l t i s w e l l known i n t h e " s e p a r a b l e " c a s e . We i n c l u d e a p r o o f f o r t h e sake o f c o m p l e t e n e s s . Lemma 5. L e t {M,a} be a c o v a r i a n t system o v e r a l o c a l l y compact, cr-compact, a b e l i a n group G . Suppose t h a t M i s a b e l i a n and in-f i n i t e . Then t h e r e i s a c o m p l e t e a - f i n i t e measure space ( f t , A , u ) , a m e a s u r a b l e a c t i o n t —> W of G on (ft,A,vO and an i s o m o r p h i s m CO CO K of L (u) w i t h M such t h a t f o r a l l f e L (u) and a l l t e G K(f o w ) = a (Kf) . f : L e t N be t h e s u b s e t of M c o n s i s t i n g of a l l x f o r w h i c h P r o o t —> a (x) i s norm c o n t i n u o u s . N i s a C - s u b a l g e b r a of M c o n t a i n i n g t 1 . F o r f a c o n t i n u o u s , c o m p a c t l y s u p p o r t e d f u n c t i o n on G and f o r x e M y = f ( t ) a t ( x ) d t - 77 -i s i n N . Hence N i s a-dense i n M . We.let 9 be t h e spec t r u m o f N and l e t IT : C(9) —>• N be t h e i n v e r s e o f t h e G e l f a n d t r a n s f o r m a t i o n . S i n c e a p r e s e r v e s N, we can f i n d homeomorphisms W f o r t e G o f 9 s u c h t h a t i r ( f o W_t) = a t ( n ( f ) ) f o r each f e .c(ft) and t e G . I t f o l l o w s t h a t t —> W i s an a c t i o n o f G on 9, . That i s , W_ , = W ° W , f o r a l l t , s e G . t+s t s S i n c e t —>• f o W i s norm c o n t i n u o u s f o r f e C(9) i t f o l l o w s t h a t to,t —>• f o W (co) i s p r o d u c t m e a s u r a b l e where we t a k e t h e B o r e l a-a l g e b r a on 9 and l e t (G,L,m) be Haar measure on G . Hence, t h e same i s t r u e f o r a B o r e l measure f u n c t i o n f on 9 . Now, l e t T be a f a i t h f u l n o r m a l s t a t e on M ( s i n c e M i s a - f i n i t e we can f i n d s u c h a s t a t e ) and l e t (9,A,\i) be t h e Radon measure on 9 w h i c h s a t i s f i e s f d y = T ( i r ( f ) ) , f o r a l l f e C(n) . 9 U s i n g t h e G.N.S. c o n s t r u c t i o n f o r x, we may assume t h a t M a c t s on a H i l b e r t space H w h i c h c o n t a i n s a c y c l i c and s e p a r a t i n g v e c t o r £Q w i t h x ( x ) = (x£ ,5 ) , f o r a l l x e M . 2 S i n c e t h e s u p p o r t o f y i s 9, C(9) imbeds i n L (y) and the map f -> Tr(fK 0 2 ° ° / e x t e n d s t o a u n i t a r y o p e r a t o r U: L (y) —> H . F o r f e L (y) we - 7 8 -2 i d e n t i f y f w i t h t h e m u l t i p l i c a t i o n o p e r a t o r £ —> f£ f o r £ e L ( y ) . Under t h i s i d e n t i f i c a t i o n we have UfU* = T r ( f ) , f o r a l l f e C(fi) . * CO Hence K: f —>• UfU i s an i s o m o r p h i s m o f L (y) w i t h M . We now show t h a t each W i s an automorphism o f (fi , A,y) . S i n c e W i s a homeomorphism i t s u f f i c e s t o show t h a t i f E c fi i s a B o r e l s e t w i t h y ( E ) = 0 t h e n y(W E) = 0 . F o r any compact s e t K c W ( E ) , W_ (K) i s compact and W (K) c E . We can f i n d a sequence f , n = 1,2,... of c o n t i n u o u s f u n c t i o n s such t h a t n and V t ( K ) ± f n ± 1 f dy —> 0 as n n That i s , x ( 7 r ( f ) ) —>• 0 as n —*- oo a n d 0 < u ( f ) < 1 f o r a l l n n — n — Hence, iT(f ) —> 0 a-weakly as n —>- °° and so n T(OI f ( f ) ) —>• 0 as n —>- °° . That i s f o W dy —>• 0 as n n - t S i n c e 1 < f W < 1 and s i n c e f o W ^ i s c o n t i n u o u s , we K ~ n - t — n - t c o n c l u d e t h a t y(K) = 0 . Hence y(W E) = 0 . We n e x t show t h a t t —>• W i s a m e a s u r a b l e a c t i o n o f G on (fi , A,y) . F o r t h i s i t s u f f i c e s t o show t h a t i f E c fi i s B o r e l w i t h y ( E ) = 0 t h e n - 7 9 -F = { ( t i ) , t ) : W 0) e E} ( w h i c h i s p r o d u c t m e a s u r a b l e ) i s y x m n u l l . But y x m ( F ) = [j l F ( o > , t ) d y ( a j ) ] d t G fi [ 1_ o W (co)dy(w)]dt Hi L G fi y(W E ) d t 0 F i n a l l y , we have i < ( f ° W S i n c e C(fi) CO i s a-dense i n L (y) and s i n c e W d e f i n e s a c o n t i n u o u s a c t i o n of G on L (y) we have CO I i K ( f o W ) = a t ( K f ) , f o r a l l f e L (y) . || P r o o f of theorem 1: Use lemma 5 t o r e p r e s e n t a as a m e a s u r a b l e a c t i o n of R on a c o m plete a - f i n i t e measure space. A p p l y theorem 4 t o c o n c l u d e t h a t t h i s m e a s u r a b l e a c t i o n i s i s o m o r p h i c t o a f l o w b u i l t under a f u n c t i o n . Lemma 2.1 now shows t h a t { M , a } i s i s o m o r p h i c t o a f l o w b u i l t under a c e i l i n g o p e r a t o r . || 

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