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Existence of normal linear positive functionals on a von Neumann algebra invariant with respect to a… Hsieh, Tsu-Teh 1971

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EXISTENCE OF NORMAL LINEAR POSITIVE FUNCTIONALS' ON A VON NEUMANN ALGEBRA INVARIANT WITH RESPECT TO A SEMIGROUP OF CONTRACTIONS by TSU-TEH HSIEH B.S., Cheng-Kung U n i v e r s i t y , Taiwan, China, 1962. M.A., The Un i v e r s i t y of Western Ontario, London, Ontario, 1967. A THESIS SUBMITTED IN.PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard The U n i v e r s i t y of B r i t i s h Columbia March 1971 In present ing t h i s thes is in p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make it f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying o f t h i s thes is fo r s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It is understood that copying or p u b l i c a t i o n o f th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion . Depa rtment The Un ivers i ty o f B r i t i s h Columbia Vancouver 8, Canada Date /far// /?/<77/ Supervisor: Dr. E. Granirer ABSTRACT Let A be a von Neumann algebra of l i n e a r operators on the H i l b e r t space H . A l i n e a r operator T (resp. a l i n e a r bounded. f u n c t i o n a l <j> ) on A i s said to be normal i f f o r any increasing net {B } of p o s i t i v e elements i n A with l e a s t upper bound B , T(B) a a i s the l e a s t upper bound of ^(B^) (resp. <J>(B) = sup ^(B^)) . Two l i n e a r p o s i t i v e functionals and ty^ on A are said to be equivalent i f iJ^(B) = 0 <=> ^ ( B ) = ^ ^ o r a n Y p o s i t i v e element B i n A Let <J>Q be a p o s i t i v e normal l i n e a r f u n c t i o n a l on A . Let S be a semigroup and, {T : s e S} an antirepresentation of S as s normal p o s i t i v e l i n e a r contraction operators on A . We f i n d i n th i s thesis equivalent conditions f o r the existence of a p o s i t i v e normal l i n e a r f u n c t i o n a l <j> on A which i s equivalent to $ and i n v a r i a n t under the semigroup {T g : s e S} ( i . e . <|)(T B) = <j>(B) f o r a l l B i n A .and s e S ). We also extend the concept of weakly-wandering sets, which was f i r s t introduced by Hajian-Kakutani, to weakly-wandering projections i n A We give a r e l a t i o n between the non-existence of weakly-wandering projections i n A and.the existence of p o s i t i v e normal l i n e a r functionals on A , i n v a r i a n t with respect to an antirepresentation {T g : s e S} of normal -homomor-phisms on A . F i n a l l y we i n v e s t i g a t e the existence of a complete set of p o s i t i v e normal l i n e a r functionals on A which are i n v a r i a n t under the semigroup {T : s e S} i i i . TABLE OF CONTENTS Page INTRODUCTION 1 CHAPTER I PRELIMINARIES 6 CHAPTER II SEMIGROUPS OF POSITIVE NORMAL CONTRACTION OPERATORS 21 CHAPTER I I I SEMIGROUPS OF NORMAL -HOMOMORPHISMS 41 BIBLIOGRAPHY 76 i v . ACKNOWLEDGEMENTS I am deeply indebted to my supervisor, Professor Edmond E. Granirer,. f o r the suggestion of th i s subject and f o r h i s help during the preparation of this t h e s i s . His patience and unselfishness i n o f f e r i n g h i s time to ensure my success i s p a r t i c u l a r l y appreciated. I also wish to express my apprecia-t i o n to Professor D.J.C. Bures f o r h i s valuable suggestions during h i s reading of the d r a f t form of th i s t h e s i s . I would l i k e to express my h e a r t f e l t thanks to my parents who enable me to obtain an education of today. I also wish to thank my wife f o r taking care of my family so that I could devote my time to my research work. Many thanks are also due to Miss Barbara K i l b r a y f o r typing this thesis with patience and care. F i n a l l y , the f i n a n c i a l assistance of the Un i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. INTRODUCTION Let (X,S,p) be a f i n i t e measure space. Let S be a semigroup of measurable maps on X into X . A subset E C X i s said 00 to be weakly-wandering i f there e x i s t s {s } . C S such that n n=l -1 0 0 {s E} - are pairwise d i s j o i n t . A measure u on S i s said to be n n=l ^ J i n v a r i a n t i f u(s ^ E) = u(E) for a l l s e S , E e S . Hajian-Kakutani [11] were the f i r s t to prove the equivalence between the existence of a f i n i t e i n v a r i a n t measure equivalent to p and the non-existence of a weakly-wandering set of p o s i t i v e p-measure, i n case S i s a group generated by one non-singular i n v e r t i b l e transformation. This r e s u l t has since been generalized and improved i n d i f f e r e n t d i r e c t i o n s by L. Sucheston [19], Neveu [16], Blum-Freidman [2], Natarajan [15], Hajian-Ito [12], Granirer [9], Sachdeva [18] and others. It i s the main purpose of t h i s thesis to extend the above r e s u l t s to a von Neumann algebra A on a complex H i l b e r t space H Let A + denote the set of a l l p o s i t i v e elements i n A , and P the set of a l l projections i n A . A l i n e a r operator T on A (resp. a l i n e a r f u n c t i o n a l <t> on A ) i s said to be normal i f f o r any increasing net {A } i n A + with l e a s t upper bound A we have T(A) be the l e a s t upper bound of T(A ) (resp. we have <J> (A) = sup (J>(A )) a a a In t h i s thesis we in v e s t i g a t e conditions which are s u f f i c i e n t f o r the existence of a p o s i t i v e normal l i n e a r f u n c t i o n a l on A i n v a r i a n t with respect to a c e r t a i n semigroup of p o s i t i v e normal l i n e a r operators on 2. In Chapter 1 we give the general d e f i n i t i o n s and notations. We a l s o quote some f a c t s on von Neumann algebras, and give t h e i r proofs or i n d i c a t e the references of i t s proof. Given a f i n i t e measure space (X,S,p) . Then L (X,S,p) i s a von Neumann algebra on the complex H i l b e r t space L^CXjS,?) . The predual of L (X,S,p) i s the Banach space L..(X,S,p) , and L.(X,S,p) 00 1 1 i s the space of a l l ultra-weakly continuous f u n c t i o n a l s on L (X,S,p) (See Dixmier [7], p. 31, Theorem 1). In case that p i s a p r o b a b i l i t y measure on S , Hajian-Ito [12] and U. Sachdeva [18] proved the following: I f S i s a l e f t amenable semigroup and {T g ; s e S} i s a representation of S as p o s i t i v e contractions on L^(X,S,p) , then the following are equivalent: (1) There e x i s t s f Q e L ^ X ^ p ) with 0 < f Q = T g f Q f o r a l l s e S ( i . e . f ^ > 0 a.e.) (2) p(E) > 0 implies M ^ J ^ l d p ) > 0 where i s the support f u n c t i o n a l f o r the set of l e f t i n v a r i a n t means on m(S) (3) p(E) > 0 implies i n f { / T ldp ; s e S} > 0 E s + °° * (4) h e L (X,S,p) , £ T h e L (X,S,p) f o r some sequence oo S oo n=l n s £ S implies h = 0 , where T i s the adjo i n t operator n of T on L (X,S,p) . S oo n I t i s the main purpose i n Chapter 2 to generalize t h i s r e s u l t to von Neumann algebras. In t h i s chapter we l e t A be a von Neumann algebra on a (complex) H i l b e r t space H . Let S denote a l e f t amenable semigroup and {T g ; s £ S} be an antirepresentation of S as p o s i t i v e normal contraction operators on A ( i . e . T = T T ). Let <b n be st t s 0 a p o s i t i v e normal l i n e a r f u n c t i o n a l on A . We prove that (Theorem 2-6) the following are equivalent: 3. (1) There ex i s t s a S-invariant p o s i t i v e normal l i n e a r f u n c t i o n a l <j) ( i . e . |(T A) = <KA) f o r a l l s e S ; A e A ) such that <J> ~ <$> ( i . e . tj)(A) = 0 i f and only i f <J>Q(A) = 0 for A e A +) (2) E e P, <j>n(E) > 0 implies inf{<j> (T E) ; s £ S} > 0 , and u u s <j)0(E) = 0 implies M ^ Q O M E ) ) = 0 (3) A e A + , (J)Q(A) > 0 implies inf{<j>0(TgA) ; s e S} > 0 , and <J>Q(A) = 0 implies Mj^  (cj)0 (T gA) ) = 0 . * Moreover, i f ^S<$>Q 1 S c e n t r a l f o r a l l s e S then <j> can be chosen c e n t r a l (where T i s the adjo i n t of T on the dual space A of s s A ). I f T*(j,0 -< <j>0 ( i . e . ,|,0(A) = 0 implies * 0 ( T g A ) = T*<|, (A) = 0 fo r A e A + ) f o r a l l s £ S and {T gE ; s e S} C {E}" f o r any E e P , then we have (Theorem 2-10) the equivalence of the following: (1) There e x i s t s a S-invariant p o s i t i v e normal l i n e a r f u n c t i o n a l (J) on A such that <j> ~ CJ>Q (2a) I f A e P ((2b) A e A + ) then ^ ( A ) > 0 implies inf{<J>0(T A) ; s e S} > 0 (3a) I f A e P ((3b) A e A + ) then $ (A) > 0 implies V * 0 ( T S A ) ) > 0 (4a) If A e V ((4b) A e A + ) and there e x i s t s a sequence 0 0 C O {s } C S such that E T A e A , then <J>n(A) = 0 . n=l n * Moreover, i f T d>„ i s c e n t r a l f o r a l l s e S then 6 can be chosen . s T0 c e n t r a l . Some r e l a t e d r e s u l t s also obtained. We strengthen somewhat our r e s u l t s f o r f i n i t e von Neumann algebras (see Theorem 2-12). In Chapter 3 we are mainly concerned with semigroups of normal *-homomorphisms on a von Neumann algebra A . We l e t S be a semigroup and {T ; s e S} an anti r e p r e s e n t a t i o n of S as normal *-homomorphisms s on A . A p r o j e c t i o n E e P i s said to be weakly-wandering i f there o o e x i s t s a sequence {s } , i n S such that T E'T E = 0 for any n n=l s. s. . 1 3 i =f j . Let AQ be a p o s i t i v e normal f u n c t i o n a l ( l i n e a r ) on A The following i s one of our main r e s u l t s i n t h i s chapter. Theorem 3-4: If S i s amenable and T E-E = E-T E f o r any E e P, s s s e S , then (1) <=> (2) and ( l 1 ) <=> (2*): (1) There ex i s t s a S-invariant p o s i t i v e normal $ e A such that <J>0-< <J> • (2) There i s no weakly-wandering p r o j e c t i o n E i n P with • 0 ( E ) > 0 . ( l 1 ) There e x i s t s a S-invariant p o s i t i v e normal § e A such that 4> 4>Q (2') If E e P, $ 0(E) = 0 => M($ 0(T gE)) = 0 , and (2) holds. Moreover, i f § i s c e n t r a l then <|> can be chosen c e n t r a l . This theorem i s s t i l l true i f we consider {T : s e S} to be a representa-s t i o n of S as normal *-homomorphisms on A . Related r e s u l t s are obtained f o r the case that A i s a f i n i t e (or a - f i n i t e f i n i t e ) von Neumann algebra (see Theorem 3-9 and C o r o l l a r y 3-11). In t h i s chapter we also i n v e s t i g a t e the s u f f i c i e n t conditions f or the existence of a complete set of S-invariant p o s i t i v e normal ( l i n e a r ) f u n c t i o n a l s on A (a set TCA i s said to be complete i f given non-zero A e A , there i s (J) e T such that (j)(A) =[ 0 ): 5. Theorem 3-14: Let S be amenable and such that E'T E = T E'E for s s any p r o j e c t i o n E e P, s e S . Then the following are equivalent: (1) There e x i s t s a complete set of S-invariant p o s i t i v e normal ( l i n e a r ) functionals on A (2) There i s no non-zero weakly-wandering p r o j e c t i o n i n A (3a) If 0 =f A e P ((3b) 0 =f A e A + ), then 0 i s not i n the strong closure of (^gA ; s c S} (Also see Theorem 3-15 and C o r o l l a r y 3-17). 6. CHAPTER 1 PRELIMINARIES In t h i s chapter we give some general d e f i n i t i o n s and notations. We also r e c a l l some f a c t s that are needed i n the next two chapters. If S i s any semigroup, m(S) denotes the space of a l l bounded complex valued functions on S with sup norm. If f e m(S) , f denotes the conjugate of f i . e . f e m(S) such that f ( s ) = f ( s ) f o r a l l s e S , where f ( s ) i s the complex conjugate of the complex number f ( s ) . Let m(S) be the dual space of m(S) i . e . the space of a l l continuous l i n e a r functionals on m(S) . An element <J> e m(S) i s sa i d to be a mean on m(S) i f the following two properties hold: (1) ty(f) = <f>(f) ' f o r a l l f e m(S) . (2) I f f i s r e a l then i n f f ( s ) <_ty(f) <_ sup f ( s ) . s s Let 1 e m(S) be such that l ( s ) = 1 f o r a l l s e S , then (2) i s equivalent to (2') <j>(l) = 1 and <f.(f)>_0 i f f >_ 0 . I t i s cl e a r that i f cj> i s a mean on m(S) then ty has norm one i . e . Let S be a semigroup and a e S We define the l e f t [ r i g h t ] t r a n s l a t i o n : Z [r ] on m(S) by a a I f ( s ) = f ( c s ) [r f ( s ) = f(s o ) ] f o r a l l a a f e m(S), a e S 7. A mean <j) on m(S) i s said to be l e f t i n v a r i a n t [right i n v a r i a n t ] {invariant} i f f o r a l l a e S , f e m(S)' we have <|>u f ) = 4>(f) [*(r f ) = <|)(f)]' U U / ) = * ( R 0 F ) = * ( f ) > • w e denote by LIM[RIM] {IM} the set of a l l l e f t i n v a r i a n t [right i n v a r i a n t ] {invariant} means on m(S) . A semigroup S i s s a i d to be l e f t amenable [r i g h t amenable] {amenable} i f LIM i s not empty [RIM i s not empty] {IM i s not empty} . If LIM (resp. RIM, IM ) i s not empty, then for r e a l f e m(S) we l e t ^ ( f ) = sup U ( f ) ; y e LIM} (resp. M R(f) = sup {<j>(f) ; <j> £ RIM} , M(f) = sup {<j>(f) J <J> e IM}) . If H i s a complex H i l b e r t space with inner product ( | ) , L(H) denotes the algebra of a l l bounded l i n e a r operators on H . For A E L(H) , l e t A* E L(H) be the adjoint of A i . e . A* £ L(H) such that (A x|y) = (x|Ay) for a l l x,y i n H . Let A e L(H) , then A i s c a l l e d hermitian i f A = A ; A i s c a l l e d unitary i f A A = AA = 1 where I i s the i d e n t i t y operator on H ; A i s c a l l e d a p r o j e c t i o n 2 * i f A = A = A ; A i s c a l l e d p o s i t i v e , denoted by A >_ 0 , i f (Ax|x) i s p o s i t i v e f o r a l l x £ H . There are at l e a s t f i v e u s e f u l topologies on L(H) ; the uniform topology, the strong topology, the weak topology, the u l t r a - s t r o n g topology, and the ultra-weak topology (see [4], p. 32). If {A } . i s a family of elements i n L(H) with ct a £ A an index set A , l e t {a ; a i s a f i n i t e subset of A} be p a r t i a l l y ordered by set i n c l u s i o n , then { I A ; a i s a f i n i t e subset of A} a aca i s a net of elements i n L(H) . I f the strong l i m i t of t h i s net e x i s t s , we denote i t by £ A - B y the same reason, i f the strong l i m i t A a CIEA 8. of the net { n A ; o i s a f i n i t e subset of A} e x i s t s we denote i t a aea by II A (where II A i s the product of elements A for a e a ). . a a a aeA aea If E and F are two projections i n L(H) , they are s a i d to be orthogonal i f EF = FE = 0 , and denoted by E J- F . I t i s c l e a r that i f two projections E and F are such that EF = 0 then E J- F . Let {E } . be a family of projections i n L(H) , denote by V E the a aeA J r J . a aeA suprema of (E } . i . e . the l e a s t p r o j e c t i o n E i n 1(H) such that a aeA J E < E for a l l a e A (note that f o r A.B i n L(H) , A < B means a — — B - A i s p o s i t i v e ) ; denote by A E the infima of {E } i . e . r '' ' . a OL aeA aeA the l a r g e s t p r o j e c t i o n F i n L(H) such that F <_ E^ f o r a l l a e A In case {E } . i s pairwise orthogonal ( i . e . E J- E. i f a f 3) we a aeA ^ & a 3 have V E = E E . I n case {E } . i s a commutative family . a . a a aeA aeA aeA ( i . e . EE„ = E E f o r a l l a,g i n A ) then A E = II E aeA aeA If A i s a subset of 1(H) , we l e t A' = {A e L(H) ; AB = BA for a l l B e A} . A subalgebra of the algebra 1(H) i s sa i d to be a ^-algebra i f i t contains A whenever i t contains A i n L(H) . We say that A i s a von Neumann algebra on a H i l b e r t space H i f A C L(H) i s a *-algebra and A" = A . The following theorem i s w e l l -known. 9. Theorem 1-A ([4], p. 43 Theorem 2 and p. 44 Co r o l l a r y 2): A *-algebra A L(H) i s a von Neumann algebra i f and only i f i t contains the i d e n t i t y operator I and i t i s closed under any one (hence a l l ) of the following .topologies; the strong topology, the weak topology, the u l t r a -strong topology, and the ultra-weak topology. If A i s a von Neumann algebra on a H i l b e r t space H , we c a l l A' f\ A the center of A , and we denoted i t by Z . I t i s cl e a r that the center of a von Neumann algebra i s again a von Neumann algebra. I f the center of a von Neumann algebra A contains only s c a l a r multiples of I i . e . Z = {Al ; A e C , the complex numbers} then A i s said to be a f a c t o r . Let A be a von Neumann algebra on a H i l b e r t space H , we l e t A denote the dual space of A i . e . the space of a l l bounded l i n e a r f u n c t i o n a l s on A . Let A + = {A E A ; A > 0} , P = (A e A ; A i s projection} , and U = {A e A ; A i s unitary} . I t i s well-known that f o r any A e A , A = A^ + i A 2 with hermitians A^ and i n A ; any hermitian A £ A , A = A^ - A^ with A^ and A^ i n A + ; A £ A i f and only i f A = B B f o r some B E A (for reference see [17]). An element E i n P A Z i s said to be a ce n t r a l p r o j e c t i o n . 10. Let A be a von Neumann algebra on a H i l b e r t space H When we say that <j> i s a f u n c t i o n a l on A we always mean that <j> e A When we say that T i s an operator on A we always mean that T i s a l i n e a r operator on A into A Let A be a von Neumann algebra on a H i l b e r t space H . Let T be an operator on A , then T i s c a l l e d p o s i t i v e i f T(A +) C A + ; T i s c a l l e d c e n t r a l i f T ( A B ) = T ( B A ) for a l l A , B i n A ; T i s c a l l e d contraction i f IITAII < IIAII f o r a l l A e A (for A e L (H) the norm of A i s defined to be |A|| = sup ||Ax|| , x e H ); T i s c a l l e d I  x |-1 normal i f f o r any increasing net {A } i n A with l e a s t upper bound A we have TA be the l e a s t upper bound of {T(A )} ; T i s c a l l e d a a *~homomorphism [*-isomorphism] i f i t i s an algebraic homomorphism [isomorphism] and such that T(A ) = (TA) . Let <j> e A , then <t> i s c a l l e d p o s i t i v e i f cf> (A) >_ 0 f o r A c A + ; <j> i s c a l l e d c e n t r a l i f <j)(AB) = (J)(BA) for a l l A, B i n A ; <}> i s c a l l e d f a i t h f u l i f ^ (A) = 0 implies A = 0 f o r A e A + ; <j> i s c a l l e d normal i f f o r any increasing net {A } i n A + with l e a s t upper bound A we have J a a d)(A) = sup <j)(A ) ; d> i s c a l l e d p o s i t i v e singular i f <b i s p o s i t i v e ot a * and f o r any p o s i t i v e normal ij; e A such that ^ <_ <j> then \p = 0 ; <j) i s c a l l e d singular i f $ i s a l i n e a r combination of p o s i t i v e singu-l a r f u n c t i o n a l s on A ; ' <j> i s c a l l e d completely add i t i v e i f for any family {E } , of pairwise orthogonal projections i n A we have <j>( E E ) = E <))(E ) . The following theorem i s well-known ([17], aeA a aeA a p. 15, Theorem 3 and p. 16 foot note (16)): 11. Theorem 1-B (Dixmier) ; Let A be a von Neumann algebra on a H i l b e r t space H , then the following are equivalent: (1) <j> £ * A i s p o s i t i v e and normal. (2) <t> £ * A i s p o s i t i v e and completely a d d i t i v e . (3) ty £ * A i s p o s i t i v e and ultra-weakly continuous Let A be a von Neumann algebra on a H i l b e r t space H Let <j> and ty are two p o s i t i v e f u n c t i o n a l s on A , then <j>^  i s sai d to be absolutely continuous with respect to ty (written ty^^ ty2 ) i f <J>2(A) = 0 implies ty^(A) = 0 f o r A e A+ ; ty^ i s s a i d to be equivalent to ty^ (written ty ~ ty2 ) i f <J>-^-< ty2 3X1,2 ^2"^ ^1 A non-negative numerical function ty , f i n i t e or i n f i n i t e , defined on A + of a von Neumann algebra A i s sa i d to be a trace on A i f i t has the following p r o p e r t i e s : (1) ty(k + B) = cf)(A) + ty(B) f o r A, B i n A + (2) ty (AA) = XtyiA) f o r A E A + , X >_ 0 (Assume 0°° = 0 ) (3) ({.(UAU-1) = ty(k) f o r A e A + , U e U . Let ty be a trace on A , then ty i s sa i d to be f i n i t e i f <j> (A) < °° for a l l A e A ; - t y i s sa i d to be f a i t h f u l i f 0 f A e A implies tj)(A) > 0 ; ty i s said to be normal i f f o r any incr e a s i n g net {A ^  i n A + with l e a s t upper bound A we have ty(A) = sup <j>(A ) . I t a i s c l e a r that a p o s i t i v e c e n t r a l f u n c t i o n a l on A i s a f i n i t e trace on A .The following p r o p o s i t i o n ([4], p. 80 Proposition 1) gives the converse. 12. Proposition 1-C: Let <j> be a trace on a von Neumann algebra A . The set of a l l A e A + with <j>(A) < °° form the p o s i t i v e part of a two sided i d e a l I C A . There i s one and only one l i n e a r f u n c t i o n a l cj> on I which coincides with <j> on I and <f>(AB) = <}>(BA) f o r A e I , B e A . Also i f (j> i s normal then A -> <j>(BA) > f o r B e l , i s ultra-weakly continuous on A In view of Proposition 1-C and Theorem 1-B, we can i d e n t i f y the set of a l l f i n i t e traces on A with the set of a l l p o s i t i v e c e n t r a l elements i n A .We can even i d e n t i f y the set of a l l f i n i t e normal (resp. f i n i t e f a i t h f u l ) traces on A with the set of a l l p o s i t i v e normal c e n t r a l (resp. p o s i t i v e f a i t h f u l c e ntral) f u n c t i o n a l s on A A von Neumann algebra A on a H i l b e r t space H i s sa i d to be f i n i t e i f f o r any non-zero A e A + there i s a f i n i t e normal trace ^ on A such that cj>(A) =f 0 ; A i s s a i d to be q - f i n i t e i f any family ^ a ^ a e / \ °^ pairwise orthogonal projections contains at most countably many elements. Hence any von Neumann algebra A on a separable H i l b e r t space H i s a - f i n i t e . The following i s a w e l l -known pr o p o s i t i o n ([4], p. 98 Proposition 9 ( i i ) ) : P r o p o s i t i o n 1-D: Let A be a von Neumann algebra on a H i l b e r t space H , then the following are equivalent: (1) There e x i s t s a f i n i t e normal f a i t h f u l trace on A (2) A i s a - f i n i t e f i n i t e ( i - e . a - f i n i t e and f i n i t e ) . 13. (3) A i s f i n i t e and Z i s o - f i n i t e . Let A be a von Neumann algebra on a H i l b e r t space H Two proj e c t i o n s E and F i n A are c a l l e d equivalent (written E = F) * * i f E = AA and F = A A for some A £ A 1 • i • Theorem 1-E: A von Neumann algebra A i s f i n i t e i f and only i f the only p r o j e c t i o n i n A which i s equivalent to I i s I i . e . i f E i s a p r o j e c t i o n i n A such that E - I , then E = I . ([4], p. 308 Theorem 1). Let A be a fa c t o r and P the set of a l l projections i n A . A function ty on P , takes on values i n [0,<*>) V {°°} , i s sa i d to be a r e l a t i v e dimension on A i f ty i s the r e s t r i c t i o n of a f i n i t e normal f a i t h f u l trace on A Proposition 1-F; Let A be a f i n i t e f a c t o r and D =j= 0 be a f i n i t e non-negative function defined on P . Then D i s a r e l a t i v e dimension on A i f i t s a t i s f i e s (1) D(E + F) = D(E) + D(F) f o r E,F i n P with E X F (2) D(UEU _ 1) = D(E) f o r any E e P , U e U ([4], p. 248 Proposition 15). In view of Proposition 1-F, any non-zero f i n i t e trace on a f i n i t e f a c t o r A i s normal and f a i t h f u l . The symbol // i s used f o r the completeness of a proof. In the following we state some fa c t s on von Neumann algebras, and we give the proof (or i n d i c a t e the references of the proof) f o r each r e s u l t . Let A denote a von Neumann algebra on a H i l b e r t space H Lemma 1-G: Let <J> e A be p o s i t i v e . Let A e A with <j> (A) > 0 Then there are E e P , c > 0 such that cE <_ A and cj> (E) > 0 . Proof: From the s p e c t r a l theorem f o r bounded hermitian operators ( [ 1 ] ) , A i s an uniform l i m i t of a net of elements of type n { Z A.E ; A. > 0 and E are projections i n {A}"} . So there . -• • 1 A . ' 1 A . 1=1 1 1 i s some A. > 0 with A.E. < A and <j> (E, ) > 0 • Since A " = A , l I A. — A. l l a l l E, e A . // A , 1 Lemma.1-H: Let ty ,ty i n A be p o s i t i v e , and such that ty(E) = 0 implies ty (E) = 0 f o r E e P . Then ty ~< ty Proof: Let A e A + with ty (A) > 0 . From Lemma 1-G there are E e P , c > 0 such that cE <_ A and ty (E) > 0 . So ty(E) > 0 , hence ty(A) >_ ty(cE) = cty(E) > 0 . So ty -< ty . // Theorem 1-1: If if i s a p o s i t i v e normal f u n c t i o n a l on A , then ty i s strongly continuous on any bounded subset of A . ([4], p. 40 Theorem l ( i i ) ) . Theorem 1-J: If T i s a p o s i t i v e normal operator on A , then T i s ultra-weakly continuous, and i s weakly continuous on any bounded subset A A of A . If there e x i s t s k _> 0 such that (TA) TA <_ kT(A A) for A e A , then T i s u l t r a - s t r o n g l y continuous and i s strongly continuous on any bounded subset of A . ([4], p. 56 Theorem 2). A P r o p o s i t i o n 1-K: Let <)> e A be p o s i t i v e and normal. Then there i s a l a r g e s t p r o j e c t i o n F i n the set of projections G such that 9(G) = 0 For t h i s F we have 9 (FA) = cj>(AF) = 0 f o r a l l A e A . ([4], p. 61 Proposition 3). The p r o j e c t i o n I-F with F as i n Proposition 1-K i s c a l l e d the support of <b , and denoted by E J . I t i s easy to see that E, 9 <j> i s the smallest p r o j e c t i o n G i n A such that 9 (G) = $(1) • I t i s known that E I = I i f and only i f <j> i s f a i t h f u l ([4], p. 61), and 9 E, i s i n the center of A i f A> i s c e n t r a l ([4], p. 82). 9 A P r o p o s i t i o n 1-L: Let A^ be the unit b a l l of A , and 9 e A be A p o s i t i v e and normal. On A^ , the convergence of <j> (A A) to zero i s equivalent to the convergence of AE^ to zero strongly. ([4], p. 62 Proposition 4). C o r o l l a r y 1-M: Let {A } be an uniformly bounded sequence i n A + n n A If <j> e A i s p o s i t i v e normal and such that E, i s i n Z , then 9 <b(A ) converges to zero implies A E converges to zero strongly, n n 9 Proof: Since {A } are uniformly bounded, there i s k > 0 such that n n IIA || < k for a l l n . Since A > 0 f o r a l l n , there e x i s t s 11 n" — n — 2 {B } such that B > 0 and B = A for a l l n . So we have n n n — n n || B || <_ k f o r a l l n . Thus the sequence {^n} i s a sequence i n x. k A B . B A A, and 4>(.y^~ ' 17~J = 9 (~ o") = ~~i 9 (^  ) converges to zero. From k k B B P r o p o s i t i o n 1-L, converges to zero strongly i . e . \\yT~ E^x|| converges to zero f o r a l l x e H . Since E e Z , we have 9 A B B ||—Tr E xII = | | - — E (-— E x) || converges to zero f o r a l l x e H . Hence 1 ^ Z 9 1 ' k (J) K. <j> A E converges to zero strongly. # n 9 P r o p o s i t i o n 1-N: Let 9,1}) be two p o s i t i v e normal f u n c t i o n a l s on A The following are equivalent: (1) <j><> (3) on the unit b a l l A^ of A , the topology given by the 1_ * 2 seminorm [^(A A)] i s stronger than the topology given by 1 A ' 2 the seminorm [ i f ( A A ) ] . ([4], p. 62 Proposition 5). C o r o l l a r y 1-N' : Let 9,4) be two p o s i t i v e normal f u n c t i o n a l s on A such that <fa —< il» . If {A } i s a uniformly bounded net i n A such T a a that iKA ) converges to zero, then 9(A ) converges to zero, a a A Proof: Let k > 0 such that IIA II < k f o r a l l a . Then {-—} 11 a" — k a i s a net i n A , the unit b a l l of A . Our c o r o l l a r y follows from Proposition 1-N. // Proposition 1-0: If A i s a f a c t o r , then any two f i n i t e normal f a i t h f u l traces on A are p r o p o s i t i o n a l . ([4], p. 90 C o r o l l a r y ) . Let N be a normed space with norm || || . If N i s complete ( i . e . every Cauchy sequence i n N has a l i m i t i n N ) then N i s c a l l e d a Banach space. A Banach space N i s c a l l e d a Banach algebra i f N i s also an algebra and such that (1) ||ab|| £ || a || ||b|l f o r a,b i n N (2) If N has an i d e n t i t y e , then ||e|| = 1 . Let N be a Banach algebra, a mapping '*' ' defined on N into N i s c a l l e d an i n v o l u t i o n on N i f '*' s a t i s f i e s the following p r o p e r t i e s : (1) (a + b ) X = a* + b* f o r a,b i n N * — & — (2) (Aa) = Aa , where A i s the complex conjugate of the complex number A , f o r a e N (3) (ab)* = b*a" for a,b i n N (4) a = a f o r a e N A Banach algebra with an i n v o l u t i o n i s c a l l e d a Banach ^-algebra. It i s easy to see that L(H) (and any ^-algebra i n 1(H) ) i s a Banach *-algebra with the i n v o l u t i o n '*' on L(H) to be the "a d j o i n t " of an element i n L(H) . It i s a l s o easy to see that i f (X,S,p) i s a measure space then L (X,S,p) i s a Banach *-algebra with the i n v o l u t i o n * — '*' to be the "conjugation" ( i . e . f = f f o r f e L (X,S,p) 18. Let N^ and ^ be two Banach ^-algebras. A mapping 9 on into N 2 i s said to be an isometric ^-isomorphism i f 9 i s an algebraic isomorphism and such that (1) i| 9(a) || = || a|| for a e H± C2) 9(a ) = 9(a)' for a e N . I f there exists an isometric ^-isomorphism between two Banach ^-algebras N-^  and > then and are said to be isomet r i c a l l y ^-isomorphic. A Banach ^-algebra N i s said to be a C -algebra i f N i s isometrically ^-isomorphic to a uniformly closed *-algebra i n L(H) for some Hi l b e r t space H If (X,S,p) i s a f i n i t e or <r-finite measure space then L „ ( X , S , p) i s a Hilbe r t space with inner product (f|g) = j f(x)g(x)dp /. X I f , for f e L^CX.S.p) , we l e t f(g)(x) = f(x)g(x) for any g e L ^ C X J S J P ) and x e X , then L O T ( X , S , P ) i s a commutative von Neumann algebra on the Hilbe r t space L2(X,S,p) . In fa c t , every commu-ta t i v e von Neumann algebra i s of the type L O T(X,S,p) for some measure space (X,S,p) Theorem 1-P: A von Neumann algebra A on a Hil b e r t space H i s commu-ta t i v e i f and only i f A i s isometrically ^-isomorphic to L^CXjSjp) for some measure space (X,S,p) . ([4], p. 117 Theorem 1 and Theorem 2). In [5] and [6] Dixmier proved the following r e s u l t s . 19. Theorem 1-Q: If A i s a f i n i t e von Neumann algebra on a H i l b e r t space H , then there i s one and only one function 7^ from A onto the center Z of A with the following properties: (1) = A i f A e Z (2) ( AA)^ = \A^ f o r A e A , X e C the s c a l a r s (3) (A + B)^ = A* + B^ for A,B i n A (4) (AB)^ = (BA)^ f o r A,B i n A (5) (AB)^ = AB^ for A e Z, B e A . (6) If A e A + then A^ e A + and A^ = 0 implies A = 0 (7) (A*]/ 7 = (A* 7)* f o r A e A . ([5], p. 249 Theorem 10 and Theorem 11). The function tj i n Theorem 1-Q i s c a l l e d the canonical c e n t r a l valued function. Theorem 1-R: Let A be a f i n i t e von Neumann algebra with the canonical c e n t r a l valued function 7^ . For each A e A , l e t to be the convex h u l l of {UAU U e U the set of a l l unitary elements i n A} . Let K be the uniform closure of K , then the element A^ i s exactly the unique element i n K C\ Z . ([5], p. 251 Theorem 12). Theorem 1-S: The canonical c e n t r a l valued function on a f i n i t e von Neumann algebra A i s strongly continuous on the un i t b a l l A-^ of A . ([5], p. 256 Theorem 17(B)). Coro l l a r y 1-S': The canonical c e n t r a l valued function 7^ on a f i n i t e von Neumann algebra A i s strongly continuous on any bounded subset N of A . Proof: Let N be a bounded subset of A , then there i s k > 0 such that ||A|| <_ k f o r a l l A e N . Thus ; A e N} C A the unit b a l l of A . Our c o r o l l a r y now follows from Theorem 1-S. // Proposition 1-T: Let A be a f i n i t e von Neumann algebra on a H i l b e r t space H . There i s a one to one correspondence between the set of a l l f i n i t e traces <j> on A and the set of a l l p o s i t i v e f u n c t i o n a l s ty on Z , the center of A . This correspondence i s defined by <j)(A) = ty(A^) f o r a l l A e A , and ty i s nothing but the r e s t r i c t i o n of ty on Z If ty i s normal (resp. f a i t h f u l ) then ty i s normal (resp. f a i t h f u l ) and the converse i s also true. ([6], p. 5 Proposition 2). Co r o l l a r y 1-T': I f A i s a f i n i t e f a c t o r , then A^ = cf>(A)I f o r a l l A e A , where ty i s the unique f i n i t e trace on A such that ty(l) - 1 ([4], p. 267). F i n a l l y , i f u and 0 are two measures, we l e t u < Q means " u i s absolutely continuous with respect to 0 and ~ u means " ^  i s equivalent to u " as i n measure theory. 21. CHAPTER 2 SEMIGROUPS OF POSITIVE NORMAL CONTRACTION OPERATORS Let A be a von Neumann algebra on a H i l b e r t space H Let S be a l e f t amenable semigroup and {T g ; s e S} an antirepresen-t a t i o n of S as p o s i t i v e normal contraction operators on A ( i . e . T „ = T T ) . In t h i s chapter we i n v e s t i g a t e conditions which are s t t s equivalent to the existence of a p o s i t i v e normal l i n e a r f u n c t i o n a l on A which i s in v a r i a n t under the semigroup {T g ; s e S} . The main theorems i n t h i s chapter are Theorem 2-6 and Theorem 2-10 which generalize Hajian-Ito's r e s u l t s i n [12] and Sachdeva's Theorem 3-3 i n [18], Our f i r s t theorem (Theorem 2-1) i s a ge n e r a l i z a t i o n of a theorem of Hewitt-Yosida i n [13] (p. 50 Theorem 1.18). As a consequence of Theorem 2-1, we (Coroll a r y 2-2) generalize a theorem of Calderon [3] (p. 1962, 2). * * Let N be a C -algebra and N the dual space of N For a e N we define the r i g h t t r a n s l a t i o n R on N by R 9(b) = 9(ba) f o r <j> e N and b e N . Let A be a von Neumann algebra on a H i l b e r t space H . Let A be the dual of A . Let A ^ C A be the space of a l l ultra-weakly continuous fu n c t i o n a l s on A , and A^~ be the space of a l l singular functionals on A . Dixmier proved i n [7] (p. 31 Theorem 1) that A i s the dual space of the Banach space . Takesaki proved i n [20] (p. 196 Theorem 3) that A i s the J _ A d i r e c t sura of A , and A , . Moreover, he proved that A . = R A A A ' R A ZQ • A AA and A , = R / n \A , where z i s a c e n t r a l p r o j e c t i o n m A * ^ i - Z f J 0 AA A AAA (considering A C A and A C A ). We state h i s theorem i n the following form (part of Theorem 2-A was proved by Nakamura [14]). Theorem 2-A: Let <j> be a p o s i t i v e f u n c t i o n a l on A . Then ty can be uniquely decomposed as a sum of a p o s i t i v e normal f u n c t i o n a l <j> , the normal part of ty , and a p o s i t i v e singular f u n c t i o n a l ty^ , the sin g u l a r part of ty . The normal part <j>^  of <j> i s nothing but the l a r g e s t p o s i t i v e normal f u n c t i o n a l on A dominated by ty . If <j> i s c e n t r a l , then both ty^ and ty are c e n t r a l . ([20] and [14]). Theorem 2-B: Let ty be a p o s i t i v e f u n c t i o n a l on A . Then ty i s singular i f and only i f for any non-zero p r o j e c t i o n E e A , there i s a non-zero p r o j e c t i o n F e A such that F < E and <f>(F) = 0 . ([21], p. 365-366). Remark: From Theorem 2-B, we have the following: A I f ty e A i s p o s i t i v e f a i t h f u l , then the normal part ty of ty i s also f a i t h f u l . For i f ty^(E) = 0 but E =f= 0 , l e t ty^ be the si n g u l a r part of ty , then by Theorem 2-B there i s non-zero p r o j e c t i o n F <_ E such that <j>2(F) = 0 , hence ty(F) = 0 . Since ty i s f a i t h f u l , F = 0 which i s a c o n t r a d i c t i o n of our choice of F The following theorem generalizes a theorem of Hewitt-Yosida i n [13] (p. 50, Theorem 1.18). Theorem 2-1: Let 9 be a p o s i t i v e f u n c t i o n a l on A . If 9 i s singu-l a r , then given any p o s i t i v e normal 9 N e A , 6 > 0 , and E e P with 9(E) > 0 , there e x i s t s i n A a p r o j e c t i o n F < E such that 9 Q ( F ) < and 9(F) = 9(E) . Proof: 9(E) > 0 , so E =}= .0 . From Theorem 2-B there i s a non-zero p r o j e c t i o n F < E such that 9 (F ) = 0 . Since 9 (E - = 9(E) > 0 by Theorem 2-B again, we have F 2 e P such that 0 < F 2 < E - F.^  and d>(F„) = 0 . Let {F } be a chain of f a m i l i e s such that f o r each a y 2 a a F = { E } i s a family of pairwise orthogonal non-zero projections with Ot Ot p E E^ < E and <b(E^) = 0 for a l l 0 , and such that {F } i s l i n e a r l y . a — y v a a a p ordered by i n c l u s i o n . Then the union F of a l l F f o r a l l a i s J a again a family of pairwise orthogonal non-zero projections with E G <_I GeF and <b(G) = 0 for a l l G e F . By Zorn's Lemma, l e t {E } be a maximal family of pairwise orthogonal projections with E E <_ E and a d)(E ) = 0 for a l l a . Then E = E E . For i f E - E E 4= 0 then a a by Theorem 2-B we have E„ e P such that 0 < E „ < E - E E and 0 0 — a a <b(E„)-= 0 , which contradicts the maximality of {E } . Thus we y 0 a a have a family {E } of non-zero pairwise orthogonal projections i n a a A with E E = E and d>(E ) = 0 for a l l a . Since 9(E) > 0 , a {E }' i s not f i n i t e . Since <brt i s p o s i t i v e normal, E 9„(E ) = 9 0(E) < a a r0 0 a 0 a Hence the set {E : <h„(E ) 4= 0} i s countable. If {E : <pn(E ) 4= 0} a 0 a ' a U a i s f i n i t e , then l e t F = E - E - {EE : <|>-(E ) 4= 0} where E i s a Q a Y0 a 1 a Q 2 4 . not i n {E : <J> (E ) 4= 0 } a 0 a ' ty(F) = <j>(E) with F < E I t i s c l e a r that <J>Q(F) = 0 < 6 and If {E : <i> (E ) + 0 } i s i n f i n i t e , oo Let G = (ZE : <b(E ) = 0 } , then a 0 a enumerate i t as {E.}. oo (J>Q(G) = 0 and 4>Q(E) = 2 ^ g ^ i ^ w n i c h i s f i n i t e . Hence there i s i = l 0 0 0 0 an integer N > 1 such that Z <i>n(E.) < 6 . Let F = G + Z E. i=N+l 1 i=N+l 1 then <J>n(F) < 6 and Remark: Theorem 2 - 1 also gives the following f a c t : I f ty i s a p o s i t i v e s i n g u l a r f u n c t i o n a l on A and ty^ a p o s i t i v e normal f u n c t i o n a l on A If E e P i s a p r o j e c t i o n such that both <j>(E) > 0 and <J>Q(E) > 0 , then there e x i s t s a non-zero p r o j e c t i o n F < E i n A such that <j>(F) = 0 but <|>0(F) > 0 . The following c o r o l l a r y i s a g e n e r a l i z a t i o n of one of Calderon's main theorems i n [ 3 ] (p. 1 9 6 0 , 2 ) . C o r o l l a r y 2 - 2 : Let ty and ty^ be two p o s i t i v e functionals on A such that 4> -< <f> and <J)Q normal. Then the normal part ty^ of ty has the property -< ^ > hence ty i s not s i n g u l a r i f ty^ =f= 0 . I f ty^ ~ ty , then tyn ~ ty Since N > 1 and E > 0 f o r a l l a F < E a 25. Proof: Let 9 = 9^  + <j>2 w n e r e <i>^  i s t n e normal part of 9 and cj> 2 is the sin g u l a r part of 9 . Let E e P such that 9^(E) = 0 Case 1: I f 9 (E) = 0 , then 9(E) = 9 (E) + <f>2(E) = 0 . Hence 9 Q(E) = 0 since <J>A-< 9 . Case 2: I f 9 2 ( E ) > 0 but 9 Q ( E ) > 0 , then by the remark a f t e r Theorem 2-1 we have a non-zero p r o j e c t i o n F < E i n A such <j>2(F) = 0 but 9 Q(F) > 0 . Since 0 <_ 9-^F) <_ cp^E) = 0 , 9 X( F) = 0 • Hence we have 9(F) = 91<F) + <j>2(F) = 0 . Thus 9 Q(F) = 0 which contradicts the choice of F . Hence 9 Q ( E ) = 0 • Thus, f o r any E e P we have 9^(E) = 0 implies 9Q(E) = 0 . From Lemma 1-H, we have 9 ^ 9 ^ If 9Q =f= 0 , then 9^  =f= 0 , hence 9 i s not si n g u l a r . I f cj>n ~ 9 , then 9 Q^ <j>1 <_ 9 ~ 9Q , so <j>0 ~ ^ . # Let A be a von Neumann algebra on a H i l b e r t space H . If T i s a contraction operator on A , we l e t T be the adjoint operator of T defined on A into A by T 9 (A) = <j>(TA) f o r a l l 9 e A , A e A . If both T and 9 are p o s i t i v e then T 9 i s p o s i t i v e . If it both T and d> are normal then T 9 i s normal. I f {T } . i s a k family of operators on A , an element 9 e A i s said to be i n v a r i a n t under {T } . i f 4(1 A) = 9(A) f o r a l l A e A, a e A . I t i s equi-a aeA a valent to say that T 9(A) = 9(A) f o r a l l a e A and A e A ( i . e . * T 9 = d> f o r a l l a e A ) . The following theorem i s a g e n e r a l i z a t i o n of a 0 0 one of Calderon's main theorems i n [-3] (p. 1961, 3). Theorem 2-3; Let <j> be a p o s i t i v e f u n c t i o n a l on a von Neumann algebra A . Let {T } , be a family of p o s i t i v e normal contraction operators a aeA on A I f d> i s i n v a r i a n t under {T } . then ty. , the normal Y a aeA T l part of d> , i s i n v a r i a n t under {T } . r Y a aeA Proof: Let ty = d>^  + ty with normal part ty^ and singular part ty^ Since ty i s p o s i t i v e , both ty^ and ty^ are p o s i t i v e . Since T^ i s * p o s i t i v e and normal f o r a l l a e A , we have T tyn i s p o s i t i v e normal a 1 * A A A and T ty. p o s i t i v e f o r a l l a e A . But Tty. +Ttyn=Tty=ty and a 2 a 1 a 2 a A A T <J>„ i s p o s i t i v e f o r a l l a e A , hence T ty. < ty f o r a l l a e A a Y2 v a Y l — T A A So, by Theorem 2-A, T tyA < ty. f o r a l l a e A , and hence T <f>_ > d>~ J a 1 — 1 a 2 — T2 A f o r a l l a e A . But, for each a e A , T <j>„(I) = d>0(T I) < d>0(I) a 2 2 a — 2 A A since T I < I , we have T d>„ = <J>„ f o r a l l a e A . Hence 1 ty = ty. a — a 2 2 a i l for a l l a e A i . e . ty. i s i n v a r i a n t under {T } . . # T l a aeA From now on A w i l l be denote a von Neumann algebra on a H i l b e r t space H , and S w i l l be a semigroup. Consider any antirepresentation {T : s e S} of S as p o s i t i v e normal contraction operators on A s Then, f o r L e A and A e A , the function s $ (T A) i s i n m(S) U U S An element <j> e A i s sa i d to be S-invariant i f <j>(TgA) = ty (A) f o r a l l A e A and s e S . A subset M C A i s said to be S-stable i f T M <C M f o r a l l s e S . s Proposition 2-4: Let (pQ e A be p o s i t i v e , then (la) <=> ( l b ) . I f S i s l e f t amenable, then (2a) <=> (2b), (3a) <=> (3b) and (la) => (2a) => (3a); (la) A E P ((lb) A £ A + ) , <p0(A) > 0 implies i n f {^(TgA): s £ S} > 0 (2a) A e P ((2b) A e A + ) , 9 q ( A ) > 0 implies M L ( 9 Q ( T G A ) ) > 0 + o o (3a) A E P ((3b) A e A ) , i f there e x i s t s {s } . C S with n n=l 0 0 E T A e A , then 9 N ( A ) = 0 ' n=l n Proof: (lb) => ( l a ) , (2b) => (2a), and (3b) => (3a) are c l e a r since P C A + . I f A E A + with 9 0 ( A ) > 0 , then there are E e P , and c > 0 such that <pA(E) > 0 and cE < A . So 9 n ( T A) > 9 (T (cE)) = C 9 a ( T E) for a l l s e S . If (la) (resp. (2a)) holds, then i n f (<f>A(T A) : s e S} >c i n f U A ( T E) : s e S} > 0 (resp. U s — U s M ( 9 CT A ) ) > cM T(. 9 n(T E)) > 0 ). So (la) => (lb) and (2a) => (2b). t i u s — ij u s o o o o I f there e x i s t s {s } , C S such that E T A e A then n n=l T s n=l n 0 0 0 0 E T E < — E T A e A (since i t i s the strong l i m i t of _ i s c n s n=l n n=l n k E T E as k -> » ). Hence, i f (3a) holds, 4>n(E) = 0 which _ i s U n=l n contradicts our choice of E . So (3a) => (3b). It i s c l e a r that (la) => (2a) since i n f { 9 q ( T S E ) : S e S} £ M L ( 9 o ( T g E ) ) f o r any E e P OO To show (2a) => (3a), l e t E e P , {s } , C S be such that n n=i o o • o o E T E e A . Then there e x i s t s K > 0 such that E T E < K l . n=l n n=l n N So E T T E < T (Kl) = KT I < K l f o r a l l s e S and any integer N s s s s ~~~ n=l n 28. Since T T E = T E , <h„(T T E = t 4>n(T E) . Hence, f o r a l l s s s s O s s s T 0 s n n n n N : N , E d)n(T T E) < Kd) (1)1 , where 1 e m(S) i s such that l ( s ) = 1 n fo r a l l s e S . So, f o r any y e LIM , we have (<f> (T E)) = N N I y U s < J> Q( Ts E ) ) = y ( Z * o ( T s T s E ) ) 1 K * 0 ( I ) f o r a 1 1 N ' H e n c e n=l n n=l n u(4> 0(T gE)) = 0 f o r a l l u e LIM . Thus \(<$> Q(T gE)) = 0 . Since (2a) holds, $ ( E ) = 0 . So (2a) = (3a). # Lemma 2-5: Let ty^ be a p o s i t i v e normal f u n c t i o n a l on A > and S l e f t amenable. Assume that; (*) If E E P , <!>Q(e) > 0 then i n f {<t„(T E ) : s e S} > 0 holds. Then there e x i s t s an S-invariant 0 s p o s i t i v e normal f u n c t i o n a l <j> on A such that <(>Q-< ty • I f , i n addi i t o n to (*), we have <b„(E) = 0 implies KU.(T E ) ) = 0 f o r (J L TU s * E e P , then even <|>Q ~ <J> • Moreover, i f tS<!>Q 1 S c e n t r a l f o r a l l s e S , then <j> can be chosen c e n t r a l . Proof: Let y e LIM be f i x e d . Define ty on A by ty (A) = y(d>n(T A)) fo r a l l A e A • Then ty i s l i n e a r on A and |^(A)j = | y (<J)Q (T gA)) | < U 0(T gA)| <.||d>0|| ||TsA|| <JU0I| ||A|| f or a l l A e A . Since y e LIM and cf>Q i s p o s i t i v e , so i s iji . Let a e S , then ty (T A) = y (<f>0(TsTaA)) = M(<f- 0( T a s A)) = y a a4>o ( Ts A ) ) = ;J ( * o ( T s A ) ) = * ( A ) f o r a 1 1 A e A . So iji i s an S-invariant p o s i t i v e f u n c t i o n a l on A By Theorem 2-3, the normal part ty^ of ty i s S-invariant. I f E e P with ijj(E) = 0 , then 0 <_ i n f {ty (T E) : s £ S} <_ ty(E) = 0 . From (*) u s we have <!)Q(E) = 0 . Hence ty ty . By Coroll a r y 2-2, the normal part i j j ^ of ip has the property §Q< ^ • Now, i f i n a d d i t i o n to (*) we have <pn(E) = 0 implies M (<j> (T E)) = 0 f o r E e P , then (j,0CE) = 0 implies ^(E) = 0 f o r E e P , hence 9Q ~ ^ . Thus, * by C o r o l l a r y 2-2, 9g " ^1 ' I f Ts^0 i S c e n t r a l f o r a 1 1 s e S ' then \p i s c e n t r a l . Hence, by Theorem 2-A, ij> i s c e n t r a l . Let <p = ij;^ , 9 i s the required f u n c t i o n a l on A . # Theorem 2-6: Let <|>Q e A be p o s i t i v e normal and S l e f t amenable. Then the following are equivalent: Cl) There ex i s t s an S-invariant p o s i t i v e normal 9 e A such that 9Q ~ 9 . (2) E E P , (f>n(E) > 0 implies i n f {9 (T E) : s e S} > 0 and U U S 9 Q(E) = 0 => \ ( 9 0 C T s E ) ) = 0 . C3) A e A + , 9 A(A) > 0 implies i n f (9 0(T gA) : s e S} > 0 and 9 Q(A) = 0 => M LC9 0CT SA)) = 0 . Moreover, i f T <t>_ i s c e n t r a l f o r a l l s e S then A> can be chosen s Y0 c e n t r a l . Proof: (3) => (2) i s c l e a r since P C A + . (2) => (1) by Lemma 2-5. I f T 9n i s c e n t r a l f o r a l l s e S then, by Lemma 2-5, 9 can be s*0 chosen c e n t r a l . Now, suppose (1) holds. Let A e A + with 9Q(A) = then 9CT A) = 9(A) = 0 f o r a l l s E S . So 9 (T A) = 0 f o r a l l s e S , hence M (9 (T A)) = 0 . Let A e A + with <f>n(A) > 0 , then i_i u s u <pCA) > 0 . Since the set {T A : s e S} i s uniformly bounded and s i n f {9(T gA) : s e S} = <pCA) > 0 , from Cor o l l a r y 1-N' we have i n f {9 (T A) : s e S} > 0 . Hence (1) => (3). // U S C o r o l l a r y 2 - 7 : Let <j> be a p o s i t i v e normal f u n c t i o n a l on A . Let S be l e f t amenable and T d)-.-^  d) for a l l s e S . Then the following S U \J are equivalent: .* (1) There e x i s t s an S-invariant p o s i t i v e normal d> e A with ( 2 ) E e P , (J>0(E) > 0 implies i n f {d, (T gE) : s e S} > 0 . (3) A e A + , 4>Q(A) > 0 implies i n f {d>0(TgA) : s e S} > 0 . * Moreover, i f T i>n i s c e n t r a l f o r a l l s e S then d) can be chosen s Y0 c e n t r a l . Proof: Since T d>„-< <b„ f o r a l l s E S , we have d>„(A) = 0 implies s 0 U 0 <j>n(T A ) = 0 f o r a l l s e S , thus M (<j)n(T A ) ) = 0 for A e A + . Invoke now Theorem 2 - 6 . # C o r o l l a r y 2 - 8 : If A admits a f a i t h f u l p o s i t i v e normal <J)Q E A , and S i s l e f t amenable then the following are equivalent: (1) There e x i s t s an S-invariant p o s i t i v e normal f a i t h f u l <J> e A ( 2 ) 0 i s not i n the strong closure of ( T S E : s £ S} for 0 f E E P . (3) 0 i s not i n the strong closure of {T gA : s e S} for 0 j A E A + . •k Moreover, i f T 6n i s c e n t r a l f or a l l s E S then A can be chosen s r0 c e n t r a l . Proof: Since 9 i s f a i t h f u l , 9 A(A) = 0 i f and only i f A = 0 for • A E A + . Since {T A : s e S} i s uniformly bounded by ||A|| for A e A + , s Theorem 1-1 and C o l l a r y 1-M we have inf{9 (T A ) : s e S} = 0 i f and u s only i f 0 i s i n the strong closure of { T gA : s e S} for A e A + + * Since c|>0(A) = 0 i f and only i f A = 0 for A e A , ^si>Q< 9Q f o r a 1 1 s E S . Apply now C o r o l l a r y 2-7 to get C o r o l l a r y 2-8. # Lemma 2-9: Let <|>Q be a p o s i t i v e normal f u n c t i o n a l on A and * _ T 9n -< 9n ^ o r a ^ s e s • Suppose that f or any E E P , S U \) {T gE : s e S} C {E}" then (1) implies (2): 00 + 00 (1) I f A e A and there are {s } , C S such that E T A e A n n=l 1 s n=l n then 9 Q(A) = 0 . (2) If E e P then 9 (E) > 0 implies i n f { 9 (T E) : s e S} > 0 . U 0 s Proof: Let E e P with d> _ (E) > 6 > 0 but inf{ct>A(T E) : s e S} = 0 , 0 0 s 00 then there are {u } _ C S such that 9„(T E) converges to zero. Let n n=l U u k n 6, = 8/2 , then there i s u such that <j>_(T E) < 6. . If k • n, 0 u 1 1 n± u ,...,u have been chosen, l e t u be such that n l n k n ( k + l ) * 0 ( T u E ) + . Z +0 ( T u T u T u E ) 6 k + l n ( k + l ) 1 = 1 n i n k n ( k + l ) x. e. j,n(T E) + Z 9_(T T E) < <S K0 u - _ i 0 u . . . u u k+1 n ( k + l ) 1 - 1 n k n i n ( k + l ) 3 2 . This can be done since T s 9 g - < <I>Q f o r a 1 1 s e S (note: By Proposition 1-N ft we have T mrt(T E) converges to zero f o r a l l s e S . So cj>-(T T E) s y 0 u 0 s u n n converges to zero f o r any s E S . Let u be the maximal of n ( k + l ) k ^k+1 {u }. _ -such that 6 _ (T E) < -—•—- f o r a l l n > n. , n N and n. ,. N x=0 0 u k + 1 — k(0) k ( i ) n 6 k + l * 0 ( T u ... u T u E ) < k T T f ° r 3 1 1 n > n k i = 1, ...,k) . Let n k n. n s. = u . Let 8 be the von Neumann algebra generated by {I,E,T E,T I : k n, s s k s E S} . Since E e {E}' and T E E {E}" for a l l s E S , we have s T E-E = E-T E f o r a l l s E S . So {T E : s E S} C{E}' , thus s s s { I , E , T E : s e S} i s commutative. Now { E , T E : S E S } C { I } ' and s s {T I : s E S} C {I}" hence {I,E,T E,T I : s E S} i s commutative, s s s So B i s a commutative von Neumann algebra. We also have T 8 C B s f o r a l l s E S since a l l T are p o s i t i v e normal and 8 i s the u l t r a -s r weak closure of the algebra generated by {I,E,T E,T I : s £ S} (see s s Theorem 1-A and Theorem 1 - J ) . So {T g : s E S} i s also an antirepresen-t a t i o n of S as p o s i t i v e normal contraction operators on 8 . By it Theorem 1-P, 8 i s i s o m e t r i c a l l y -isomorphic to L o o(X,S,p) of a measure space (X,S,p) . So = ^0^8 induces a p o s i t i v e normal f u n c t i o n a l on L o o(X,S,p) , again denoted by ipA , and T g induces a p o s i t i v e normal contraction operator on L o o(X,S,p) f o r a l l s E S , again denoted by T g . Let Y e and Xy (the c h a r a c t e r i s t i c function of Y ) be the element i n L o o(Y,S,p) which corresponds to E , and v be the measure which corresponds to I/Ja given by [8] Theorem IV-8-16. 33. Then we have T|) (f) = / fdv f o r a l l f e L^CX.S.p) , so ^ ( E ) = ^ ( E ) = ^ Q ( X Y ) = / X ydv and ^(TgE) = / T g X y d v f o r a l l s e S . Let oo n h = (x Y " ^ ^ T s ...s X Y ) + e L>,S,p) n=l i = l n i and l e t B e 8 be the corresponding element of h i n 8 O A Then d>Q(B) = ip (B) = 4>Q(h) = / hdv . We want to show that <f>Q(B) > 0 0 0 oo and that there exists {a } , C S f o r which E T B e A • I t xs n n=l T a n=l n i+k s u f f i c i e n t to show that [ hdv > 0 and E T „ h < 1 f o r a l l J s .. • . s . — J = l J ! j >_ i >^  1 and f o r a l l k >_ 0 . Since, then, we can l e t i = 1 and 0 0 oo k -> °° , we w i l l get E T h e L (X,S,p) , hence E T B e 8 C A. S « . . S 1 ° ° . S • • - o , n=l n 1 n=l n I Since oo n / ( x - h)dv < / E E T g > s X y d v n=l i = l n i N < \ ! . \ T s ...s X Y dv n=l i = l n i co n ± E , . E > ( T s ...s.E> n=l i = l n i < E 6 = 6 . n=l i+k and d>Q(E) = / X y d v > S , so / hdv > 0 . Let k = 0 , then E T g ...s/h 3 4 . i+k = T h < T Y <T 1 < 1 f o r any i . Assume that E T h <_ 1 s. — s . A Y — s — i = i s ...s i l l J x J -1-i+k Q+l f o r any i >_ 1 and k = k» . We need to show E T, „ h <_ 1 for 0 . . s .. . . s. j=i 3 i any i > 1 . Since i+k Q+l i + k 0 + 1 ' S T s ...s. h= }. ( T s . T s . . . . s 1 + 1 h ) j = i j 1 J=l 1 J 1+1 i + l + k 0 = T (h + E T h) s i j=i+l S j " , S i + l i + l + k 0 i t i s enough to show h + E T h < 1 . Let N -{ x e X j=i+l S j 1+1 h(x) = 0} and N 2 = {x e X : h(x) > 0} . On N± i+l+k Q i + 1 + k 0 h + E T h = E T sj - ' - S i + l j=i+l S j < 1 n by assumption. On N_ , we have h > 0 , so h - x Y _ E z T s ...s XY 2 1 n=l £=1 n I and i + 1 + k 0 - n h + E T h < h + E E T h j=i+l S j ' * 1 + 1 ~ n = l * = 1 n * A oo n < h + E E T X Y n=l 1=1 Sn"Sl Y 1 Xy £ 1 35. -f CO So we have B e A , {a } . C S such that E T B e A but ff>„(B) > 0 n n=l , a 0 n=l n which contradicts to (1). So (1) => (2). // Remark: The idea of the proof of Lemma 2-9 came from the f a c t that a commutative von Neumann algebra i s i s o m e t r i c a l l y -isomorphic to L (X,S,p) f o r some measure space (X,S,p) . The proof of / hdv > 0 and E T h e L (X,S,p) follows Sachdeva's proof of Theorem 3-3 s • • • b i 0 0 n=l n i i n [18] which i n turn i s i n s p i r e d by the idea i n the proof of Theorem 2 i n Granirer [9]. The following theorem i s a ge n e r a l i z a t i o n of Hajian-Ito's r e s u l t s i n [12] and U. Sachdeva's theorem i n [18] (Theorem 3-3). Theorem 2-10: Let <|>Q e A be p o s i t i v e normal, S l e f t amenable and T V _ < rf>_ f o r a l l s e S . I f f o r any E e P , {T E : s e S} C {E}" s 0 0 s then the following are equivalent: (1) There e x i s t s an S-invariant p o s i t i v e normal <f> e A such that <j>Q ~ <j> (2a) I f A e P ((2b) A e A + ) then <j> (A) > 0 => i n f {$ (T A) : u U S S E S} > 0 . (3a) If A e P ((3b) A e A + ) then <|>n(A) > 0 => M (ty (T A)) > 0 U JJ U S (4a) I f A e V ((4b) A e A ) and there e x i s t s {s } _ C S with n n=l CO E T A e A then <f>n(A) = 0 . I s u n 36. Moreover, i f T <b_ i s ce n t r a l f o r a l l s e S then <f> can be chosen s K0 ce n t r a l . Proof: By Lemma 2-9, Corol l a r y 2-7 and Prop o s i t i o n 2-4. # Remark: In case A = L (X,S,p) f o r a p r o b a b i l i t y measure space (X,5,p) , S a l e f t amenable semigroup of p o s i t i v e l i n e a r contraction operators on L 1(X,S,p) and {T g : s e S with I f ( g ) = f(sg) f o r f e L^X.S.p) , g e L 1(X,S,p)} (consider L r o(X,5,p) = L 1(X,S,p) and T g the adjoint operator of s ), our Theorem 2-10 reduces to Hajian-Ito's r e s u l t s i n [12] and U. Sachdeva's theorem i n [18] (Theorem 3-3). (Note: Dixmier proved i n [7], p. 31, Theorem 1, that a von Neumann algebra A i s the dual space of the Banach space A ^ ) Since there i s a p a r t i c u l a r transformation on a f i n i t e von Neumann algebra A , c a l l e d the canonical c e n t r a l valued function (see Theorem 1-Q, Theorem 1-R, Theorem 1-S and Proposition 1-T), we w i l l discuss our r e s u l t s above f o r f i n i t e von Neumann algebras. We s t i l l l e t S be a semigroup and {T g : s e S} an antirepresentation of S as p o s i t i v e normal contraction operators on a von Neumann algebra A Lemma 2-11: Let A be a f i n i t e von Neumann algebra with the canonical c e n t r a l valued function ^ . Let <J>Q be a p o s i t i v e c e n t r a l f u n c t i o n a l on A and T a p o s i t i v e contraction operator on A such that T ty^ i s c e n t r a l . Then we have <j> ((TA)^) = cj> (T(A^)) for a l l A e A . 37. Proof: Since both < p n and T are p o s i t i v e , so T c p n i s a p o s i t i v e :entral f u n c t i o n a l on A . By Proposition 1-T, we have ^ ( ( T A ) ^ ) - 9 0(TA) = T * 9 Q ( A ) = T % Q ( A ^ ) = 9 ( ) ( T ( A ^ ) ) for a l l A e A . # Theorem 2-12: - Let A be a f i n i t e von'Neumann algebra with the canonical c e n t r a l valued function H , and l e t <j>A e A be p o s i t i v e normal c e n t r a l , * * Let S be l e f t amenable with T cb„ -< d>_ f o r a l l s e S . Suppose T m A s U 0 s (J i s c e n t r a l f o r a l l s e S and Z i s S-stable, then the following are equivalent: (la) There e x i s t s an S-invariant p o s i t i v e normal c e n t r a l <j> e A such that cj> ~ 9Q (lb) There exists an S-invariant p o s i t i v e normal \p z I with (2a) If A z V C\ Z ((2b) A e Z + , (2c) A e P , (2d) A e A + ) then ((A) > 0 implies i n f { 9 o ( T g A ) : s E S} > 0 . (3a) I f A e V C\ 1 ((3b) A e Z + , (3c) A c ? , (3d) A e A + ) then 9 Q(A) > 0 implies M^ (<j>n (T gA)) > 0 4- C O (4a) If A e V HI ((4b) A £ Z ) , and there e x i s t s {s } . <_ S n n=l C O with E T A e A then 9-(A) = 0 . I S u n Proof: From Pro p o s i t i o n 2-4 we have (2c) <=> (2d) and (3c) <=> (3d). From Pro p o s i t i o n 1-T and Lemma 2-11 \<re have (la) <=> ( l b ) , and 9 0(A) = 90(A^) . 9 0CT gA) = 9 ( ) C C r G A ) ^ ) = 9 Q (T gA^) f o r a l l s e S , A z A . So (2b) <=> (2d) and (3b) <=> (3d). I f Z i s S-stable, then, since Z i s commutative, by Theorem 2-10 (lb) <=> (2a) <=> (2b) <=> (3a) <=> (3b) <=> (4a) <=> (4b). // 38. Remark: I f {T g : s e S} i s any antirepresentation of S as c e n t r a l * A* operators on A , then Tg^g I s c e n t r a l for any c e n t r a l §Q e A and s e S . C o r o l l a r y 2-13: I f A i s a a - f i n i t e f i n i t e von Neumann algebra, S i s l e f t amenable with {T g : s E S} an antirepresentation of S as p o s i t i v e normal c e n t r a l contraction operators on A and i f Z i s S-stable then the following are equivalent: (la) There e x i s t s an S-invariant p o s i t i v e normal f a i t h f u l c e n t r a l * <j> e A (lb) There e x i s t s an S - i n v a r i a n t p o s i t i v e normal f a i t h f u l ty e 2 (2b) 0 + A E V CM ((2b) 0 =}= A e Z + , (2c) 0 =f= A e P , (2d) 0 =(= A E A + ) then 0 i s not i n the strong closure of {T A : s E S} . s (3a) A e P H Z ((3b) A E Z + ) , i f there e x i s t s {s } C S n n=l CO with E T A E A then A = 0 . n=l n Proof: Since A i s a - f i n i t e f i n i t e , there i s a p o s i t i v e normal f a i t h f u l c e n t r a l ty e A . Hence i>g(A) = 0 <=> A = 0 f o r A E A . Hence i n f {<j)n(T A ) : s e S} = 0 i f and only i f 0 i s i n the strong closure of (J s {T A : s e S} f o r A E A + (since the set {T A : s E S} i s uniformly s s bounded by ||A|| , and apply Theorem 1-1 and Co r o l l a r y 1-M) . Thus our Corol l a r y 2-13 follows from Theorem 2-12. // 3 9 . Proposition 2-14: Let A be a von Neumann algebra, S a l e f t amenable semigroup with an antirepresentation ( T g : s E S} of p o s i t i v e contraction operators on A . Let ty^ be a p o s i t i v e normal f u n c t i o n a l on A , and <t> an S-invariant p o s i t i v e normal f u n c t i o n a l on A such that ty^ ~ ty Then f o r each y E LIM , the function ty defined by ty(A) = u(<j>g(T A)) fo r a l l A e A i s an S-invariant p o s i t i v e normal f u n c t i o n a l on A with ty ~ ty . I f T ty i s c e n t r a l f o r a l l s E S then ty i s c e n t r a l . Proof: C l e a r l y ty i s an S-invariant p o s i t i v e f u n c t i o n a l on A since <J>Q i s p o s i t i v e and y i s l e f t i n v a r i a n t mean on m(S) . I f E e P such that *.(£) = 0 then <fc(T E) = <fc(E) = 0 f o r a l l s e S . So tyn(T E) = 0 T u s u s fo r a l l s E S . Also we have A Q ( E ) > 0 implies i n f {cj) 0(T gE) : s £ S} > for E E P , since ty ~ ty^ and ty i s S-invariant p o s i t i v e normal. So we have ty ~ <j>Q , since i n f { ( J) 0(T sE) : s e S} <_ ty(E) f o r E E P . Now l e t { E a ) a be a family of pairwise orthogonal p r o j e c t i o n s . Since both ty and tyn are p o s i t i v e , so both {E : ty (E ) > 0} and {E : ty(E ) > 0} are countable. Let {E } °°_ = {E : A (E ) > 0} U {E : ty(.E ) > 0} , n n=l a U a a a. CO CO CO then Z ty(E ) = Z ty(E ) and Z tyn (E ) = Z d>A(E ) . . Let G = Z E a T T n U a ^ 0 n i n a n=l " a n=l n=l and F = Z E - G , then <j> (F) = 0 hence ^(F) = 0 i . e . ty(Y. E ) = ^ (G) Let A = G - ? E, , then A converges to zero strongly. So tyn(A ) n . . . k n O n k=l converges to zero. Since ty i s S-invariant and ty ~ ty^ , by Corol l a r y 1-N' we have ' K ^ g ^ ) = < f ) ^ n ^ converges to zero uniformly on s . Hence, by Co r o l l a r y 1-N' again, ^o^s^ r P converges to zero uniformly on s Thus iJ>(A ) converges to zero. Since ty(A ) = ty(G) - Z ty(E ) we have n n i T k=l KG) = E if>(E ) • Hence i>(Z E ) = i|>(G) = E i|>(E ) = Z f(E ) . , n a .. n a n=l a n=l a Thus \p i s po s i t i v e and completely additive, hence, by Theorem 1-B, * i s p o s i t i v e normal. I f T cp i s central for a l l s e S , then s u <j>0(Ts(AB)) = T*9()(AB) = T*cp0(BA) = (T g (BA)) for a l l s e S and A, B e A . So (p 0(T g(AB)) = ^ Q ( T Q ( B A ) ) as elements i n m(S) . Hence ip(AB) = i^(BA) for a l l A, B i n A . i . e . \p i s central. // CHAPTER 3 SEMIGROUPS OF NORMAL *-HOMOMORPHISMS The main purpose of the f i r s t part of this chapter i s to show that the connection between the nonexistence of weakly-wandering sets of p o s i t i v e measure and the existence of i n v a r i a n t measures, f i r s t found by Hajian-Kakutani i n [11] s t i l l holds true f o r von Neumann alge-bras. Weakly-wandering sets w i l l be replaced by weakly-wandering projections i n t h i s case. The r e s u l t s of Hajian-Kakutani [11] were generalized by L. Sucheston [19] to any non-singular transformation and then i n turn by E. Granirer i n [9] (Hajian-Ito i n [12] and U. Sachdeva i n [18]) to l e f t amenable semigroups of nonsingular transformations ( c o n t r a c t i o n s ) . In the second part of th i s chapter we give conditions which are s u f f i c i e n t f o r the existence of a complete set of S-invariant p o s i t i v e normal functionals on a von Neumann algebra A In t h i s chapter we. w i l l assume that {T : s e S} i s an s antirepresentation of a semigroup S as normal *-homomorphisms on a von Neumann algebra. The r e s u l t s i n Chapter 2 remain true i n th i s case. The main r e s u l t s of the chapter are Theorem 3-4, Theorem 3-14, Coro l l a r y 3-7,. Theorem 3-15 and Cor o l l a r y 3-.17. 42. Let A be a von Neumann algebra on a Hilbert space H . The following Proposition 3-1 is well known and we bring i t s proof for the sake of completeness. "k 4 - 4 -Proposition 3-1: Let T be a -homomorphism on A . Then T(A ) C_ A , T(P) C ? , and ||TA|| <. ||A|| for a l l A e A . If TI = I then T(U) C U . Proof: A e A + <=> A = B*B for some B e A => T(A) = T(B)*T(B) >_ 0 . Hence T(A +) C A + . If E e P then E 2 = E = E* , hence T(E.)T(E) = T(E 2) = T(E) = T(E*) = T(E) * . So T(P) C V . Suppose A e A and ||A|| <_ 1 , then A*A <_ I . So T(A)*T(A) <_ TI <_ I => J| TA|| <_ 1 . Hence || TA ] <J|A|| for a l l A e A . If TI = I and U e U then U*U = UU* = I , hence T(U) T(U) = T(U U) = T(I) = I = T<UU ) = T(U)T(U) . So T(U) C U . # Proposition 3-2: Let A be a f i n i t e von Neumann algebra with the canonical central valued function *y • • Let T be a -homomophism on A with TI = I . Then ^» T = T o tj i f a n c j only i f Z is T-stable (i.e. T ( Z ) d Z ). Proof: If tj o T = T ° kf , then, letting A e Z , we have T(A) = T(A^) = (TA)^7 e Z . Hence Z is T-stable. Suppose Z is T-stable, then o o o T(Z) C Z . For A e A we have T(K ) C ^ , where is the convex h u l l of {UAU ^ : U e U} . Since A^ is an uniform limit of h elements in K (see Theorem 1-R), so T(A ) is an uniform limit of fl elements i n K . Hence T(A V) e K f\ T(Z) C K T A H Z , where K f o r A e A , i s the uniform closure of K . But (TA)^ i s the only element i n K ^ H Z ( [ 5 ] , p. 2 5 1 Theorem 1 2 ) , we have T(A^) = (TA)^ Hence T o tj = tq 0 T . // Remark: If A i s a f i n i t e f a c t o r and T i s a -homomorphism on A such that TI = I , then the center Z of A i s T-stable, hence T o tj = tj o T . If ty i s the unique f i n i t e trace on A such that 4 ( 1 ) = 1 , then by C o r o l l a r y 1-T', A^ = ty(A)I f o r a l l A e A . Hence <j>(TA)I = (TA)^ = T(A^) = T(<j)(A)I) = A (A) I f o r a l l A e A . Hence ty(TA) = 4>(A) f o r a l l A e A i . e . d> i s i n v a r i a n t under T . I f ty i s any f i n i t e trace on A , then by Proposition 1-F and P r o p o s i t i o n 1 - 0 ty i s a multiple of ty by a p o s i t i v e number, hence ty i s i n v a r i a n t under T . Hence a l l f i n i t e traces on a f i n i t e f a c t o r are i n v a r i a n t under any * -homomorphism which leaves I f i x e d . * Remark 3 - 3 : Proposition 3 - 1 implies that any -homomorphism T on A i s a p o s i t i v e contraction operator on A . If ty^ i s a c e n t r a l f u n c t i o n a l on A and T i s a -homomorphism on A , then T A Q ( A B ) = < j , 0 ( T ( A B ) ) = <j) 0(TATB) = ty^ ( T B T A ) = ^ ( T C B A ) ) i . e . T * Q ( A B ) = T J Q ( B A ) f o r a l l A , B i n A . Hence T <j>^  i s c e n t r a l . Let S be a semigroup and { T : s e S} be an antirepresentation of S as normal -homomor-s phism on A . Then a l l r e s u l t s i n Chapter 2 remain true and we can replace the condition that "T tyn i s c e n t r a l for a l l s e S" or "T i s s r 0 s c e n t r a l f o r a l l s e S" by the condition "ty^ i s c e n t r a l " . 44. From now on we l e t S be a semigroup with {T g : s e S} an antirepresentation of S as normal -homomorphisms on the von Neumann algebra A (acting on the H i l b e r t space H ). A p r o j e c t i o n E i n A i s s a i d to be weakly-wandering i f there e x i s t s a sequence ^ s n ^ n - i ^ ~ - ^ • CO such that {T g ®' n-± a family of pairwise orthogonal proj e c t i o n s , n We w i l l f i n d a r e l a t i o n between the nonexistence of weakly-wandering projections and S-invariant p o s i t i v e normal functionals on A The following i s Proposition 1 of [9]. Propo s i t i o n 3-A: Let S be a r i g h t amenable semigroup and u e RIM. Let f e m(S) be r e a l and u(f) = 0 . Then f o r any a^,...,a e S n and 6 > 0 there i s some a e S such that f(cr) + E f(aa.) < 6 i = l 1 Theorem 3-4: Let (p^ be a p o s i t i v e normal f u n c t i o n a l on A . Let S be amenable and E'T E = T E-E f o r any E e P , s e S . Then s s (1) <=> (2) and (1') <=> (2'): * (1) There ex i s t s an S-invariant p o s i t i v e normal <J> e A such that 9 Q-< 9 • (2) There i s no weakly-wandering p r o j e c t i o n E E P with 9Q( E) > U A (1') There ex i s t s an S-invariant p o s i t i v e normal <p e A such that 9Q - 9 • (2*) If E E P then 9 Q(E) = 0 => M(<J> (T gE)) = 0 , and (2) holds. Moreover, i f (J> i s ce n t r a l then <j> can be chosen c e n t r a l . 45. Proof: Suppose ( 1 ) holds. Let E e P be a weakly-wandering p r o j e c t i o n , and l e t {s }OT . C S be such that {T E}°° are pairwise orthogonal, n n=l s n - l n „ CO N Then E T E E A and, since <j> i s S-invariant, N<j)(E) = <j> ( E T E) _ s * s n=l n n=l n <_ d> ( E T E) < co for a l l integers N > 0 => <j>(E) = 0 => <f>N(E) = 0 . I s u n Hence ( 1 ) => ( 2 ) . Suppose ( 1 ' ) holds. Let E e P with <|>Q(E) = 0 then <f>(T E) = <|>(E) = 0 f o r a l l s e S => <Ji0(T E) = 0 for a l l s e S => M(<})Q(T gE)) = 0 . Since ( 1 ' ) implies ( 1 ) and ( 2 ) , ( 1 ' ) implies ( 2 ' ) . Now suppose that ( 2 ) holds. Let p £ IM and define ij,(A) = u(d>n(T A ) ) f o r a l l (J s A e A . Then ty i s an S-invariant p o s i t i v e f u n c t i o n a l on A , and ty i s c e n t r a l i f ty^ i s c e n t r a l . We prove now that *Q-< ^ • Suppose E E P such that ty(E) = 0 but <f>Q(E) > <5 > 0 . Let <$k = 6/2 . Since the function d>rt(T E) i s r e a l and u(<j>n(T E)) = ty(E) = 0 , then, by T 0 s U s Pro p o s i t i o n 3-A, there i s s^ e S such that <J>Q(TS E ) < 6^ . Let P , = E-T E and F, = { P . } , then E - E © T < E , where E <B F 1 s^ 1 1 1 — i means the sum of a l l elements i n F^ (from now on we l e t E €B F to mean the sum of a l l elements of a family F of pairwise orthogonal p r o j e c t i o n s ) . We also have ( ^ ( P ^ <. *Q(T s E) < 6 1 . Since 0 <_ y (<J>Q (T g (E - E (D V±))) <_ y (<j) ( T g E ) ) = 0 and <j) 0(T g(E - I @ F ^ ) i s r e a l , there i s , by Propo-s i t i o n 3-A, S e S such that * 0 C T s 2 ( E - E ® F l ) } + • 0 C T s 2 s 1 C E - i e F l } ) < 6 2 Let P = (E - E © F-^'T, (E - E $ F j ) and F 2 = ¥± U { P 2 > Let 46. P 2 1 = (E - F ^ ' T ^ E - E F 2 ) and F ^ - F 2 U f P 2 1 > = F ^ ( ^ - ^ 1 Then, by ( i 2 ) , we have W + + 0 ( P 2 1 ) < 62 (1P Since 0 <_ y(<|.0(Ts(E - E © F 2 1 > ) ) i P ^ ^ ) ) = 0 and 9n(T s(E - E © F ^ ) ) i s r e a l , by Proposition 3-A we again have s^ E S such that ^ 0 ( T s 3 ( E " Z ® F 2 1 } ) + * 0 ( T s 3 s 2 ( E " Z ® F 2 1 » + * 0 ( T s 2 s 2 S l ( E " Z® hl» < 6 3 Let P = (E - E © F . J ' T ( E - E © F 0 1 ) and F = F , L/ { P,} . Let i 21 s^ 21 3 21 3 P32 " ( E " Z ® F 3 ) ' T s 3 s 2 ( E " E ® F 3 } a n d F32 = F3 ^ { P 3 2 } = F21 U ^ ' V Let P 3 2 1 (E - E © F ^ ^ E - E © F ^ ) and F 3 2 1 = F 3 2 U ^ = F 2 1 U { p 3 » P 3 2 ' P 3 2 i } * T h e n ' b y ^ 3 ^ ' w e h a v e * 0 ( P 3 ) + W + *0 ( P321 ) K S3 ( ± 3 ) I f s 1,s„,...,s have been chosen with F , l N a family of ± z n n(.n— I; . . .Zl pairwise orthogonal projections, a l l of them les s than or equal to E (where ^i» F2i' F321 a S ^ n t^' e a b o v e - ' > a n c ' k-1 W + .f1 • o ( P k ( k - l ) . . . i ) < 6k ^ 47. where k = 1,2,...n , F k ( k _ 1 } > < ^ = F ( k_D ( k_ 2) . . . 21 { P k ' P k ( k - l ) , ' " '' Pk(k-1) £^ with £ = l,2,...k , then we can choose s n + 1 e s > DY Prop o s i t i o n 3-A, since 0 <. y (<j>0 (T g (E - E © F n ( n _ 1 ) 2 1 ) ) ) i ^ ^ " ^ = 0 and <i>_(T (E - I © F , l N i s r e a l , such that T0 s n(n-l;...21 9 n ( T s + (E - Ee F n ( n _ 1 ) - f _ 2 1 ) ) + E , Q ( T s _ (E - Z © F n ) ) n+1 £=1 n+1 n £ < 6n+l ( i n + l } Let P , = (E - E © F , -.x 9 n ) ' T (E - I © F , , N 9 , ) and n+1 n(n-l)...21 s n4.^ n ( n - l ; . . . 21 F , = F , 1 X „, U {P ^} • Let P, .... = (E - Z © F ,-, ) "T (E -n+1 n(n-l)...21 ^ 1 n+1 (n+l)n n+1 s n + i s n E © F and F, ,, >. = F , ,^  0 1 U C p . n p / ,,x } • L e t n+1 (n+l)n n(n-l)...21 w n+1, (n+l)n P(n+l)n...£ = ( E " E ® F(n+l)n... ( £ + ! ) ) ' T s n + 1 s n . . . s / E " E ® F(n+l)n...(£+1)> a n d F(n+l)n...£~ F n ( n - l ) . . . 2 1 U { P n + l ' P ( n + l ) n " - " P ( n + l ) n . . . £ } where £ = l,2,...,n . Then, by ( i n + ] _ ) » w e have ^ 0 C P n + l ) + 4 • 0 ( P C n + l ) n . . . i ) < 6 n + l ^ and F, , N ~, i s a family of pairwise orthogonal p r o j e c t i o n s , a l l of (n+l;n... z l them are less than or equal to E . In this way we get a countable i n -f i n i t e family F = (p, /i I N i = 1,2,...,k and k = 1,2,...} o o k(k-l) ... £ of pairwise orthogonal p r o j e c t i o n s , a l l of them are less than or equal to CO CO E , and * 0(E<S F J E [ E V P k ( k - l ) . . .£> ] < J ; « k = « • Also, K. X X/ — X K. X oo from our construction of F , we have a sequence {s } n C S such co n n n = l t h 3 t Pk(k-1)...£ = (E - E © F k ( u _ 1 ) . . . t t + 1 ) ) - T S k S k _ i > > . 8 A ( E " E ® Fk(k-1) (£+!)'* W ± t h £ = 1' 2> • • •» ^ k~ 1) » a n d \ = ( E " E ® ^ k-D.-.z '^X^ - E ® FCk-i)...2i> ' where F k ( k _ n ) . . . £ " F ( k - i ) . . . 2 1 ^ { P k ' P k ( k - l ) ' ' " ' ' P k ( k - 1 ) . . ' k = 1,2,... . Now l e t G = E - E Q F^ , then G <_ E i s a p r o j e c t i o n i n A . Since 4, (E) > 6 and <j>0(£© F J < 6 ,4, (G) = <j>0(E) -CO <{>„(E© F ) > 0 . We claim that {T G} . are pairwise ortho-^0 °° s . ..s.s, n=l n 2 1 gonal. Let j > i be two integers with i >_ 1 , then, since T G = T ..•T G and T G = T .••T T . . . T G , S . - - - S . S . S , S . S . . . . S S S ^ S . S . .. S , 1 2 1 1 1 j 2 1 1 1 l + l k T G-T G= T ...T (G'T ...T G) S . * « * S _ S _ . S 4 « « » S _ S - S - S , S * , - > s, 1 2 1 j 2 1 1 1 l + l j = T (G-T G) 1 2 1 j i + l So we need only to show that G-T G = 0 s , • • • s . _ J 1+1 Case 1: If j = i + 1 , then O i G ' T G <_ T G ^ T (E -Si+1 Si+1 i + l *Q ha-i)...2i> a n d 0 l G ' T s i + 1 G - G - E " E ® F i ( i - D . - . 2 i • So 0 < G-T G - Si+1 < (E - Effi F . ^ ^ . ^ - T ^ C E - Ee F 1 ( 1 _ 1 ) > > > 2 1 ) P.. < F o 1+1 — ° But 0 < G-T G<_G = E - E © F which i s orthogonal to E © , so Si+1 G-T G = 0 . Si+1 . Case 2: If j > i + 2 , then 0 <_ G'T G s . • • • s . -3 i + l <_ T G 3 i + l < T (E - E © F. n+iO — S . . . . S . . T j ( j - 1 ) . . . ( i + 2 ) 3 l + l a n d 0 1 G - T s . . s . ^ G ^ G ^ ( E " E 8 F 3 ( 3 - D . . . ( i + 2 ) } * S ° 3 i + l 0 < G'T G 3 i + l < (E - I S F . ^ j , . . . ( 1 + 2 ) ) ' V . . . B . + 1 ( E - *® F ] ( j-l)...Ci42)' - P J 0 - l ) . . . ( i + U 1 E ® F - • 5 0 . But 0 < G-T G < G = E - E © F which i s orthogonal to E <£ F — S .... S . . . — oo ° oo J 1+1 So G-T G = 0 . Hence T G i s orthogonal to T G S . • . • S , _. S....S-S-j i + l i 2 1 j 2 1 CO for any integers i 4 j with i , j >_ 1 , i . e . {T G} are s • • . s _ s - in j. n 2 1 pairwise orthogonal. Thus we" obtain a weakly-wandering p r o j e c t i o n G e P with <PQ(G) > 0 . That contradicts to (2). So we have 9Q~< ^ If (2') holds, then (2) holds and i f 9 (E) =0 f o r E z P we have M((p (T E)) = 0 , hence tJj(E) = 0 . So (2') => 9 ~ ip . Now apply U S U C o r o l l a r y 2-2 and Theorem 2-3, the normal part ij> of ^ i s an S-invariant p o s i t i v e normal f u n c t i o n a l on A such that 9Q-^  ^ (resp. tp^  ~ ^ f o r the case of (2') holds). Moreover, i f <f>A i s c e n t r a l then i s c e n t r a l , hence by Theorem 2-A, i s c e n t r a l . Let 9 = ty^ , then (2) => (1) and (2') => (1'). # Remark: (a) The idea of the proof of Theorem 3-4 i s i n part i n s p i r e d by the proof of Theorem 2 of Granirer [9]. (b) Let A be a von Neumann algebra on a H i l b e r t space H . Let S be a semigroup with {T g ; s e S} an antirepresentation of S as -homomorphisms on A . I f 9 i s an S-invariant p o s i t i v e normal f u n c t i o n a l on A with support E . Then <p(E,) = 9 (I) and 9 9 d>(T E J = d)(E ) = 9(1) for a l l s e S . Since E i s the smallest s 9 9 ip p r o j e c t i o n i n A with d>(E) = 9(I) , we have T E > E J f o r a l l 9 s 9 — 9 s e S . Hence 1 E ' E , = E •T E, = E , for a l l s E S . So, at l e a s t , s 9 9 9 s 9 9 there e x i s t s a p r o j e c t i o n E i n A such that T E -E = E "T E f o r 9 S 9 9 9 S 9 a l l s e S . 51. (c) Let A,S and {T g : s e S} be as i n (b). Let 9 be an S-invariant p o s i t i v e normal f u n c t i o n a l on A . Let E e P and s E S such that T G E = I , then 9(E) = 9(T GE) = <j>(I) , hence E > . If there i s a p o s i t i v e normal f u n c t i o n a l 9 on A such that <f>Q-< 9 » then E I < E ; . I f E > E then 9(E) = 9(I) hence <p N(E) = <pn(I) . 9Q — -9 — 9 U U Also 9 ( T E ) = 9(E) = 9(1) for a l l s e S , hence 9 n(T E ) = 9 (I) = 9Q(E) for a l l s E S . Thus <J>N i s S-invariant on the subalgebra generated by a l l projections G > E — 9 Remark 3-5: In Theorem 3-4, i f T <j)_ 9- f o r a l l s E S then we can s Y0 ^ Y0 replace M by and S needs to be only l e f t amenable. In this case, (1) <^ > (2) <=> (1') <=> (2'). I f 9 i s c e n t r a l then 9 can be chosen c e n t r a l . Proof: It i s clear that (1') => (1). The proof f o r CD => C2) i s the * same as i n Theorem 3-4. Since T 4. 7 L f o r a l l S e S , i f E e P s 0 0 we have that 9 QCE) = 0 implies M ^ ^ O ^ E ) ) = 0 . Hence (2*) and (2) coincide. Therefore we need only to show C2) => (1'). This w i l l be c l e a r by C o r o l l a r y 2-7 and the following lemma. # Lemma 3-6: Let 9^  e A be p o s i t i v e normal. Assume that Tg(j)n_< 9^  f o r a l l s E S and E'T E = T E - E f o r any E E P , s E S . Then the s s following are equivalent: (1) I f E E P then 9 QCE) > 0 implies inf' U n C ^ E : s E S} > 0 . C.2) There i s no weakly-wandering p r o j e c t i o n E E P with 9QCE) > 0 Proof: I f E £ P i s a weakly-wandering p r o j e c t i o n , then there i s a CO CO sequence {s } . C~ S such that {T E} _ are pairwise orthogonal, n n=l s n=l n CO 00 So Z 4>Q(tS E ) = ^ £ T s E) < co , which implies 4>Q(Ts E) converges n=l n n=l n n to zero, hence i n f {<j)n(T E) : s e S} = 0 and (1) => (2). Suppose (2) u s holds. Let E e P be such that A (E) > 6 > 0 but i n f {<j>n(T E) : U u s s e S} = 0 . Let 6 k = 6/2 . Since i n f { ^ ( l ^ E ) : s e S} = 0 , th ere i s {u } _, C S such that (b„(T E) converges to zero. Hence n n=l Y0 u n there i s u such that <{>„(T E) < 6 -. • Follow the proof f o r Theorem TU u 1 1 n^ 3-4 step by step with the replacement of Prop o s i t i o n 3-A by the property & CO T A>rx-< <t>^  f ° r a l l s e S . We get a sequence {u } , .. which has s Y0 Y0 n^ k=l the same property as the sequence ^ s n ^ n - l ^ n t* i e P r o ° f °^ Theorem 3-4. Also we get the same family F^ of pairwise orthogonal pr o j e c t i o n s which we got i n the proof of Theorem 3-4 (replace the s^ i n Theorem 3-4 by u here, k = 1,2,... ). Let G = E - F , we have, as i n n k the proof of Theorem 3-4, that G i s a weakly-wandering p r o j e c t i o n i n A with 4>Q(G) > 0 which contradicts to (2). Thus, f o r E £ P , 6 (E) > 0 implies i n f {4 n(T E) : s e S} > 0 i . e . (2) => (1). # U U S Remark: (a) Lemma 3-6 i s a ge n e r a l i z a t i o n of Theorem 6 i n [2j (p. 305). The proof i n [2] i s i n c o r r e c t . For a correct proof see J9J ( l a s t theorem of section 1). Remark 3-5 generalizes the "remark" a f t e r Theorem 2 i n [9]. The idea of our proofs f o r Remark 3-5 and Lemma 3-6 are i n s p i r e d by the remark of Granirer ([9]) a f t e r Theorem 2, and the l a s t theorem of sec t i o n 1 of [9]. (b) In the proof of (2) => (1) i n Theorem 3-4 (resp. i n Lemma 3-6) we proved that i f there i s a pr o j e c t i o n E i n A such that <KE) = 0 but ( p 0 ( E ) > 6 > 0 (resp. 9 ( )(E) > 6 > 0 but i n f (cj> n(T gE) : s e S} = 0 ), where (A) = u(<f>n(TgA)) f o r A i n A and u i s a f i x e d i n v a r i a n t mean on m(S) , then there i s a weakly-wandering p r o j e c t i o n G such that <PQ(G) > 0 . I t w i l l be much eas i e r to prove the existence of such a G i f we replace the condition that " E-I E = I E'E f o r any E e P , s s s e S " by a strong condition that ' {T gE : s e S} C {E}" f o r any E e P '. The proof ( i n s p i r e d by the proof of Theorem 2 i n [9]) i s as follows: k Let 6 k = 6/2 . Since <P 0(T SE) i s r e a l and y ( 9 0(T gE)) = 0 , by Proposition 3-A there i s s e S such that 9 n ( T E) < & ( i n case of 1 CO Lemma 3-6, there i s a sequence ^ u n ^ n - ] _ ^ ^ such that 9Q^ u E) converges n to zero, hence there i s s. = u such that 6n(T E) < 6. ). If 1 n^ 0 s^ 1 s.. , s„,. . . ,s, (s. = u , s. = u s, = u i n case of Lemma 3-6). 1 2 k 1 n, 2 n„ k n, 1 2 k * have been chosen, then by Proposition 3-A (by T s9Q-< f ° r a ± ± s e S and Proposition 1-N i n case of Lemma 3-6) there i s s( k4-^) e ^ ^ n case of Lemma 3-6, s ,.. N = u ) such that fc (k+D n ( k + i ) cp (T E) + E <p (T E) < 6 . Let U (k+1) i = l U S ( k + l ) S k ' ' - S i co j i F = V V [(T E)-E] , then F i s a p r o j e c t i o n and 1 ., S • • • s . x=l n x oo n co 9 n(F) <_ E E c p n (T E) < E 6 =5 . The i n e q u a l i t y i s true n=l 1=1 n l n=l because of the fa c t that (T E ; i = 1,2,...n and n = 1,2,...} are S • • • s * n l commutative (the commutativity of this family comes from {T gE ; s e S} G ( E } " ) 54. Let G = E - F , then G i s a weakly-wandering p r o j e c t i o n with ( J , Q ( G ) > U The f a c t that Y Q ( G ) > 0 i s c l e a r since 4>Q(E) > <5 and <j>Q(F) < <5 • Consider any j > i > 1 , we have T G-T G = s....s„s s . • • • s _ s i 2 1 j 2 1 T (G-T G ) . Since T G = T ( E - F ) s . . . . s 2 s 1 ^ S j . . . s ( i + 1 ) 1 S j - " S ( i + l ) S j - " S ( i + l ) <_ T E <_ F and F i s orthogonal to G , we have S j ' " S ( i + l ) G'T G = 0 . Thus T G orthogonal to T G J (i+l) x 2 1 J 2 1 oo f o r any j > i > 1 , hence {T G} are pairwise orthogonal S • • • S „ S XI—X n 2 1 i . e . G i s a weakly-wandering p r o j e c t i o n . // Co r o l l a r y 3-7; Let ^ e A be p o s i t i v e normal. Let S be l e f t * amenable with E-T E = T E-E and T A -< A f o r any E E P , s e S s s s u u Then the following are equivalent: (1) There e x i s t s an S-invariant p o s i t i v e normal A e A such that Y ~ YQ (2a) I f A e P ((2b) A e A + ) then T Q ( A ) > 0 implies i n f {i>AT A) : s e S} > 0 . 0 s (3a) I f A e P ((3b) A e A + ) then d, (A) > 0 implies W T s A ) ) > 0 (4a) I f A e P ((4b) A e A + ) and there e x i s t s {s }°° . C S such n n=l CO that E T g A e A then A Q(A) = 0 . n=l n (5) Tliere i s no weakly-wandering p r o j e c t i o n E z P such that 4>0(E) > 0 . Moreover, i f A^ i s c e n t r a l then A can be chosen c e n t r a l . 55 Proof: Suppose (4a) holds and l e t E be a weakly-wandering p r o j e c t i o n oo oo i n A . Let {s } -, C S be such that {T E} , are pairwise ortho-n n=l s n=l n CO gonal, then £ T g E e A . So, by (4a), 9 Q ( E ) = 0 . Hence (4a) => (5) n=l n Now our Corollary 3-7 follows from Lemma 3-6, Remark 3-3, Corollary 2-7 an Proposition 2-4. // Theorem 3-8: Let <pn z A be p o s i t i v e normal. Let S be l e f t amenable and E-T E = T E-E f o r any E e P , s e S . Then the following are s s equivalent: * (1) There e x i s t s an S-invariant p o s i t i v e normal <J> e A with 9 0 ~ <j) . (2a) I f A e P ((2b) A e A + ) then 9 Q ( A ) = 0 <=> ^ ( ^ Q (T gA)) = 0 Moreover, i f ^ i s c e n t r a l then <j) can be chosen c e n t r a l . Proof: (2b) => (2a) i s c l e a r since P C A + . Suppose (2a) holds. Let a e S and E z P with 9 Q ( E ) = 0 , then y ( 9 0 ( T G E ) ) = 0 f o r a l l u e LIM . Now, f o r a l l u e LIM , u (cf>n(T (I E))) = u(cp n(T E ) ) = y ( £ 0 9 A ( T S E ) ) = y ( 9 n ( T S E ) ) = 0 so M L ( 9 ( ) (T g ( T ^ E ) ) ) = 0 . Hence * 9 0 ( T O E ) = 0 for a l l a z S . So T g 9 0 - < 9 Q f o r a l l s e S . Let 00 E e P be a weakly-wandering p r o j e c t i o n , then there e x i s t s ^ n ' ^ i ^ ^ 00 00 such that {T E } are pairwise orthogonal, hence E T E z A . S Tl-* J . - S n n=l n By Proposition 2-4, since (2a) holds, 9 Q ( E ) = u • Hence there i s no weakly-wandering p r o j e c t i o n E e P with 9 Q ( E ) > 0 . From Remark 3-5, we have (2a) => ( 1 ) . Suppose (1) holds. From Theorem 2-6 we have 4>n(A) > 0 implies i n f (<j> (T A) : s e S} > 0 and * n(A) = 0 implies u u s u M (<}> (T A)) = 0 for A e A + . Apply Proposition 2-4 again, we have J_I u s d>„(A) = 0 <=> M (<J>„(T A)) = 0 for A e A + . Hence (1) => (2b). Moreover, U L 0 s by Remark 3-3 and Theorem 2-6, i f TQ i s ce n t r a l then dp c a n °e chosen c e n t r a l . # Remark: In case A = L (X,S,p) of some f i n i t e measure space (X,5,p) , and S i s a semigroup of measurable maps on X X and T f (x) = f (sx) fo r a l l s e S , f e L (X,S,p) and x e X , our Theorem 3-8 reduces to Theorem 3 of E. Granirer [9]. The above proof follows the idea i n [9]. Theorem 3-9: Let A be a f i n i t e von Neumann algebra and $Q a p o s i t i v e normal c e n t r a l f u n c t i o n a l on A . Let S be amenable and Z S-stable. Then (1) <=> (2a) <=> (2b) and (I') <=> (2*a) <=> (2'b): (1) There e x i s t s an S-invariant p o s i t i v e normal c e n t r a l | e A with TQ <j) (2a) There i s no weakly-wandering p r o j e c t i o n E e P A Z ((2b) E e P ) with T Q ( E ) > 0 . (1 T) There e x i s t s an S-invariant p o s i t i v e normal c e n t r a l <j) e A such that <j>Q ~ <j> (2'a) I f E e P n Z ((2'b) E E P ) then d, (E) = 0 => M ^ O ^ E ) ) = 0 and (2a) (resp. (2b)) hold. 5 7 . Proof: Assume (1) holds. Let E be a weakly-wandering p r o j e c t i o n i n CO . 0 0 A then there e x i s t s {s } , C S such that {T E} . are pairwise n n=l s n=l n CO 00 00 orthogonal, hence £ T E e A . So « > ( p ( Z T E ) = E <p(T E) = _ -1 s - s s n=l n n=l n n=l n N = lim E <j>(Ts E) = lim N<j>(E) => 9(E) = 0 hence 9 Q(E) = 0 . So (1) => N-*» n=l n N-*» (2b) => (2a). I f ( l f ) holds, then 9 (E) => <p(T E) = 9(E) f o r a l l vj S s e S => 9 0(T gE) = 0 f o r a l l s e S => u (<j>n (T gE)) = 0 for a l l u £ IM => M(<j> (T E)) = 0 . Also (1') => (1). So ( l 1 ) => (2'b) => (2'a). u s Suppose (2a) (resp. (2'a)) holds. Since Z i s commutative and S-stable, * there e x i s t s an S-invariant p o s i t i v e normal i> £ Z such that 9Q - N <|> (resp. <pQ — ^ ) on Z by Theorem 3-4. Let 9(A) = ty(A^) f o r A E A , then 9 i s a p o s i t i v e normal c e n t r a l f u n c t i o n a l on A . Let a E S , from Lemma 2-11 we have ^(T^A) = ^((T^A)^) = \p (T (A^)) = ip(A^) = 9(A) f o r a l l A E A . Hence 9 i s S-invariant. Let E E P , then 9(E) = 0 <=> if ) ( E ^ ) = 0 => VQCE^) = 0 <=> 9 Q(E) = 0 since $Q-< 4> on Z (resp. 9(E) = 0 <=> i>(E^) = 0 <=> 9 Q(E^) = 0 <=> 9Q ( E ) = 0 since \p _ 9n on Z ). Hence c p n - < 9 (resp. < p n _ 9 ). Thus (2a) => (1) and (2'a) => (1'). // Remark 3-10: In Theorem 3-9, i f we assume that T <!>-•-< d>„ for a l l s y0 r0 s E S , then S need only be l e f t amenable and (1) <=> (2a) <=> (2b) <=> C l 1 ) <=> (2'a) <=> (2'b). M would have to be replaced by M^ i n t h i s case. 58. Proof; Replace Theorem 3-4 by Remark 3-5 i n the above proof, and using the f a c t (2a) <=> (2'a) i n Remark 3-5. // Coroll a r y 3-11: Let A be a a - f i n i t e f i n i t e von Neumann algebra. Let S be l e f t amenable and Z S-stable. Then the following are equivalent: (1) There e x i s t s an S-invariant p o s i t i v e normal f a i t h f u l c e n t r a l A e A (2a) There i s no non-zero weakly-wandering ((2b) central) p r o j e c t i o n i n A Proof: Since A i s a - f i n i t e f i n i t e , there i s a p o s i t i v e normal f a i t h -f u l c e n t r a l ^ e A . The Co r o l l a r y follows from Remark 3-10. ?/ Prop o s i t i o n 3-12: Let A be a f i n i t e von Neumann algebra and <J>Q a p o s i t i v e normal c e n t r a l f u n c t i o n a l on A . Let T I = I f o r a l l s e S " s and Z be S-stable, then the following are equivalent: oo 4- co (1) I f A e A and there i s {s } , C S with £ T A e A then n n=l . s n=l n A 0(A) = 0 . CO + CO (2) I f A e Z and there i s {s } , C S with E T A e Z then n n=l . s n=l n 4>0(A) = 0 . Proof: (1) => (2) i s cl e a r since Z + C A + . Suppose (2) holds. Let be the canonical c e n t r a l valued function on A . Let A e A + and CO suppose there i s {s }°°_ C S with. E T A e A Then, from n IT— J. n s n=l n 59 Proposition 3-2 and Theorem 1-S, we have E T ( A ^ ) = E (T A / 7 n=l n n=l n = ( E T A ) e Z . By (2) we have A ( A ) = A ( A ^ ) = 0 . Hence IS u u n (2) => (1). // Let A be a von Neumann algebra on a H i l b e r t space H . 1 ft subset J C A i s s a i d to be complete i f for any non-zero A e A , there i s A e J such that A (A) 4 0 . Let S be a semigroup and ft {T g : s e S} an antirepresentation of S as normal -homomorphisms on A P r o p o s i t i o n 3-13: Let J be a set of p o s i t i v e functionals on A Then J i s complete i f and only i f for any non-zero p r o j e c t i o n E e A there i s A e J such that <j>(E) > 0 . Proof: Since P (2. A + , the condition i s obviously necessary. Now suppose f o r any 0 4 E e P we have A e J such that A(E) > 0 . Let 0 T A e A + , then from the proof of Lemma 1-G we have c > 0 and 0 4 E e P such that cE <_ A . Since E 4 0 , there i s A e J such that A(E) > 0 and hence A(A) >_A(cE) = cA(E) > 0 . So J i s complete Theorem 3-14: Let S be amenable and such that E'T E = T E'E for-any s s p r o j e c t i o n E e A and s e S . Then the following are equivalent: (1) There i s a complete set of S-invariant p o s i t i v e normal functionals on A 60. (2) There i s no non-zero weakly-wandering p r o j e c t i o n i n A (3a) If 0 4 A e P (resp. (3b) 0 4 A E A + ) , 0 i s not i n the strong closure of {T gA : s E S} Proof: (3b). => (3a) since P C A + . Suppose (3a) holds. Let E E P CO co be a weakly-wandering p r o j e c t i o n and {s } n C S such that {T E} 3 n n=l s n=l n oo are pairwise orthogonal p r o j e c t i o n s . Then E T E E A , and n=l n 0 0 0 0 || E T Ex ||2 = E || T Ex || for a l l x e H . Thus )|T EX|| converges I s _ s s n n=l n n to zero f o r any x s H i . e . T E converges to zero strongly. Hence by (3a), E = 0 . So (3a) => (2). Suppose (2) holds. Let 0 4 A E A then there are p o s i t i v e number c and non-zero E E P such that cE <_ A . Since E 4 0 , there i s x A E H such that Ex A = x n 4 0 Let <j>0(A) = (Ax Q|x n) f o r a l l A E A , where ( | ) i s the inner product of the H i l b e r t space H . Then 9^  i s a p o s i t i v e normal f u n c t i o n a l on A (see [4], p. 54, Theorem 1) with ^ Q ^ ) > u • I t i s cl e a r that there i s no p r o j e c t i o n F s A such that ^QO ? ) > 0 and F i s weakly-wandering. So, by Theorem 3-4, there i s an S-invariant p o s i t i v e normal f u n c t i o n a l 9 on A such that 9 Q 9 • Thus 9(E) > 0 , and 9Q(A) >_ ctp(E) > 0 . Hence there i s a complete set of S-invariant p o s i t i v e normal functionals on A and (2) =>(1). Suppose (1) holds. Let A E A + and {s } C_ S be a net such that T A converges to a a s a zero strongly. Then f o r any p o s i t i v e normal f u n c t i o n a l f on A we have I(J(T A) converges to zero. In p a r t i c u l a r 9 ( T A) converges to a a zero f o r any S-invariant p o s i t i v e normal f u n c t i o n a l 9 on A But 61. <j>(T A) = <j>(A) f o r a l l s , so <j> (A) = 0 f o r any S-invariant a p o s i t i v e normal f u n c t i o n a l <j> on A . By (1) we have A = 0 . Hence (1) => (3b). // Remark: The equivalence of (3a) and (3b) i s true f o r any von Neumann algebra A and any semigroup S . For i f 0 4 A e A+ be such that there i s a net {s } C S such that T A converges to zero strongly, then a s a T A converges to zero weakly. So (T Ax|x) converges to zero f o r s " s a a any x e H . Since A 4 0 , there are c > 0 and 0 4 E E P such that cE < A . Hence, f o r each a, c|JT Ex [j = c(T E X | T E X ) = c(T Ex|x) s s s s a a a a _< (T Ax|x) f o r a l l x E H . Thus || T Ex jj converges to zero f o r any s s a • a x E H . Hence T E converges to zero strongly but E 4 0 a Theorem 3-15: Let A be a von Neumann algebra and S be amenable. Then (*) or (**) implies the equivalence of (1), (2), (3a) and (3b). (*) Z i s S-stable. (**) E " T G E = T G E - E f o r any E £ P , s E S . Cl) There i s a complete set of S-invariant p o s i t i v e normal c e n t r a l functionals on A (2) A i s f i n i t e , and there i s no non-zero weakly-wandering p r o j e c t i o n i n A (3a) A i s f i n i t e , and i f 0 4 A E P ((3b) 0 4 A £ A + ) then 0 i s not i n the strong closure of (T gA : s E S} Proof: The proof (3a) => (2) and (1) => (3b) are the same as i n the proof of Theorem 3-14 (see the d e f i n i t i o n of a f i n i t e von Neumann algebra). (3b) => (3a) by P C A + . Suppose (2) holds. Let 0 4 A e A + then there i s a p o s i t i v e normal c e n t r a l f u n c t i o n a l c p n on A such that 9 Q ( A ) > U (note that, by P r o p o s i t i o n 1-C and Theorem 1-B, we can i d e n t i f y the set of f i n i t e normal traces on A with the set of p o s i t i v e normal c e n t r a l functionals on A ). By (2), there i s no weakly-wandering p r o j e c t i o n F e A such that 9 Q ( F ) > 0 . If (*) holds, then from Theorem 3-9 there i s a S-invariant p o s i t i v e normal c e n t r a l f u n c t i o n a l cj> on A such that 9 Q . . < 9 • So cp(A) > 0 . I f (**) holds, then from Theorem 3-4 there i s an S-invariant p o s i t i v e normal c e n t r a l f u n c t i o n a l (p on A such that 9Q-< 9 • So <J>(A) > 0 . Thus we have a complete set of S-invariant p o s i t i v e normal c e n t r a l functionals on A . Hence (2) => (1). # Remark 3-16: Let A be a von Neumann algebra and Z i t s center. Let T be a -homomorphism on A which i s onto. For A e A , there i s A± e A such that T(A 1) = A . Hence, i f B e Z then T(B)-A = T(B)T(A^) = T(BA 1) = T(A B) = T(A 1)T(B) = A*T(B) . Hence T(Z) C Z i . e . Z i s T-stable. Let S be a semigroup. If {T : s e S} i s an antirepresen-* t a t i o n of S as normal -homomorphisms on A onto A , then Z i s S-stable. Hence the condition " Z i s S-stable" can be omitted from the hypotheses i n both Chapter 2 and this chapter (chapter 3} i f we consider an antirepresentation of a semigroup S as normal -homomorphisms on A onto A (note that a -homomorphism on A onto A must carry I to I , so the condition " T I = I for a l l s e S " can be omitted). C o r o l l a r y 3-17: If A i s a f i n i t e von Neumann algebra on a H i l b e r t space H , and S an amenable semigroup with an antirepresentation {T g : s e S} of S as normal -homomorphisms on A onto A Then the following are equivalent: (1) There i s a complete set of S-invariant f i n i t e normal traces on A (2a) There i s no non-zero weakly-wandering p r o j e c t i o n (resp. (2b) no non-zero weakly-wandering c e n t r a l projection) i n A (3a) I f 0 4 A e P (resp. (3b) 0 + A e A + ) then 0 i s not i n the strong closure of {T A : s e S} s Proof: By Theorem 3-15, Remark 3-16, Theorem 3-9, Proposition 1-T, and the paragraph a f t e r Proposition 1-C. // Remark 3-18: (a) I f we replace both the l e f t amenability of S and the antirepresentation of S as operators on A by r i g h t amenability of S and representation of S as operators on A r e s p e c t i v e l y (also replace and LIM by M^ and RIM r e s p e c t i v e l y ) then a l l our r e s u l t s i n Chapter 2 and Chapter 3 remain true. Hence Theorem 3-4, Theorem 3-9, Theorem 3-14, Theorem 3-15, and Cor o l l a r y 3-17 remain true i f we consider {T g : s e S} as a representation of S as the corresponding operators on A i n these theorems (note: an analogue of p r o p o s i t i o n 3-A i s : Let S be l e f t amenable and y e LIM . If f e m(S) i s r e a l and y ( f ) = 0 . Then f o r any a ,a„,...,a i n S and 6 > 0 , there i s a e S such n that f(o") + E. f(a.a) < <S ). Lemma 3-6 and Proposition 3-12 also i = l 1 remain true i f we replace the "antirepresentation of S " by "represen-t a t i o n of S ", since we do not need the amenability of S i n Lemma 3-6 and P r o p o s i t i o n 3-12. ft (b) Since -automorphism on a von Neumann algebra A i s normal, a l l r e s u l t s i n Chapter 2 and Chapter 3 remain true (the conditions " Z i s S-stable" and " T I = I f o r a l l s e S " can be omitted) i f we l e t s S be an amenable group with an antirepresentation ( or representation) * {T g : s e S} of S as -automorphisms on A Let S be a l e f t amenable semigroup. Define, f o r r e a l f e m(S) lim sup f ( s ) = i n f sup f ( s t ) . This d e f i n i t i o n reduces to the usual S s t lim sup i n case S = {1,2,...} with a d d i t i o n . I f f i s p o s i t i v e then l i m sup f ( s ) = i n f j| £ f|| s Let (X,5,p) be a measure space with f i n i t e measure p . Let S be a l e f t amenable semigroup of measurable maps on X such that ps p f o r a l l s e S , where ps ^"(E) = p (s ^E) f o r E e 5 . Then L (X,S,p) i s a von Neumann algebra (commutative) on the H i l b e r t space CO L (X,5,p) . For f E L c o(X,S,p) , s e S , we define T g : L^CXS.p) -> L o o(X,S,p) by T f(x) = f(sx) (since ps _ 1-< p so T g f e L o o(X,5,p) ), ft then T . i s a normal -homomorphism on A with T 1 = 1 , where s s l( x ) = 1 f o r a l l x e x . Thus {T g : s e S} i s an antirepresentation ft of S as normal -homomorphisms on the von Neumann algebra L (X,S,p) 6 5 . with T 1 = 1 f o r a l l s e S . The f u n c t i o n a l 4> on L (X,S,p) s 0 0 0 defined by Y Q C ^ ) = / fdp i s a p o s i t i v e normal f a i t h f u l f u n c t i o n a l on L^CXjSjp) . The existence of an S-invariant p o s i t i v e normal f u n c t i o n a l A on L^CXjS^) such that <j> ~ <J>Q i s equivalent to the existence of a f i n i t e i n v a r i a n t measure u ~ p It has been shown by Hajian-Kakutani [11] that: Given the f i n i t e measure space (X,S,p) and S = {s 1 1 : n _> 1} , where s i s a measurable map on X X with ps p . The condition "p(E)-> 0 => l i m sup(s nE)> 0 n->oo f o r E e S " i s not s u f f i c i e n t f o r the existence of a f i n i t e i n v a r i a n t measure u ~ p . Hence i n the von Neumann algebra L (X,S>p) with the antirepresentation {T g : s e S where T f(x) = f(sx) f o r a l l x e X} n * of S = {s : n >_ 1} as normal -homomorphisms on L (X,S,p) with T 1 = 1 f o r a l l n > 1 , the condition " duCXrJ > 0 implies n — Y0 A E s l i m sup r n ( T xw) > u i ° r E e S " i s not s u f f i c i e n t f o r the existence S \) S I J of an S-invariant normal p o s i t i v e f u n c t i o n a l <j> on L (X,S,p) such that CO <j> ~ TQ (The projections i n ^ ( X j S j p ) are j u s t the c h a r a c t e r i s t i c functions Y i n L (X,5,p) f o r E e S ). Thus, even f o r a commutative h i co a - f i n i t e (a commutative von Neumann algebra i s f i n i t e ) von Neumann algebra A with a p o s i t i v e normal f a i t h f u l f u n c t i o n a l AQ on A , the condition "(*) T Q ( E ) > 0 implies lim gsup r Q ( T g E ) > 0 for E e P" i s not s u f f i c i e n t f o r the existence of an S-invariant p o s i t i v e normal f u n c t i o n a l <j> on A such that <J>Q ~ <J> , where S i s a l e f t amenable semigroup and {T g : s e S} i s an antirepresentation of S as normal -homomor-phisms on A . But there i s a subclass of l e f t amenable semigroup f o r which the condition (*) i s s u f f i c i e n t . That i s the class of extremely 66. l e f t amenable (ELA) semigroups (a semigroup S i s c a l l e d ELA i f there i s a m u l t i p l i c a t i v e l e f t i n v a r i a n t mean u on m(S) i . e . u(fg) = u(f)u(g) fo r f, g e m(S) ). Theorem 3-19: . Let A be a von Neumann algebra and <j>n a p o s i t i v e normal f u n c t i o n a l on A . Let S be an ELA semigroup with an antirepresenta-t i o n {T : s e S} as normal -homomorphisms on A . Suppose T E*E = s s E'T gE f o r any E e P , s E S , then the following are equivalent: (1) There e x i s t s an S-invariant p o s i t i v e normal f u n c t i o n a l <p on A such that <J>Q ~ <j> (2a) I f A e P ((2b) A e A + ) then 9 (A ) = 0 <=> lim g sup 9 0(TgA) = 0 . I f < p n i s c e n t r a l then <j> can be chosen c e n t r a l . I f one of (1), (2a) and (2b) holds, then there e x i s t s an S-invariant p o s i t i v e normal f u n c t i o n a l on A with <f ~ 4>Q ( i f 9Q i s c e n t r a l then 4> can be chosen central) which s a t i s f i e s : If A,B e A are such that c p n ( T AT B) = 9 (T A ) c p (T B) f o r u s s u s u s a l l s £ S , then i|)(AB) = ^(A)iJi(B) . Proof: Since S i s ELA, by [10] p. 68, we have M (f) = l i m sup f ( s ) XJ s f o r r e a l f E m(S) . Hence the conditions (2a) and (2b) are equivalent to the conditions (2a) and (2b) of Theorem 3-8 r e s p e c t i v e l y . Hence the f i r s t part of our theorem follows from Theorem 2-8. Now i f one of (1), (2a) and (2b) holds, then there i s an S-invariant p o s i t i v e normal f u n c t i o n a l <p on A such that < p Q „ 9 . Let u be a m u l t i p l i c a t i v e l e f t i n v a r i a n t mean on m(S) , then Proposition 2-14 implies that iJ>(A) = u(<j>n(T A)) f o r a l l A E A i s an S-invariant p o s i t i v e normal f u n c t i o n a l on A such that ty ~ ty^ and ty i s c e n t r a l i f ty^ i s c e n t r a l . Now l e t A,B c A such that cj)- (T AT B) = ty (T A)ty (T B) for a l l s e S u s s u s u s then ty (AB) = y (TAB) )) = y (_tyQ ( T A T s B ) ) = y (tyQ (TA)d> 0 (T gB) ) = y(<fr 0(TA)) y( YQ(T gB)) = i|i(A)ij;(B) , since y i s m u l t i p l i c a t i v e . // Remark: Theorem 3-19 i s a ge n e r a l i z a t i o n of Theorem 7 i n Granirer [9] Example 1: Let (X,S,p) be a f i n i t e measure space, and S a l e f t amenable semigroup of non-singular measurable maps on X in t o X ( i . e . ps p f o r a l l s e S , where ps ^ i s a measure on S defined by ps -1 (Y) = p(s "hr) f o r Y e S ). Let y be a f i n i t e i n v a r i a n t measure on S ( i - e . y ( s _ 1 Y ) = y(Y) f o r a l l s e S and Y e S ) such that y ~ P • The algebra L Q o(X,5,p) i s a von Neumann algebra on the H i l b e r t space L 2(X,S,p) . Define tyQ on L^X.S.p) by tyQ(£) = / fdp for a l l f e L o o(X,S,p) , then ty^ i s a p o s i t i v e normal f a i t h f u l f u n c t i o n a l on L o o(X,S,p) . So L o o(X,S,p) i s a commutative a - f i n i t e von Neumann algebra. The projections i n L (X,5,p) are the c h a r a c t e r i s t i c functions OO X Y f o r Y E S . For each s e S and f e L o Q(X,S,p) , we define f on X by fgCx) = f(sx) f o r a l l x e X . Then the f a c t that ps \< p g implies that f e L (X,S,p) . For each s e S , we define T on S °o L (X,S,p) into L (X,S,p) by T S f = f f o r a l l f e L (X,S,p) , co co S oo S • * then T i s a normal -homomorphism on L (X,S,p) (note that the 0 0 i n v o l u t i o n on L (X,5,p) i s the 'conjugation'). It i s cl e a r that 0 0 S t t S S T = T T f o r s,t i n S . Hence {T : s e S} i s an antirepresenta-t i o n of S as normal -homomorphisms on L (X,S,p) . I t i s cl e a r that 68. *0(.XY) = POO and (^Ay) = p(s 1Y) for any s E S , Y E S . Since u i s equivalent to p and in v a r i a n t under S , we have p(Y) > 0 implies i n f {p(s *hf) : s £ S} > 0 f o r Y E S i . e . ^ ( X y ) > u implies g i n f { ^Q ( T x y ) : s E S} > 0 for any p r o j e c t i o n x y i n L^CX.S.p) By Lemma 3-6, .there i s no weakly-wandering p r o j e c t i o n Xy x n L^CXjSjp) -1 s * such that l^ C x y ) > 0 (note that ps -< p => ( T ) ipQ-< ipQ f o r a l l s £ S ). Let M be a non-commutative von Neumann algebra on a H i l b e r t space H . Let t j ; ^ be a normal p o s i t i v e non-zero f u n c t i o n a l on M Let A = M x L Q o(X,S,p) , the product of M and L (X,S,p) , then A i s a non-commutative von Neumann algebra on the H i l b e r t . space H © L2(X,S,p) , the H i l b e r t sum of H and L2(X,S,p) (see [4], p. 21). An element i n A i s of form (A,f) where A E M and f e L o o(X,S,p) . Define 9 n on A by cp 0((A,f)) = ^ ( A ) + ^ ( f ) f o r (A,f) e A , then cf>0 i s a p o s i t i v e normal f u n c t i o n a l on A . For each s £ S , we define T s s * on A into A by T g ( ( A , f ) ) = (A,T f) , then T g i s a normal -homomor-phism on A . Hence {T g : s £ S} i s an antirepresentation of S as normal -homomorphisms on A such that T s<p n-< <pn f o r a l l s E S (note that, since ( T S ) ^ Q - < ^ Q and T*<j>n ((A, f) ) = 9 ( ) ( (A,T Sf)) = ^ (A) + ^ Q ( T S f ) , we have t S ? Q " < $Q )• A p r o j e c t i o n i n A i s of form (E,Xy) where E and _xY are projections i n M and L o o(X,S,p) r e s p e c t i v e l y . I f 0 4 E i s a p r o j e c t i o n i n M , then (E,Xy) i s not a weakly-wandering p r o j e c t i o n i n A (since T g ( (E, X y ) ) ( ( E , X y ) ) = (E, T S X y - T ^ ) 4 (0,0) f o r any t,s e S). A projections ( 0 , x v ) i n A i s weakly-wandering i f and only 69. i f Xy 1 S a weakly-wandering p r o j e c t i o n i n L^CXjSjp) (under {T : s e S) ). Hence there i s no weakly-wandering p r o j e c t i o n ( E , X y ) i n A such that < J ) Q ( ( E , X Y ) ) > 0 . Since L^CXjSjp) i s commutative, T S ( ( E , X y ) ) commutes with (E,Xy) i ° r a n Y p r o j e c t i o n (E,Xy) i n A and s e S . By Corol l a r y 3-7,- there i s an S-invariant p o s i t i v e normal f u n c t i o n a l A on A such that <f> ~ <J)Q . In t h i s example there e x i s t s an S-invariant p o s i t i v e normal f u n c t i o n a l on A and A i s non-commutative. Moreover, since i|> i s f a i t h f u l , a p r o j e c t i o n Xy ^ n l ^CX jS jp ) i s zero i f and only i f ij; (xy) = 0 » hence there i s no non-zero weakly-wandering p r o j e c t i o n i n L (X,S,p) . Therefore there i s no non-zero weakly-wandering p r o j e c t i o n i n A . By Theorem 3-14, there i s a complete set of S-invariant normal p o s i t i v e functionals on A Example 2: Let (X,S,p) be a f i n i t e measure space, and l e t S , ^ , s {f : s e S} and {T : s E S} be as i n Example 1. Let A be the von Neumann algebra of a l l 2 x 2 matrices f h e g with e n t r i e s i n L (X ,S ,p ) For each s e S , we define T on A into A by s f f h Vf T Sh" - — f h s s e g T Se T Sg_ e g s °s_ then T i s a normal -homomorphism on A . Hence {T : s £ S} i s s s an anti r e p r e s e n t a t i o n of S as normal -homomorphisms on A 70. Let t be the trace on the algebra of a l l 2 x 2 complex matrices. Define 4. = t ill. on A by 0 r 0 Vo f h e g = t if, A(f) if)Q(h) then <pA i s a p o s i t i v e normal f a i t h f u l f u n c t i o n a l on A . The existence of a p o s i t i v e normal f u n c t i o n a l 9 on A such that 9 ~ <f>n and i n v a r i a n t under {T g : s e S} i s equivalent to the existence of a p o s i t i v e normal f u n c t i o n a l if) on L^CXjSjp) such that if) ~ if)A and i n v a r i a n t under g {T : s e S} . Since the existence of such an if) i s equivalent to the non-existence of a non-zero weakly-wandering p r o j e c t i o n i n L (X,S,p) g (with respect to {T : s e S} ), the existence of an S-invariant p o s i t i v e normal f u n c t i o n a l 9 on A such that 9 "0 i s equivalent to the non-existence of a non-zero weakly- wandering p r o j e c t i o n i n s L (X,S,p) (with respect to {T : s e S} ). Now, i f there e x i s t s an S-invariant p o s i t i v e normal f u n c t i o n a l 9 on A such that 9 ~. <pA , then t h i s r e a d i l y implies the non-existence of a non-zero weakly-wandering p r o j e c t i o n i n A (see the proof of Theorem 3-4 ((1) => ( 2 ) ) ) . Conversely, i f there i s no non-zero weakly-wandering p r o j e c t i o n i n A , then there i s no non-zero weakly-wandering p r o j e c t i o n i n L (X,S,p) with respect to {T : s e S} (For i f x y ^ u i s a weakly-wandering p r o j e c t i o n i n • 8 TXv 0" L (X,S,p) with respect to {T : s e S} , then 0 0 i s a non-zero weakly-wandering p r o j e c t i o n i n A ) . So the non-existence of a non-zero weakly-wandering p r o j e c t i o n i n A implies the existence of an S-invariant p o s i t i v e normal f u n c t i o n a l A on A such that d> ,. 6 We show now that there are projections E i n A such that T E-E ^ E'T E i f T E ^ E . Let g be any p o s i t i v e element i n s s s 2 g S L (X,S,p) , l e t f = - f r , then 0 < f < 1 and 0 < f = — — r < 1 . c ° ' r g+1 — — S g +1 1 1 S Let E f h h 1-f 2 2 where h = f (1-f) , then E i s a p r o j e c t i o n i n Now T E = E i s equivalent to f = f (hence equivalent to g = g s s s Since and E'T E s T E-E = s f f + hh s s hf + h (1-f) s s f f + hh s s fh + h ( l - f ) s s fh + h ( l - f ) s s hh + ( 1 - f ) ( 1 - f ) s s hf + h (1-f) s s hh + ( 1 - f ) ( 1 - f ) s s i t i s c l e a r that T E-E = E'T E i f and only i f s s fh + h ( l - f ) = hf + h (1-f) s s s s 1 1 1 1 1 1 1 1 i . e . f f 2 ( l - f ) 2 + f 2 ( l - f ) 2 ( l - f ) = f 2 ( l - f ) 2 f + f 2 ( l - f ) 2 ( l - f ) s s s s s s 1 1 1 1 1 I 1 1 1 i I 1 i . e . f 2 ( l - f ) 2 [ f 2 f 2 + ( l - f ) 2 ( l - f ) 2 ] = f 2 ( l - f ) 2 [ f 2 f 2 + ( l - f ) 2 ( l - f ) 2 s s s s s s 11 1 1 1 I I I i . e . [ f 2 f 2 + ( l - f ) 2 ( l - f ) 2 ] - [ f 2 ( l - f ) 2 - f 2 ( l - f ) 2 ] = 0 (a) s s s s 72. 1 1 1 1 Since 0 <_ f < 1 and 0 <_ f < 1 , [ f 2 f 2 + ( l - f ) 2 ( l - f ) 2 ] > 0 . Hence s s s 1 1 1 1 ( a ) holds i f and only i f f 2 ( l - f ) 2 - f 2 ( l - f ) 2 = 0 s s 1 I I I i . e . f Z ( l - f Y = f / ( l - f ) / s s i . e . f ( 1 - f ) = f (1-f) s s i . e . f - f f = f - f f s s s i . e . f = f g ( i . e . g g = g ) i . e . T E = E s Hence T E-E 4 E*T E i f T E 4 E . s s s g Now i f T g i s not the i d e n t i t y on A then T i s not the i d e n t i t y on L ^ X j S j p ) , hence there i s p o s i t i v e g e L^CXjSjp) such that g g 4 g Let f, E as above then T E»E 4 E*T E . Hence there are projections s s E i n A such that T E-E 4 E*T E f o r some s e S . s s This example shows that the condition that " T E-E = E-T E s s fo r any s z S and E e P " i n the hypothesis of our main theorems i n Chapter 3 can be relaxed. We do though not know at t h i s time how to prove a stronger v e r s i o n of i t . 73. Example 3: Let H be a H i l b e r t space and L (H) be the algebra of a l l bounded l i n e a r operators on H Let H. = H f o r a l l p o s i t i v e integers CO i . Consider the product A = II L(H.) (see [4], p. 21), then A e A i = l 1 i f and only i f A = {A^}^^ such that A_^  e L(H./) and sup ||A || < 0 0 i CO A i s a von Neumann algebra on the H i l b e r t space E © H. , the H i l b e r t 1=1 1 sum of ^ j ^ i - i Let S = {1,2,...} be the set of p o s i t i v e integers with i o j = max {i,j} , then S i s a commutative semigroup under o . Hence S CO i s an amenable semigroup. For n e S , l e t T (A) = {B.}. -. such that n i i — l B. = 0 i f i < n and B. = A. i f i > n i . e . TA=(0,0,...,0,A,,, l — x i n n+1 * A n +2>*'-) • Then i s a normal -homomorphism on A . Hence {T : n e S } i s a set of normal -homomorphisms on A . Let m, n i n n S , then Case 1: m < n T T (A) = T ({B. ; B. = 0 i f i < n , B. = A. i f i > n}) m n m x i — x x = {C. ; C. = 0 i f i < m , C. = B. i f i > m} X X — X X = {C. ; C. = 0 i f i < n , C. = A. i f i > n} x x — x x T T (A) = T '({B. ; B. = 0 i f i < m , B. = A. i f i > m}) n m n x x — x x = {C. ; C. = 0 i f i < n , C. = B. i f i > n} x x — x x = {C. ; C. = 0 i f i < n , C. = A. i f i > n} X X — 1 X T (A) = T (A) = T (A) = {C. ; C. = 0 i f i < n , C. = A. i f i > m°n n°m n x x — x x Hence T T = T T = T = T m n n m m°n n°.Ti Case 2: m > n T T (A) = T ( { B . ; B . = 0 i f i < n , B . = A. i f i > n} ) m n - m i x — 1 1 = {C. ; C. = 0 i f i < m , C. = B . i f i > n} i i — l l = {C. ; C. = 0 i f i < m , C. = A. i f i > m} i i — l l T T (A) = T ( { B . ; B . = 0 i f i < m , B . = A. i f i > m}) n m n l l — l l = {C. ; C. = 0 i f i < n , C. = B . i f i > n} i i — I I = {C. ; C. = 0 i f i < m , C. = A. i f i > m} i i — i i T (A) = T (A) = T (A) = {C. ; C. = 0 i f i < m , C. = A. i f i > m°n n°m m 1 l l — l i Hence T T = T T = T = T m n n m m°n nom Thus {T : n e S} i s a representation of S (also an antirepresentation n * of S since S i s commutative) as normal -homomorphisms on A . A CO p r o j e c t i o n E e A i s of form where E_^ are pr o j e c t i o n s i n CO 0 0 .. ,|2 L(H.) f o r a l l i . Let & > 0 and x e E © H. , then E | | x . | | < ~ 1 i = l 1 i = l 1 0 0 2 There i s N > 0 such that E ] | x . || < 6 . For n > N , we have i=N 1 CO T E = { B . : B . = 0 i f i < n , B . = E . i f i > n} T Ex = { B . x . } . ^ , n i l — ' i i J ' n L i i i = l 0 0 CO 0 9 2 so ]|T EX | | = E | | B . x . | | = E | | E . x . | | . Since E . <_ I , hence n i = l i=n 0 0 oo l l E ^ i l l 2 1 j l ^ l i 2 • Thus we have j | T n E x | | 2 = E | | E . x . | | 2 < E || X . f < S , i=n ^ 1 = n since n > N . Since 5 > 0 i s a r b i t r a r y , j | T^Ex|| converges to zero f o r any x e Z 0 H. i . e . T E converges to zero strongly. Since i = l {T^E : n E S} i s a bounded set, a l l normal p o s i t i v e functionals <J>Q have the property that ACT^E) converges to zero, for a l l E E P In p a r t i c u l a r i f <j> i s S-invariant then 4(E) = ty(T^E) i s zero f o r a l l projections E i n A Hence <f> i s the zero f u n c t i o n a l i f <j> i s S-invariant. This proves that there i s no non-zero S-invariant p o s i t i v e normal f u n c t i o n a l on A . I t i s c l e a r that any p r o j e c t i o n C O E = {E.}. , such that E J = 0 f o r a l l but f i n i t e i i s a weakly-1 i = l i wandering p r o j e c t i o n . 76. Bibliography [1] Bachman, G. and N a r i c i , L., "Functional A n a l y s i s " , Academic Press 1966. [2] Blum, J^R. and Friedman, N., "On i n v a r i a n t measures for classes of transformations", Z. Wahrscheinlichkeitstheorie verw. Geb. 8 (1967) 301-305. [3] Calderon, M., Alberto, P., "Sur les mesures i n v a r i a n t e s " , C.R. Acad. S c i . P a r is (1955) V. 240, 1960-1962. [4] Dixmier, J . , "Les Algebres D'operateurs Dans L'espace H i l b e r t i e n " , Paris 1957, 1st e d i t i o n . [5] Dixmier, J . , "Les anneaux d'operateurs de classe f i n i e " , Ann. Ec. Norm. Sup., t.66 (1949) 209-261. [6] Dixmier, J . , "Applications tf dans l e s anneaux d'operateurs", Compos. Math., t.10 (1952) 1-55.. [7] Dixmier, J . , "Forms l i n e a i r e s sur un anneaux d'operateurs", B u l l . Soc. Math. Fr., t. 81 (1953) 9-39. [8] Dunford, N. and Schwartz, J.T., "Linear operators I", Interscience, New York, 1958. [9] Granirer, E.E., "On f i n i t e equivalent i n v a r i a n t measures f o r semi-groups of transformations", To appear i n Duke Math. J . [10] Granirer, E.E., "Functional a n a l y t i c properties of extremely amenable semigroups", Trans. AMS 137 (1969) 53-75. [11] Hajian, A.B. and Kakutani, S., "Weakly wandering sets and i n v a r i a n t measures", Trans. AMS 110 (1964) 136-151. [12] Hajian, A. and Ito, Y., "Weakly wandering sets and i n v a r i a n t measures fo r a group of Transformations", Journal of Math, and Mechanics, Vol. 18,.No. 12 (1969) 1203-1216. 77. [13] Hewitt and Yosida., " F i n i t e l y a d d i t i v e measures", Trans. AMS 72 (1952) 46-66. [14] Nakamura, M., "A proof of a theorem of Takesaki", K6dai Math. Sem. Rep. 10 (1958) 189-190. [15] Natarajan, S., "Contributions to Ergodic Theory", Thesis, The " Indian S t a t i s t i c a l I n s t i t u t e , Calcutta, 1968. [16] Neveu, J . , "Sur l'existence de mesures invariantes en theorie ergodique", CR. Acad. Scie. P a r i s , 260 (1965) 393-396. [17] Rickart, C.E., "General theory of Banach algebra", Van Nostrand. Princeton, N.J., 1960. [18] Sachdeva, U., "Research Fundation", Thesis, The Ohio State U n i v e r s i t y 1970. [19] Sucheston, L., "On existence of f i n i t e i n v a r i a n t measures", Math. Z., 86 (1964) 327-336. [20] Takesaki, M., "On the conjugate space of operator algebra", Tohoku Math. J . 10 (1958) 194-203. [21] Takesaki, M., "On the s i n g u l a r i t y of a p o s i t i v e l i n e a r f u n c t i o n a l on operator algebra", Proc. Japan Acad., t. 35 (1959) 365-366. 

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