UBC Theses and Dissertations

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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., M.A.,  Cheng-Kung U n i v e r s i t y , Taiwan, China, 1962.  The U n i v e r s i t y  o f Western O n t a r i o ,  London, O n t a r i o ,  A THESIS SUBMITTED IN.PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  the Department of MATHEMATICS  We a c c e p t t h i s to the r e q u i r e d  The  t h e s i s as conforming standard  U n i v e r s i t y of B r i t i s h March 1971  Columbia  1967.  In  presenting this  thesis  an advanced degree at  further  agree  fulfilment  the U n i v e r s i t y of  the L i b r a r y s h a l l make I  in p a r t i a l  it  freely  of  the  requirements  B r i t i s h C o l u m b i a , I agree  available  for  that permission for extensive copying o f  of  this  representatives. thesis for  It  financial  i s understood that c o p y i n g o r gain s h a l l  written permission.  Depa rtment The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  /far//  /?/<77/  not  that  r e f e r e n c e and s t u d y . this  thesis  f o r s c h o l a r l y purposes may be granted by the Head o f my Department by h i s  for  or  publication  be allowed without my  Supervisor:  Dr. E. G r a n i r e r  ABSTRACT  Hilbert  Let  A  be a von Neumann a l g e b r a  of l i n e a r operators  space  H  .  ( r e s p . a l i n e a r bounded.  functional net is  {B } a a  A l i n e a r operator  <j> ) on  A  i s s a i d to be normal i f f o r any i n c r e a s i n g  o f p o s i t i v e elements i n  t h e l e a s t upper bound o f  iJ^(B) = 0 <=>  Let S  ^(B)  <J>Q  =  normal p o s i t i v e l i n e a r  functional  the semigroup s e S first  ).  A  {T  g  o  r  a n  Y  ty^  <J>(B) = sup on  A  ^(B^))  .  B  in A  A  an a n t i r e p r e s e n t a t i o n  of  on  f o r the e x i s t e n c e  which i s e q u i v a l e n t  to  A  .  We f i n d  . Let S  A  o f a p o s i t i v e normal $  and i n v a r i a n t under  ( i . e . <|)(T B) = <j>(B)  by H a j i a n - K a k u t a n i ,  and.the e x i s t e n c e  with respect  for a l l B  in  A .and  to weakly-wandering p r o j e c t i o n s  A  {T  o f p o s i t i v e normal l i n e a r  to an a n t i r e p r e s e n t a t i o n .  {T  g  f u n c t i o n a l s on  : s e S}  F i n a l l y we i n v e s t i g a t e the e x i s t e n c e  p o s i t i v e normal l i n e a r semigroup  linear  We a l s o extend the concept o f weakly-wandering s e t s , which was  introduced  phisms on  as  i n this  We g i v e a r e l a t i o n between t h e n o n - e x i s t e n c e o f weakly-wandering in  Two  equivalent  f u n c t i o n a l on  contraction operators  : s e S}  B , T(B)  a r e s a i d to be  p o s i t i v e element  {T : s e S} s  conditions  <j> on  (resp.  and ^  w i t h l e a s t upper bound  be a p o s i t i v e normal l i n e a r  be a semigroup and,  thesis equivalent  ^  A  ^(B^)  linear positive functionals if  T  on the  : s e S}  f u n c t i o n a l s on  A  o f normal  in  A  projections A  , invariant  -homomor-  o f a complete s e t o f  w h i c h a r e i n v a r i a n t under the  iii.  TABLE OF CONTENTS  Page  INTRODUCTION  1  CHAPTER I  PRELIMINARIES  6  CHAPTER I I  SEMIGROUPS OF POSITIVE NORMAL CONTRACTION  OPERATORS  CHAPTER I I I  BIBLIOGRAPHY  SEMIGROUPS OF NORMAL  21  -HOMOMORPHISMS  41  76  iv.  ACKNOWLEDGEMENTS  I am deeply for  the s u g g e s t i o n  this  thesis.  my s u c c e s s  indebted  to my s u p e r v i s o r , P r o f e s s o r Edmond E. G r a n i r e r , .  o f t h i s s u b j e c t and f o r h i s h e l p d u r i n g  H i s patience  and u n s e l f i s h n e s s i n o f f e r i n g h i s time to ensure  i s p a r t i c u l a r l y appreciated.  I a l s o w i s h to e x p r e s s my a p p r e c i a -  t i o n to P r o f e s s o r D.J.C. Bures f o r h i s v a l u a b l e s u g g e s t i o n s reading  o f the d r a f t form o f t h i s  I would l i k e  the p r e p a r a t i o n o f  thesis.  to express my h e a r t f e l t  e n a b l e me t o o b t a i n an e d u c a t i o n  during h i s  o f today.  thanks to my p a r e n t s  who  I a l s o w i s h t o thank my w i f e  f o r t a k i n g c a r e o f my f a m i l y so t h a t I c o u l d devote my time t o my  research  work.  Many thanks a r e a l s o due to Miss B a r b a r a K i l b r a y f o r t y p i n g thesis with patience  Finally,  this  and c a r e .  the f i n a n c i a l a s s i s t a n c e o f the U n 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.  semigroup o f measurable maps on to  be weakly-wandering  -1 E} n n=l  X  into  i f there e x i s t s  0 0  {s  are pairwise d i s j o i n t . ^ J  invariant  i f u ( s ^E) = u(E)  [11] were the f i r s t finite  .  {s } . C S n n=l u  on  S  . Hajian-Kakutani  t o prove the e q u i v a l e n c e between t h e e x i s t e n c e of a  s e t of p o s i t i v e  p  and t h e n o n - e x i s t e n c e o f a  p-measure, i n case  L. Sucheston [19], Neveu [16], Blum-Freidman [12], G r a n i r e r  [ 9 ] , Sachdeva  S  i s a group  set  A  +  of a l l p r o j e c t i o n s i n A  a linear functional  be t h e l e a s t In  result  d i r e c t i o n s by  [2], Natarajan [15],  [18] and o t h e r s .  A  on a complex  Hilbert  denote the s e t o f a l l p o s i t i v e elements i n A  increasing net  This  i s the main purpose o f t h i s t h e s i s to extend t h e above  r e s u l t s t o a von Neumann a l g e b r a Let  i s said  i s s a i d t o be  fora l l s e S , E e S  has s i n c e been g e n e r a l i z e d and improved i n d i f f e r e n t  It  be a  such t h a t  g e n e r a t e d by one n o n - s i n g u l a r i n v e r t i b l e t r a n s f o r m a t i o n .  Hajian-Ito  S  E C X  A subset  00  A measure  i n v a r i a n t measure e q u i v a l e n t t o  weakly-wandering  X  Let  {A }  <t> on in A  upper bound o f  .  A +  )  A l i n e a r operator i s said  T  , and on  A  H P  the  (resp.  to be normal i f f o r any  w i t h l e a s t upper bound T(A ) a  space  ( r e s p . we have  A  we have  <J> (A) = sup (J>(A ) ) a a  t h i s t h e s i s we i n v e s t i g a t e c o n d i t i o n s which a r e s u f f i c i e n t  e x i s t e n c e 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  T(A)  A  f o r the  invariant with  r e s p e c t t o a c e r t a i n semigroup o f p o s i t i v e normal l i n e a r o p e r a t o r s on  2.  In Chapter 1 we g i v e the g e n e r a l  d e f i n i t i o n s and n o t a t i o n s .  We a l s o quote some f a c t s on von Neumann a l g e b r a s ,  and g i v e t h e i r  proofs  or i n d i c a t e the r e f e r e n c e s o f i t s p r o o f . Given a f i n i t e measure  space  (X,S,p)  a von Neumann a l g e b r a on the complex H i l b e r t predual  of  L (X,S,p)  i s the Banach space  .  space  .  , and  measure on  S  , Hajian-Ito  following:  If  S  S  p  L (X,S,p)  i s a probability  [12] and U. Sachdeva [18] proved the  i s a left  r e p r e s e n t a t i o n of  I n case t h a t  The  1  c o n t i n u o u s f u n c t i o n a l s on  [ 7 ] , p. 31, Theorem 1 ) .  is  L.(X,S,p)  1  t h e space o f a l l u l t r a - w e a k l y  (See D i x m i e r  L (X,S,p)  L^CXjS,?)  L..(X,S,p)  00  is  Then  amenable semigroup and  as p o s i t i v e c o n t r a c t i o n s on  {T  g  ; s e S}  L^(X,S,p)  isa  , then  the f o l l o w i n g a r e e q u i v a l e n t : (1)  There e x i s t s s e S  (2)  f  L^X^p)  Q e  (i.e. f^ > 0  p(E) > 0  with  f u n c t i o n a l f o r the s e t of l e f t p(E) > 0  implies  (4)  h e L (X,S,p) ,  + oo  £  *  T  £ S  implies  g  for a l l  Q  i s the support  i n v a r i a n t means on  m(S)  S} > 0  e  h e L (X,S,p) oo  S  n=l s  = T f  where  i n f { / T ldp ; s E s  °°  Q  a.e.)  M^J^ldp) > 0  implies  (3)  0 < f  f o r some sequence  n  h = 0  , where  T  i s the a d j o i n t  operator  n of  T  on  S  L (X,S,p) oo  .  n It  i s t h e main purpose i n Chapter 2 to g e n e r a l i z e t h i s r e s u l t  Neumann a l g e b r a s .  In t h i s  on a (complex) H i l b e r t semigroup and  {T  g  c h a p t e r we l e t A  space  ; s £ S}  normal c o n t r a c t i o n o p e r a t o r s  H  .  Let  S  be a von Neumann a l g e b r a denote a l e f t  be an a n t i r e p r e s e n t a t i o n o f on  A  (i.e. T st  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 the f o l l o w i n g a r e e q u i v a l e n t :  A  to von  .  = T T t s  amenable S  ).  We p r o v e t h a t  as p o s i t i v e  L e t <b  be  n  0 (Theorem 2-6)  3. (1)  There e x i s t s a  S-invariant  <j) ( i . e . | ( T A) = <KA) <J> ~ <$> (2)  p o s i t i v e normal l i n e a r f u n c t i o n a l  for a l l s e S  ( i . e . tj)(A) = 0  i f and only  A e A  ;  ) such t h a t  i f <J>(A) = 0  for A e A ) +  Q  E e P, <j> (E) > 0 i m p l i e s inf{<j> (T E) ; s £ S} > 0 u u s <j) (E) = 0 i m p l i e s M^QOME)) = 0  , and  n  0  (3)  A e A and  <J>(A) = 0 Q  * i f ^ <$>Q  Moreover,  (where  A  If  for  A e A  E e P  1  T  T*(j, -< <j> 0  +  )  0  (2a)  If  A  A  g  Mj^ (cj) ( T A ) ) = 0 0  T  s £ S  .  g  central f o r a l l s e S  then  <j>  can be chosen  on the d u a l space  s  implies  of  * ( T A ) = T*<|, (A) = 0 0  g  { T E ; s e S} C {E}"  and  A  f o r any  g  (Theorem 2-10) the e q u i v a l e n c e o f the f o l l o w i n g : 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> ~ CJ>Q  such t h a t  P ((2b) A e A  e  0  0  for a l l  on  implies  inf{<j> (T A) ; s e S} > 0 ,  ( i . e . ,|, (A) = 0  There e x i s t s a (J)  implies  i s the a d j o i n t o f  s  , then we have (1)  S  S  central ).  , (J)Q(A) > 0  +  )  +  then  ^(A) > 0  implies  inf{<J> (T A) ; s e S} > 0 0  (3a)  If  A  V * 0 (4a)  If  (  P  e  T  A  ((3b) A e A )  A e V ((4b) A e A  $ (A) > 0  implies  +  )  and t h e r e e x i s t s a sequence  00  {s }  C  *  Moreover, i f T d>„ .s 0 T  our  then  > 0  )  S  CO  central.  )  +  S  such t h a t  E T A e A n=l n  , then  i s central f o r a l l s e S  then  Some r e l a t e d r e s u l t s a l s o o b t a i n e d .  r e s u l t s f o r f i n i t e von Neumann a l g e b r a s  6  <J>(A) = 0 n  .  can be chosen  We s t r e n g t h e n somewhat  (see Theorem 2-12).  In Chapter 3 we a r e m a i n l y concerned w i t h semigroups o f normal *-homomorphisms on a von Neumann a l g e b r a and  {T  s  ; s e S}  an  A  antirepresentation of  . S  We l e t S as normal  be a semigroup *-homomorphisms  on  A  .  A projection oo  e x i s t s a sequence i  .  =f j  .  Let  The f o l l o w i n g  A  {s } , n n=l  Q  If  S  <J>-< <J>  > 0  0  (l )  that (2')  Moreover,  4>  (J) e T  S-invariant  projection  A  $ e A  E e P,  such  that  §  E  in  P  with  § e A  such  4>Q  (2) h o l d s .  0  i s central  then  t r u e i f we  <|>  A  can be chosen  consider  is a finite  {T A  s  : s e S} to be a r e p r e s e n t a -  .  Related r e s u l t s are obtained  3-11).  In t h i s c h a p t e r we  also  c o n d i t i o n s f o r the e x i s t e n c e of a complete s e t  p o s i t i v e normal  (linear) functionals  to be complete i f g i v e n non-zero (j)(A) =[ 0  central.  (or a - f i n i t e f i n i t e ) von Neumann  (see Theorem 3-9 and C o r o l l a r y the s u f f i c i e n t  ,  g  as normal *-homomorphisms on  such t h a t  f o r any  p o s i t i v e normal  and  i s said  f o r any  3  p o s i t i v e normal  S-invariant  0  S-invariant  TCA  1  E = 0  ( l i n e a r ) on  (2*):  ( l ) <=>  s.  chapter.  E e P, $ ( E ) = 0 => M ( $ ( T E ) ) = 0  f o r the c a s e t h a t  of  E'T  .  T h i s theorem i s s t i l l  investigate  i n this  If  if  S  s. 1  (2) and  There e x i s t s a  1  T  T E-E = E-T E s s  There i s no weakly-wandering • (E)  algebra  such t h a t  i f there  •  0  t i o n of  S  i s amenable and  There e x i s t s a  (2)  in  be a p o s i t i v e normal f u n c t i o n a l  , then (1) <=> (1)  i s s a i d to be weakly-wandering  i s one o f our main r e s u l t s  Theorem 3-4: s e S  E e P  ):  on  A e A  A  (a s e t , there i s  5.  Theorem 3-14:  Let  any p r o j e c t i o n  E e P, s e S  (1)  S  be amenable and such t h a t .  Then the f o l l o w i n g  There e x i s t s a complete s e t o f ( l i n e a r ) f u n c t i o n a l s on  (2) (3a)  0 =f A e P  ((3b)  0 =f A e A  and C o r o l l a r y  (^ A g  +  p o s i t i v e normal  ; s c  3-17).  projection i n  ), then S}  for  equivalent:  A  i n the s t r o n g c l o s u r e of ( A l s o see Theorem 3-15  are  S-invariant  There i s no non-zero weakly-wandering If  E'T E = T E'E s s  0  i s not  A  6.  CHAPTER 1 PRELIMINARIES  In t h i s c h a p t e r we g i v e some g e n e r a l d e f i n i t i o n s and n o t a t i o n s . We a l s o r e c a l l some f a c t s t h a t a r e needed i n the next two c h a p t e r s . If  S  i s any semigroup,  complex v a l u e d f u n c t i o n s on denotes all f(s)  S  m(S)  denotes  w i t h sup norm.  If  t h e c o n j u g a t e o f f i . e . f e m(S) such t h a t  s e S .  , where  L e t m(S)  to be a mean on  be the d u a l space o f  m(S)  m(S)  i . e . the space o f a l l  An element  < J > e m(S)  i s r e a l then  be such t h a t  i s said  hold:  .  i n f f ( s ) <_ty(f) <_ sup f ( s ) . s  1 e m(S)  f(s) = f(s) for  i f the f o l l o w i n g two p r o p e r t i e s f  If f  m(S)  .  (1) ty(f) = <f>(f) ' o r a l l f e m(S)  (2)  f e m(S) , f  f ( s ) i s t h e complex c o n j u g a t e o f the complex number  c o n t i n u o u s l i n e a r f u n c t i o n a l s on  Let  the space o f a l l bounded  s  l(s) = 1  fora l l s e S  , then (2) i s  e q u i v a l e n t to (2')  <j>(l) = 1  and  i f f >_ 0  <f.(f)>_0  I t i s c l e a r t h a t i f cj> i s a mean on  Let [right]  S  be a semigroup  translation:  Z [r ] a  a  on  and  m(S)  a e S  m(S)  I f(s) = f(cs) [r f(s) = f(so)] a a  .  then  ty  has norm one i . e .  We d e f i n e the l e f t  by  f o r a l l f e m(S), a e S  7.  A mean  <j)  on  m(S)  i s said a e S  {invariant} i f f o r a l l <|>u  LIM[RIM] {IM}  .  A semigroup  amenable] {amenable} i f  f  e m(S)  M (f) R  .  we  = sup  If  that  , let  (A x|y)  0  S  F  IM  )  = *(f)> [right  = sup  i s a complex H i l b e r t  w  e  denote by  A*  =  E L(H)  (x|Ay)  I  amenable  i s not empty] {IM  (Ax|x)  in  A  is called  A  H  A  f o r a l l x,y ;  H  is  (resp.  {<j>(f) J <J> e IM})  the a d j o i n t o f  = A  [right  .  space w i t h i n n e r p r o d u c t  i s the i d e n t i t y o p e r a t o r on  2 A  be  {invariant}  i s not empty, then f o r r e a l  U ( f ) ; y e LIM} , M(f)  •  invariant]  left  [RIM  )  invariant]  have  i s s a i d to be  i s not empty  ^ ( f ) = sup  c a l l e d hermitian i f  where  if  H  R  invariant  ( r e s p . RIM,  {<j>(f) ; <j> £ RIM}  = *(  we  [right  denotes the a l g e b r a o f a l l bounded l i n e a r o p e r a t o r s on  A E L(H)  is  LIM  let  If L(H)  LIM  invariant  f e m(S)'  the s e t of a l l l e f t  m(S)  not empty}  ,  left  [*(r f ) = <|)(f)]' U U / )  f ) = 4>(f)  means on  to be  .  ;  A  i.e.  A*  |  H  unitary i f  )  .  £ L(H)  A e L(H)  Let  (  ,  For such  , then  A =1  A A = AA  i s called a projection  * = A = A  ;  A  i s called positive,  i s positive for a l l x £ H  t o p o l o g i e s on  L(H)  weak t o p o l o g y ,  the u l t r a - s t r o n g  .  denoted  by  A >_ 0  There are a t l e a s t  ; the u n i f o r m t o p o l o g y , t o p o l o g y , and  five  the ultra-weak  i s a f a m i l y o f elements i n i s a f i n i t e subset of A}  o r d e r e d by  { I  then  ; a is a finite  A  useful  the s t r o n g t o p o l o g y ,  (see [ 4 ] , p. 32). If {A } . A , l e t {act ;a £ aA an index s e t set i n c l u s i o n ,  , if  the  topology L(H) with be p a r t i a l l y  subset  of  A}  a  aca i s a n e t o f elements i n we  denote i t by  £  L(H)  A A  CIEA  -By a  .  I f the s t r o n g l i m i t the same r e a s o n ,  of t h i s net  i f the s t r o n g  exists,  limit  8.  of the net  { n  aea (where  by  II A . a aeA  If  E  if  EF = FE = 0  and  F  projections {E  } . a aeA  < E a —  B - A  ; o i s a f i n i t e subset of II A a aea  , and denoted by and  F  J  L(H)  E J- F  a r e such t h a t r  (E } . a aeA  for a l l  e x i s t s we denote i t  i s the p r o d u c t o f elements  a r e two p r o j e c t i o n s i n  E  A}  a  .  then  J  i s pr o s i t i v e ) ;  ''  (note  t h a t f o r A.B  denote by  the l a r g e s t p r o j e c t i o n  '  F  in  A E . a aeA  L(H)  ).  E J- F  . Let  , denote by  i . e . the l e a s t p r o j e c t i o n  a e A  for a e a  a  I t i s c l e a r t h a t i f two  L(H)  J  A  , they a r e s a i d t o be o r t h o g o n a l  EF = 0  be a f a m i l y o f p r o j e c t i o n s i n  suprema of E  A  in  E  in  V E the . a aeA 1(H) such t h a t  L(H) , A < B —  the i n f i m a o f  such t h a t  {E } OL  F <_ E^  means  aeA  for a l l  i.e. a e A  {E } . i s p a i r w i s e o r t h o g o n a l ( 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 f a m i l y . 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 In case  &  If for a l l  A  B e A}  i s a subset o f .  i f i t contains  say  A  A C L(H) known.  , we l e t  A s u b a l g e b r a o f the a l g e b r a  ^-algebra that  1(H)  A  A' = {A e L(H) ; AB = BA 1(H)  whenever i t c o n t a i n s  A  i s s a i d to be a in  i s a von Neumann a l g e b r a on a H i l b e r t space  is a  *-algebra  and  A" = A  .  L(H)  .  We  H i f  The f o l l o w i n g theorem i s w e l l -  9.  Theorem 1-A ( [ 4 ] , p. 43 Theorem 2 and p. 44 C o r o l l a r y 2 ) : A  *-algebra  A  L(H)  the i d e n t i t y o p e r a t o r  i s a von Neumann a l g e b r a i f and o n l y i f i t c o n t a i n s  I and i t i s c l o s e d under any one (hence a l l ) o f  the f o l l o w i n g . t o p o l o g i e s ; the s t r o n g s t r o n g topology,  If call  A' f\  A  A  topology,  and the u l t r a - w e a k  the weak t o p o l o g y ,  topology.  i s a von Neumann a l g e b r a on a H i l b e r t space  the c e n t e r o f  A  the u l t r a -  , and we denoted i t by  Z  .  H  , we  It i s  c l e a r t h a t the c e n t e r o f a von Neumann a l g e b r a i s a g a i n a von Neumann algebra.  I f the c e n t e r o f a von Neumann a l g e b r a  s c a l a r multiples of then  A  i s said  Let we l e t  A  I  A  t h a t f o r any  A £ A An  ,  and  A  .  Let  U = {A e A  in  +  A  ; A 2  PAZ  i s said  H  ,  i . e . the space o f a l l bounded  = {A E A ; A > 0} i s unitary}  with hermitians  A £ A , A = A^ - A^  i f and o n l y i f A = B B E  A  A e A , A = A^ + i A  any h e r m i t i a n  element  only  , the complex numbers}  be a von Neumann a l g e b r a on a H i l b e r t space  denote the d u a l space o f  i s projection}  ;  = {Al ; A e C  contains  to be a f a c t o r .  l i n e a r f u n c t i o n a l s on  A  i.e. Z  A  with  A^  , P = (A e A ; .  A^ and  A  I t i s well-known and A^  in in  A  +  ;  f o r some B E A ( f o r r e f e r e n c e see [ 1 7 ] ) . to be a c e n t r a l p r o j e c t i o n .  10.  Let  A  be a von Neumann a l g e b r a  When we say t h a t  < j >  i s a f u n c t i o n a l on  When we say t h a t  T  i s an o p e r a t o r on  l i n e a r o p e r a t o r on  Let  A  A  into  T  i s called central  i f T(AB) = T(BA)  , then  T  we always mean t h a t  i s defined  T  isa  to be  fora l l  fora l l  |A|| =  normal i f f o r any i n c r e a s i n g n e t  on a H i l b e r t space  H  . Let  +  i f IITAII < IIAII  contraction  TA  A  < j >e A  i s called positive i f T(A ) C A  A  A  we always mean t h a t  be a von Neumann a l g e b r a  be an o p e r a t o r on  norm o f  A  H  A  T  called  on a H i l b e r t space  sup  A  e  A,B in A  ( f o r A e L (H)  A  ||Ax|| , x e H  );  II x |-1  {A }  in A  be the l e a s t upper bound o f  ;  T  the  i s called  w i t h l e a s t upper bound  we have  a  *~homomorphism [*-isomorphism] i f i t i s an a l g e b r a i c homomorphism  is if  c a l l e d p o s i t i v e i f cf> (A) >_ 0 <j)(AB) = (J)(BA)  ^ (A) = 0 anyJ  fora l l  implies  A = 0  increasing net  {A }  aa  {T(A )} a  T(A ) = (TA) for A c A  A, B  in  for A e A in A  .  +  A  ; ; < j >  +  ;  T is  A  [isomorphism] and such t h a t  +  Let  ;  T  i s called  < j >e A  , then < t >  ; < j > i s called central  +  < } >  i s called faithful i f  i s c a l l e d normal i f f o r  w i t h l e a s t upper bound  A  we have  d ) ( A ) = sup <j)(A ) ; d> i s c a l l e d p o s i t i v e s i n g u l a r 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 t h a t ^ <_ <j> then \p = 0 ; <j)  i s called singular i f $  l a r f u n c t i o n a l s on family <j>( E aeA p.  {E } E ) = a  ,  E aeA  A  singu-  ;' < j > i s c a l l e d c o m p l e t e l y a d d i t i v e i f f o r any  of pairwise <))(E )  i s a l i n e a r combination o f p o s i t i v e  .  orthogonal projections  The f o l l o w i n g  A  we have  theorem i s well-known  a  15, Theorem 3 and p. 16 f o o t note  in  (16)):  ([17],  11.  Theorem 1-B (Dixmier) ; H  be a von Neumann a l g e b r a  <> j £  (2)  * A  <t> £ A  (3)  ty £  i s p o s i t i v e and normal.  * i s p o s i t i v e and c o m p l e t e l y a d d i t i v e .  * A  i s p o s i t i v e and u l t r a - w e a k l y  Let  A  and  ty  be a von Neumann a l g e b r a  Let  <> j  said  to be a b s o l u t e l y <J>(A) = 0 2  equivalent  to  A  on a H i l b e r t space  continuous w i t h r e s p e c t  i m p l i e s ty^(A) = 0 ty^  (written  ty  to  for A e A  ~ ty  )  2  ty A  H  A  , then  <j>^ i s  ty  (written  ty^^  ty  2  ; ty^ i s s a i d to be  +  o f a von Neumann a l g e b r a  +  continuous  a r e two p o s i t i v e f u n c t i o n a l s on  A non-negative numerical f u n c t i o n on  on a H i l b e r t space  , then the f o l l o w i n g are e q u i v a l e n t : (1)  if  Let A  i f <J>-^-< ty  ^2"^ ^1  3X1,2  2  , f i n i t e or i n f i n i t e ,  defined A i f  i s s a i d to be a t r a c e on  i t has t h e f o l l o w i n g p r o p e r t i e s : (1) ty(k + B) = cf)(A) + ty(B) f o r A, B  (2) ty (AA) = XtyiA)  ty  for  all  -1  be a t r a c e on A e A  tj)(A) > 0 in  A  +  , X >_ 0  +  ({.(UAU ) = ty(k) f o r A e A  (3) Let  for A E A  A  , then  ty  in  +  A  +  (Assume  0°° = 0  , U e U .  i s s a i d t o be f i n i t e  i f <>j (A)  ; - t y i s s a i d to be f a i t h f u l i f 0 f A e A  ; ty i s s a i d t o be normal  w i t h l e a s t upper bound  A  )  implies  i f f o r any i n c r e a s i n g n e t we have ty(A) = sup  < °°  <j>(A )  {A ^ . It  a is on  c l e a r t h a t 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  .The following proposition  converse.  A  is a finite  trace  ( [ 4 ] , p. 80 P r o p o s i t i o n 1) g i v e s t h e  )  12.  P r o p o s i t i o n 1-C: set  of a l l  A e A  sided i d e a l I  Let  < j > be a t r a c e on a von Neumann a l g e b r a  with  +  I C A  .  .  Also  ultra-weakly  There i s one and o n l y < j > on  I  and  i f (j> i s normal then  c o n t i n u o u s on  one l i n e a r f u n c t i o n a l  <f>(AB) = <}>(BA)  A -> <j>(BA)  for A e I  > for B e l  finite  in  A  finite central  t r a c e s on  .We  A  f a i t h f u l ) t r a c e s on  A  if  ^  on  any f a m i l y  A a  a e  /\  A  °^ p a i r w i s e  H i l b e r t space  known p r o p o s i t i o n  P r o p o s i t i o n 1-D: H  A e A  cj>(A) =f 0  a t most c o u n t a b l y many elements. a separable  on a H i l b e r t space  non-zero  such t h a t  ^ ^  H  ;  there  +  A  A  H  There  (2)  A  i s a finite  i s s a i d to be  normal  q-finite contains  Hence any von Neumann a l g e b r a  i s a-finite.  on  (ii)):  e x i s t s a f i n i t e normal f a i t h f u l (i-e.  A  The f o l l o w i n g i s a w e l l -  be a von Neumann a l g e b r a  i s a-finite finite  (resp.  i s said  on a H i l b e r t space  , then the f o l l o w i n g a r e e q u i v a l e n t : (1)  , is  A  orthogonal projections  ( [ 4 ] , p. 98 P r o p o s i t i o n 9  Let  ,  w i t h the s e t o f a l l p o s i t i v e normal  A von Neumann a l g e b r a  trace  on  the s e t o f  the s e t o f a l l f i n i t e normal  ( r e s p . p o s i t i v e f a i t h f u l c e n t r a l ) f u n c t i o n a l s on  i f f o r any  c> j  w i t h the s e t o f a l l p o s i t i v e c e n t r a l elements  can even i d e n t i f y  to be f i n i t e  The  A  In view o f 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 all  .  <j>(A) < °° form the p o s i t i v e p a r t o f a two  which c o i n c i d e s w i t h  B e A  A  t r a c e on  A  a - f i n i t e and f i n i t e ) .  13.  (3)  A  is finite  Let  A  Two p r o j e c t i o n s if  E = AA  *  Z  is  and  F  * F = A A  in A  •  a projection i n  A  H  (written  E = F)  A £ A i •  A von Neumann a l g e b r a  only p r o j e c t i o n i n A  on a H i l b e r t space  are c a l l e d equivalent  f o r some  1  Theorem 1-E:  o-finite.  be a von Neumann a l g e b r a E  and  and  A  i s finite  which i s e q u i v a l e n t  such t h a t  E - I  to  , then  I  i f and o n l y is  E = I  I  i f the  i.e. i f E  .  is  ([4], p. 308  Theorem 1 ) .  Let A  .  A  A function  be a f a c t o r and  ty  on  P  P  , takes on v a l u e s  s a i d t o be a r e l a t i v e dimension on f i n i t e normal f a i t h f u l  P r o p o s i t i o n 1-F;  t r a c e on  Let  A  A  in  [0,<*>) V  {°°}  ,  is  i f ty i s the r e s t r i c t i o n o f a  A  A  be a f i n i t e  non-negative f u n c t i o n defined on  the s e t o f a l l p r o j e c t i o n s i n  on  P  .  f a c t o r and Then  D =j= 0  be a f i n i t e  D  i s a relative  P  with  dimension  i f i t satisfies  (1)  D(E + F) = D(E) + D(F)  (2)  D ( U E U ) = D(E) _1  f o r E,F  f o r any  in  E X F  E e P ,U e U  ([4], p. 248 P r o p o s i t i o n 1 5 ) . In view o f P r o p o s i t i o n 1-F, any non-zero f i n i t e A  i s normal and f a i t h f u l .  t r a c e on a f i n i t e  factor  The  symbol // i s used f o r the completeness o f a p r o o f .  In t h e f o l l o w i n g we s t a t e some f a c t s on von Neumann a l g e b r a s , and we g i v e t h e p r o o f  ( o r i n d i c a t e t h e r e f e r e n c e s o f t h e p r o o f ) f o r each  L e t A denote a von Neumann a l g e b r a on a H i l b e r t space  result.  Let < J >e A  Lemma 1-G:  Then t h e r e a r e E e P  Proof:  i s an u n i f o r m  A.E  . -• • 1  1=1 i s some all  , c > 0  such  that  cE <_ A  with  ; A. > 0 A .  1  A. > 0 l  E, e A A ,  and  1  '  .  limit E  <j> (A) > 0  and cj> (E) > 0  From the s p e c t r a l theorem f o r bounded h e r m i t i a n  ([1]), A n { Z  Let A e A  be p o s i t i v e .  H  .  operators  o f a n e t o f elements o f type a r e p r o j e c t i o n s i n {A}"}  .  So t h e r e  A .  with  //  1  A.E. < A I A. — l  and  <j> (E, ) > 0 A. l  •  Since  A" = A  ,  1  Lemma.1-H:  L e t ty ,ty  i m p l i e s ty (E) = 0  Proof:  for E e P  Let A e A  E e P , c > 0  in A  +  be p o s i t i v e , and such t h a t ty(E) = 0 .  w i t h ty (A) > 0  such t h a t  c E <_ A  hence ty(A) >_ty(cE)= cty(E) > 0  Theorem 1-1: ty  is  Theorem  If  if  .  From Lemma 1-G t h e r e a r e  and ty (E) > 0  .  So ty(E) > 0  . Soty-<ty.  on any bounded subset o f  ,  //  A  i s a p o s i t i v e normal f u n c t i o n a l on  s t r o n g l y continuous l(ii)).  Then ty ~< ty  A  .  , then ( [ 4 ] , p. 40  Theorem 1-J:  If  ultra-weakly  T  i s a p o s i t i v e normal o p e r a t o r on  c o n t i n u o u s , and  i s weakly c o n t i n u o u s on  A  any  , then  A  .  I f there T  exists  AeA  , then  on any  bounded subset of  k _> 0  such t h a t  (TA)  A  .  ( [ 4 ] , p.  56  A  TA  i s u l t r a - s t r o n g l y continuous and  is  bounded subset  A  of  T  <_ kT(A  A)  for  i s strongly  continuous  Theorem 2 ) .  A  Proposition  1-K:  Let  <)> e A  a largest projection  F  For  9 (FA)  this  F  Proposition  we  is  normal.  i n the s e t of p r o j e c t i o n s  projection  s u p p o r t of  <b  the s m a l l e s t  known t h a t  p o s i t i v e and  = cj>(AF) = 0  for a l l  G  Then t h e r e such t h a t  AeA  .  is  9(G)  ( [ 4 ] , p.  = 0 61  3).  The the  have  be  E  , and  with  F  denoted by  E  projection = I  I  I-F  i f and  G  in  as  only  if  if  A>  .  9  J  A  i n Proposition  is called  I t i s easy to see  such t h a t <j>  1-K  that  9 (G) = $(1)  is faithful  •  E, <j>  It is  ( [ 4 ] , p. 61),  and  9 E,  i s i n the  center  of  A  i s central  ( [ 4 ] , p.  82).  9 A  Proposition  1-L:  Let  A^  be  the u n i t b a l l of  A  ,  and  9 e A  be  A  p o s i t i v e and  normal.  i s equivalent Proposition  Corollary  to the  On  A^  , the  convergence of  convergence of AE^  <j> (A A)  to  to zero s t r o n g l y .  zero ( [ 4 ] , p.  4).  1-M:  Let  {A  } n n  be  an u n i f o r m l y  bounded sequence i n  A  If <b(A  <j> e A n  )  i s p o s i t i v e normal and  converges to zero i m p l i e s  such t h a t A E n 9  E,  9  is in  converges to z e r o  Z  , then strongly,  A  +  62  Proof: IIA 11  {B  Since  || < k n" — } n n  || B  {A } n n  for a l l n such  || <_ k  that  B  .  Since  > 0 n —  for a l l n  A  and  .  B  n  2=  > 0 —  for a l l  n  , there  A  for a l l  n  .  n  n  Thus the sequence  {^n} k  x.  A  A,  and  k > 0  a r e u n i f o r m l y bounded, t h e r e i s  B . B 4>(.y^~ ' 17~  J  such  that  exists  So we  have  i s a sequence i n  A =  9 (~ o") k k  ~~i 9 (^ )  =  converges to z e r o .  B P r o p o s i t i o n 1-L,  converges to zero s t r o n g l y i . e .  converges to zero f o r a l l  x e H  .  Since  E  e Z , we  From  B \\yT~ E^x||  have  9 A ||—Tr 1  ^Z  A E n 9  B B E xII = ||-— E (-— 9 ' k (J) K.  converges to z e r o s t r o n g l y .  P r o p o s i t i o n 1-N: The  E x) || converges to zero f o r a l l <j>  1  Let  9,1})  x e H  .  Hence  #  be two  p o s i t i v e normal f u n c t i o n a l s on  A  f o l l o w i n g are e q u i v a l e n t : (1)  <j><>  (3)  on the u n i t b a l l  A^  [^(A A)]  2  , the t o p o l o g y g i v e n by  i s s t r o n g e r than the t o p o l o g y  1  the  g i v e n by  2  A '  the seminorm  A  1_  *  seminorm  of  [ i f ( A A ) ]  .  ( [ 4 ] , p. 62 P r o p o s i t i o n 5 ) . C o r o l l a r y 1-N' : such t h a t that  iKA  <fa —< il» T  a  )  Let .  9,4) If  be  {A } a a  two  p o s i t i v e normal f u n c t i o n a l s on  i s a u n i f o r m l y bounded net i n  converges to z e r o , then  9(A  a  )  converges to z e r o ,  A  A such  Proof:  Let  i s a net i n  k > 0 A  If  A  N  e v e r y Cauchy sequence i n  N  Let is  for a l l .  a  .  then any two f i n i t e normal f a i t h f u l  ( [ 4 ] , p. 90  Corollary).  N  has a l i m i t  A Banach space  N  || || .  If  in  ) then  N  N  i s complete N  is  i s c a l l e d a Banach a l g e b r a  i s a l s o an a l g e b r a and such t h a t (1)  ||ab|| £ || a || ||b|l  (2)  If  N  N  for  a,b  has an i d e n t i t y  in  e  (1)  (a + b ) *  (2)  (Aa)  = a* + b *  X  —  A  .  '*' ' d e f i n e d on  , for  s a t i s f i e s the f o l l o w i n g  for  a,b  in  (4)  a  a e N  for  properties:  N  A  i s the complex c o n j u g a t e o f the complex  a,b  in  N  A Banach a l g e b r a w i t h an i n v o l u t i o n i s c a l l e d a Banach I t i s easy to see t h a t  L(H)  (and any  * - a l g e b r a w i t h the i n v o l u t i o n L(H)  a measure space then  N  a e N  for  o f an element i n  into  '*'  , where  ( a b ) * = b*a"  Banach  N  —  (3)  = a  ||e|| = 1  if  &  = Aa  number  N  N  , then  be a Banach a l g e b r a , a mapping  c a l l e d an i n v o l u t i o n on  .  ^-algebra i n '*'  on  L(H)  ^-algebra. 1(H)  L (X,S,p)  i s a Banach  to be the " c o n j u g a t i o n "  (i.e.  f  )  (X,S,p)  is  * - a l g e b r a w i t h the i n v o l u t i o n  —  = f  is a  to be the " a d j o i n t "  I t i s a l s o easy to see t h a t i f  *  '*'  A {-—} k a  Then  Our c o r o l l a r y f o l l o w s from  be a normed space w i t h norm  c a l l e d a Banach space. if  A  i s a factor,  are p r o p o s i t i o n a l .  Let (i.e.  II < k a" —  //  P r o p o s i t i o n 1-0: A  11  , the u n i t b a l l of  P r o p o s i t i o n 1-N.  t r a c e s on  IIA  such t h a t  for  f e L  (X,S,p)  18.  Let N  2  N^  and ^  be two Banach  i s s a i d to be an i s o m e t r i c  ^-algebras.  A mapping  ^-isomorphism i f  9  9  on  into  i s an a l g e b r a i c  isomorphism and such that (1)  i| 9(a) ||  C2)  9(a ) = 9(a)'  = || a||  for a e for a e N  I f there e x i s t s an i s o m e t r i c N-^ and  > then  A Banach  ^-algebra  ^-isomorphism between two Banach  and N  are s a i d to be i s o m e t r i c a l l y  i s s a i d to be a  C -algebra i f N  ^-isomorphic.  i s isometrically  *-algebra i n L(H) f o r some H i l b e r t  p)  (X,S,p)  i s a f i n i t e or < r - f i n i t e measure space then  i s a H i l b e r t space w i t h inner product  (f|g) = j f(x)g(x)dp  /.  X  If, for f  g e  ^-algebras  H  If L„(X,S,  ±  .  ^-isomorphic to a uniformly closed space  H  e  L^CX.S.p)  L^CXJSJP)  and x e  , we l e t f ( g ) ( x ) = f ( x ) g ( x )  , then  X  L  Neumann algebra on the H i l b e r t space  O T  space  i s a commutative von  (X,S,P)  L2(X,S,p)  t a t i v e von Neumann algebra i s of the type  f o r any  . In f a c t , every commu-  L (X,S,p) OT  f o r some measure  (X,S,p)  Theorem 1-P:  A von Neumann algebra  t a t i v e i f and only i f A some measure space  A  on a H i l b e r t space  i s isometrically  (X,S,p)  .  H  i s commu-  ^-isomorphic to L^CXjSjp) f o r  ( [ 4 ] , p. 117 Theorem 1 and Theorem 2 ) .  In [5] and [6] Dixmier proved the f o l l o w i n g r e s u l t s .  19.  Theorem 1-Q:  If  H  , then there  Z  of  A  i s a f i n i t e von Neumann a l g e b r a  i s one and o n l y one f u n c t i o n  w i t h the f o l l o w i n g  (1)  = A  i f A  e  (3)  (A + B ) ^ =  (4)  (AB)^ = (BA)^ f o r A,B  (5)  (AB)^ = AB^  (7) ([5],  If  A e A  (A*]/  for A e A , X e C  A* + B^  then  +  in  A^ e A  +  for A e A  7  c a l l e d the canonical  Let  h u l l of  {UAU  A  ^7  U e U  the unique element i n  A  Neumann a l g e b r a  A  .  A^ = 0  implies  .  A  A = 0  . The f u n c t i o n  F o r each  tj  i n Theorem 1-Q  function.  A e A ,  with the canonical  let  to be the convex  the s e t o f a l l u n i t a r y elements i n A}  K  C\ Z  The c a n o n i c a l  von  A  be a f i n i t e von Neumann a l g e b r a  be the u n i f o r m c l o s u r e o f  Theorem 1-S:  onto the c e n t e r  the s c a l a r s  in  and  c e n t r a l valued  c e n t r a l valued function  K  f o r A,B  p. 249 Theorem 10 and Theorem 11).  Theorem 1-R:  A  f o r A e Z, B e A  = (A* )*  7  ^7 from  Z  ( A ) ^ = \A^ A  on a H i l b e r t space  properties:  (2)  . (6)  is  A  K  , then the element  .  A^  . Let  i s exactly  ( [ 5 ] , p. 251 Theorem 1 2 ) .  c e n t r a l valued function  i s strongly  on a f i n i t e  c o n t i n u o u s on the u n i t b a l l  A-^ o f  ( [ 5 ] , p. 256 Theorem 1 7 ( B ) ) .  C o r o l l a r y 1-S':  The c a n o n i c a l  von  Neumann a l g e b r a  of  A  .  A  c e n t r a l valued function  i s strongly  ^7  on a f i n i t e  c o n t i n u o u s on any bounded s u b s e t  N  Proof: that of  Let  N  ||A|| <_ k A  .  be a bounded subset o f for a l l A e N  space  H  finite  traces  If  .  Let  < j >  all A e A ty  A  i s normal  on  A  A e A  , where  ([4],  p. 267).  .  , and  ty  T h i s correspondence i s d e f i n e d by  If ty  means  A  then  i s a finite  and  ty  i s normal  (resp.  on  <j)(A) = ty(A^) Z  faithful)  ( [ 6 ] , p. 5 P r o p o s i t i o n 2 ) .  factor,  i s t h e unique f i n i t e  i f u  ty  i s n o t h i n g b u t the r e s t r i c t i o n o f ty on  (resp. f a i t h f u l )  Finally,  ~ u  ball  //  and the s e t o f a l l p o s i t i v e f u n c t i o n a l s  A  C o r o l l a r y 1-T':  " u  the u n i t  such  be a f i n i t e von Neumann a l g e b r a on a H i l b e r t  and t h e converse i s a l s o t r u e .  means  ; A e N} C A  k > 0  There i s a one to one correspondence between the s e t o f a l l  , the c e n t e r o f  for  Thus  , then t h e r e i s  Our c o r o l l a r y now f o l l o w s from Theorem 1-S.  P r o p o s i t i o n 1-T:  Z  .  A  0  then  t r a c e on  A^ = cf>(A)I A  for a l l  such t h a t ty(l) - 1  a r e two measures, we l e t u < Q  i s a b s o l u t e l y continuous w i t h respect to " ^ i s e q u i v a l e n t to  0  and  u " as i n measure t h e o r y .  21.  CHAPTER 2  SEMIGROUPS OF POSITIVE NORMAL CONTRACTION  Let Let  S  A  be a von Neumann a l g e b r a on a H i l b e r t space  be a l e f t  t a t i o n of  S  T „ = T T ) st t s  OPERATORS  amenable  semigroup and  {T  ; s e S}  g  an a n t i r e p r e s e n -  as p o s i t i v e normal c o n t r a c t i o n o p e r a t o r s on .  H  A (i.e.  I n t h i s c h a p t e r we i n v e s t i g a t e c o n d i t i o n s which a r e  e q u i v a l e n t t o the e x i s t e n c e o f 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 i n v a r i a n t under the semigroup  {T  g  ; s e S}  .  The main  theorems i n t h i s c h a p t e r a r e Theorem 2-6 and Theorem 2-10 which  generalize  H a j i a n - I t o ' s r e s u l t s i n [12] and Sachdeva's Theorem 3-3 i n [ 1 8 ] , first  theorem  (Theorem 2-1) i s a g e n e r a l i z a t i o n o f a theorem o f H e w i t t -  Y o s i d a i n [13] (p. 50 Theorem 1.18).  As a consequence o f Theorem 2-1,  we ( C o r o l l a r y 2-2) g e n e r a l i z e a theorem o f C a l d e r o n  Let For  a e N  9(ba)  N  * C - a l g e b r a and  be a  we d e f i n e the r i g h t  f o r <j> e N  a Hilbert  space  and H  .  b e N  Let  N  *  translation .  Let  A  t h e d u a l space o f R  on  N  be t h e space o f a l l s i n g u l a r f u n c t i o n a l s on [7] (p. 31 Theorem 1) t h a t  A  by  N  R 9(b) =  be a von Neumann a l g e b r a on  be the d u a l o f  A  [3] (p. 1962, 2 ) .  A  .  Let  the space o f a l l u l t r a - w e a k l y c o n t i n u o u s f u n c t i o n a l s on  .  Our  A  .  A ^ CA A  be  , and A ^ ~  Dixmier proved i n  i s the d u a l space o f t h e Banach space  T a k e s a k i p r o v e d i n [20] (p. 196 Theorem 3) t h a t  A  i s the  A  J_  direct  A,  sura of  A,  and  A  Moreover, he p r o v e d t h a t  A  •  A,  and  .  '  = R  A  Z  A Q  A  R  =  AA  \A  /n  *  , where  z  i s a central projection  ^ i - Z f J  A C A  (considering following  Let  A C  and  <j>  A  ).  We  singular part  of  of  ty  ty  then b o t h  Theorem 2-B: singular  Let  , .  and a p o s i t i v e s i n g u l a r The normal p a r t  ty^  and  ty  i f and o n l y  p r o v e d by Nakamura A  Then  ty  <j>^ A  of  functional <j>  are c e n t r a l .  ([20] and  A  F < E  and  can  <j>  ty^  ,  , the  If  <> j  is  [14]).  .  Then  ty  is  E e A , there i s a  i f f o r any non-zero p r o j e c t i o n such t h a t  ty  i s n o t h i n g b u t the  dominated by ty .  be a p o s i t i v e f u n c t i o n a l on  non-zero p r o j e c t i o n F e A  .  [14]).  o f a p o s i t i v e normal f u n c t i o n a l  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 central,  s t a t e h i s theorem i n the  be a p o s i t i v e f u n c t i o n a l on  be u n i q u e l y decomposed as a sum the normal p a r t  A  AAA  form ( p a r t of Theorem 2-A was  Theorem 2-A:  m  0  A  AA  p.  R  A.  <f>(F) = 0  .  ([21],  of  ty  365-366).  Remark:  From Theorem 2-B,  we have the  following:  A  I f ty e A  i s positive faithful,  also f a i t h f u l . part  of  such that  ty  F o r i f ty^(E) = 0  , then by Theorem 2-B <j> (F) = 0 2  then the normal p a r t  , hence  which i s a c o n t r a d i c t i o n  The f o l l o w i n g  but  E =f= 0  ty^  ty(F)  = 0  .  theorem g e n e r a l i z e s  Since  ty  is  be the s i n g u l a r  t h e r e i s non-zero p r o j e c t i o n  o f our c h o i c e o f  [13] (p. 50, Theorem 1.18).  , let  ty  F <_ E  is faithful,  F = 0  F  a theorem o f H e w i t t - Y o s i d a i n  Theorem 2-1:  9 be a p o s i t i v e f u n c t i o n a l on  Let  lar,  then g i v e n any p o s i t i v e normal  9(E)  > 0  and  9(F) = 9(E)  , there  exists i n A  projection  F  < E  so  .  y  F  ={E} Ot p  Ot  E E^ < E . a —  a projection  Let  E = } = .0  .  we have  {F }  F  e P  2  y  , and  F < E  i s singu-  E e P  with  9Q( )  such t h a t  F  <  .  9 (E -  Since  such t h a t  i s a non-zero  0 < F  = 9(E) > 0  < E - F.^ and  2  be a c h a i n o f f a m i l i e s such t h a t f o r each  aa  i s a family of pairwise and  9  If  From Theorem 2-B t h e r e  9 (F ) = 0  such t h a t  by Theorem 2-B a g a i n , d>(F„) = 0 2  , 6 > 0  e A  N  .  .  9(E) > 0 ,  Proof:  9  A  <b(E^) = 0 a  for a l l  a  o r t h o g o n a l non-zero p r o j e c t i o n s w i t h 0  , and such t h a t  {F } a a  v  i s linearly  p  o r d e r e d by inclusion. J  Then t h e u n i o n  a g a i n a f a m i l y of p a i r w i s e and  <b(G) = 0  for a l l  of a l l  F  for a l l  a  a is  o r t h o g o n a l non-zero p r o j e c t i o n s w i t h  G e F  maximal f a m i l y o f p a i r w i s e  F  .  By Zorn's Lemma, l e t {E }  orthogonal projections with  EE  E GeF  G <_I  be a <_ E and  a d)(E ) = 0  for a l l  a  .  by Theorem 2-B we have <b(E„)-= 0 0  have a f a m i l y with  EE  {E }  a a  = E  E = E E a  E„ e P 0  .  such t h a t  0 < E „ < E - E E 0 — a  the m a x i m a l i t y o f  o f non-zero p a i r w i s e  and  For i f E - E E a  4= 0  then  and  a  , which c o n t r a d i c t s  y  A  Then  d>(E ) = 0  {E }  .  aa  Thus we  orthogonal projections i n  for a l l  a  .  9(E) > 0 ,  Since  a {E  }' a a  i s not f i n i t e .  Hence t h e s e t is finite,  Since  r  <b 0  r t  {E : <h„(E ) 4= 0} a 0 a '  then l e t  F = E - E  a  i s p o s i t i v e normal, i s countable. - {EE  Q  E 9„(E ) = 90(E) < 0 a 0  a  a  If  : <|>-(E ) 4= 0} 0 a Y  1  {E : <p (E ) 4= 0} a U a n  where  E  a  is Q  24.  not  in  {E  : <J> (E ) 4= 0 } 0 a '  a  ty(F) = <j>(E)  with  enumerate  i t as  (J> (G) = 0  and  It  i s c l e a r that  If  F < E oo  {E  Let  {E.}.  <J>Q(F) = 0 < 6  : <i> (E ) + 0 }  G = (ZE  a  : <b(E 0  a  and  is infinite,  ) = 0}  , then  oo Q  4>Q( ) E  2 ^ g ^ i ^ i=l  =  w  n  i  c  h  i  s  finite.  Hence t h e r e i s  00  an i n t e g e r  N > 1  such t h a t  00  Z <i> ( .) < 6 i=N+l E  n  .  Let  F = G +  1  then  <J>(F) < 6  N > 1  Remark:  If  and  Theorem  singular E e P  E  > 0  a  for a l l a  2 - 1 also gives  f u n c t i o n a l on  A  and  i s a projection  <|>(F) > 0  ty^  4> -< <f>  property then  ty  n  Let and  -< ^ ~ ty  If  ty  i s a positive  a p o s i t i v e normal f u n c t i o n a l on  such t h a t b o t h  <j>(E) > 0  F < E  in  and A  corollary i s a generalization  theorems i n [ 3 ] (p. 1 9 6 0 ,  Corollary 2 - 2 : that  fact:  <J>Q(E) > 0  such t h a t  A ,  <j>(F) = 0  .  0  The f o l l o w i n g main  1  F < E  the f o l l o w i n g  then t h e r e e x i s t s a non-zero p r o j e c t i o n but  E.  and  n  Since  Z i=N+l  ty  and  > hence  ty  2 ) .  ty^  <J)Q normal.  o f one of C a l d e r o n ' s  be two p o s i t i v e f u n c t i o n a l s  Then the normal p a r t i s not s i n g u l a r  ty^  i f ty^ =f= 0  of .  on ty  A  such  has the  I f ty^ ~ ty ,  25.  Proof:  9 = 9^ + <j>  Let  w  n  e  r  is the s i n g u l a r p a r t o f  9  Case 1:  , then  9 (E)  If  = 0  Q  Case 2:  <i>^ i s  e  t  2  9 (E) = 0  since  <J>-< 9  but  2  e  Let E e P  9(E) =  such t h a t  2  9 (E) > 0  9 (F) > 0  we have  9^(E)  = 0  then  9 ^ Q  F  Hence  implies , then  9^ =f=  1  A  0  of  T  defined  A e A  .  I f both  A T  into and  A 9  A  Hence  E e P  we have  9^9^ If  cj> ~ 9 , n  #  on a H i l b e r t space  , we l e t  by  •  which c o n t r a d i c t s  i s not s i n g u l a r . .  0  0  =  2  F  Thus, f o r any  9  <j> (F)  X  Q  be a von Neumann a l g e b r a  on  Hence  9 ( ) = 0  •  <j> ~ ^  i s a c o n t r a c t i o n o p e r a t o r on  ,  From Lemma 1-H, we have  , hence  , so  Q  .  such  9 (F) = 0  E  .  in A  Thus  9Q( ) = 0  9Q(E) = 0  <j> <_ 9 ~ 9  Let  .  2  .  F < E  0 <_ 9-^F) <_ cp^E) = 0  1  9Q =f= 0  If  Since  9(F) = 9 <F) + <j> (F) = 0  the c h o i c e o f  T  .  Q  2  , then by t h e remark a f t e r  Q  Theorem 2-1 we have a non-zero p r o j e c t i o n but  cj>  and  9^(E) = 0  (E) + <f> (E) = 0  9  9  normal p a r t o f  .  A  If 9 (E) > 0  .  n  T  be the a d j o i n t  T 9 (A) = <j>(TA)  a r e p o s i t i v e then  H  for a l l  T 9  . If operator  9 e A  ,  i s positive. If  it  both  T  and  d> a r e normal then  T 9  i s normal.  If  {T }  .  is a  k  f a m i l y o f o p e r a t o r s on under  {T } . a aeA  valent  to say t h a t  * Ta 9 = d> f o r a l l  A  , an element  i f 4 ( 1 A) = 9(A) a T 9(A) = 9(A)  a e A )  .  9 e A  i s s a i d to be i n v a r i a n t  for a l l  A e A, a e A  for a l l  a e A  and  .  I t i s equi-  A e A (i.e.  The f o l l o w i n g theorem isa 0 generalization of 0  one o f C a l d e r o n ' s main theorems i n [-3] (p. 1961,  3).  Theorem 2-3; A  .  on  A  Let  {T } , a aeA  If  part of r  Y  Proof: Since  Let  d>  d>  < 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  be a f a m i l y o f p o s i t i v e normal c o n t r a c t i o n  operators  i s i n v a r i a n t under  Y  , i s i n v a r i a n t under  {T } . a aeA  i s p o s i t i v e , both  ty^  ty. l  T  , the normal  {T } . a aeA  L e t ty = d>^ + ty w i t h normal p a r t ty  then  ty^  and s i n g u l a r p a r t ty^  and ty^ a r e 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 *  T ty. p o s i t i v e f o r a l l a 2  and A  T <J>„ a 2  all a e A  since for  .  T I < I a —  all  A  A  T ty a 1 A  A  T ty. < a l — Y  a e A  {T  s  H  T d>„ = <J>„ a 2 2  : s e S}  of  Then, f o r L  e A  S  < j >e A  all  and  A e A  T M <C M s  for a l l  T <f>_ > d>~ a 2 — 2 T  0  .  Hence  .  A e A  A  1 ty = ty. a i l  #  on a H i l b e r t  C o n s i d e r any a n t i r e p r e s e n t a t i o n  as p o s i t i v e normal c o n t r a c t i o n o p e r a t o r s and  a e A A  , and hence  {T } . a aeA  , the f u n c t i o n  U An element  fora l l  w i l l be denote a von Neumann a l g e b r a  w i l l be a semigroup. S  ty  fora l l a e A  i . e . ty. i s i n v a r i a n t under l  , and  T  and  0  T  space  A  a  a e A , T <j>„(I) = d> (T I) < d> (I) a 2 2 a — 2  A  A  A  a 2  n  a 1  A  But, f o r each  From now on  i s p o s i t i v e normal  n  Tty. +Tty =Tty=ty  But  T ty < ty. f o r a l l a 1 — 1  , we have  a e A  .  , hence  v  So, by Theorem 2-A, J for  a e A  i s positive fora l l a e A  Y  , we have  s  $ (T A)  on  A  is in  m(S)  U S i s s a i d to be s e S s e S  . .  A subset  S-invariant M C A  i f <j>(T A) =ty(A) f o r g  i s s a i d to be  S-stable  i f  P r o p o s i t i o n 2-4:  Let  (p  e A  Q  S  i s l e f t amenable, then (2a)  =>  (3a); (la)  (2a)  A E P  ((lb)  s £ S}  > 0  A e P  ((2b)  be  p o s i t i v e , then ( l a ) <=>  <=>  A £ A  +  A e A  +  (2b),  (3a)  <=>  ) , <p (A) > 0  9 q  (A)  and  implies  0  ) ,  (3b)  > 0  implies  A E P  ((3b)  If  (la) =>  inf  M ( L  (2a)  {^(TgA):  9 Q  (T A)) > 0 G  oo  +  (3a)  (lb).  AeA  ) , i f there  exists  {s  } n  .C n=l  S  with  00  E n=l  Proof: P C  A  .  +  inf M  ti  (  If  If  A  9  u  k E _ i n=l  9 N  (A)  = 0  '  : s e S}  exists  and  .  >c —  {s  oo  , C n n=l }  .  So  If (la) (resp. A  > 0 S  ).  =>  (3a)  , then t h e r e  cE < A  i n f U ( T E) U s  9n  (3b)  > 0  0  A  s e S  ( 2 a ) , and  9 (A)  <p (E) > 0  T  there  =>  with  +  CT A ) ) > c M ( . ( T E)) s — ij u s  00  E _ i n=l  A E A  for a l l  (<f> (T A) U s  , then  ( l a ) , (2b)  such t h a t  ( T E)  9 a  A e A n  ( l b ) =>  c > 0 C  T  are  (T  > 0  ( l a ) =>  A)  E e P >  ,  and  (T (cE)) =  9  then  (resp.  ( l b ) and  oo  such t h a t  clear since  (2a)) h o l d s ,  : s e S} So  9 n  are  E T A e A T s n=l n  (2a) =>  (2b).  then  00  T  E < — E s c n n=l n  T  s  E  as  T A e A s n  k -> »  ).  ( s i n c e i t i s the s t r o n g  Hence, i f (3a) h o l d s ,  ( l a ) =>  (2a)  choice  since  of  inf {  E 9 q  .  So  (3a)  =>  U  S  L  9 o  which  n  (3b).  ( T E ) : S e S} £ M (  of  4> (E) = 0  n  c o n t r a d i c t s our  limit  It i s c l e a r that  (T E))  f o r any  g  E e P  OO  To  show (2a) =>  oo  E n=l So  (3a), l e t  E e P  , {s  n  }  ,C n=i  S  be  E e A  that  oo  •  T  such  .  Then there  exists  n N E T T E < T ( K l ) = KT I < K l s s s s ~~~ n=l n  K > 0  such t h a t  for a l l  s e S  E n=l and  T  E < Kl  .  n any  integer  N  28.  Since  N  T T s s N E  ,  n  E = T s  n  E , <h„(T T s O s s  E n  = t  4>n(T E) 0 s  .  1  i s such t h a t  s n  T  Hence,  for a l l  : d) (T T  E) < Kd) (1)1  n  , where  m(S)  e  l(s)= 1  n for a l l s e S  .  So, f o r any y e LIM , we have  N  N  I y U <J> ( s n=l n T  s  E ) )  0  g  holds,  inf  y  (  *o n=l  Z  ( T  s s T  .  Assume t h a t ;  t o (*), we have  , then even , then  <|>Q ~ <J>  g  •  Then  0  y (<f> (T T A)) = M ( < f - ( A e A  a  .  Theorem with  ty  s  i s positive,  s  (3a).  f  o  ra  1  1  '  N  \(<$> (T E)) Q  H  e  T  So iji i s an  2-3,  e  .  Since  (2a)  #  e  A  > and S  then  0  <(>Q-< ty •  such t h a t implies  KU.(T E ) ) L U s  If,  in  = 0for  T  *  A a s  Define  t S  ty  <!>Q  1  central  S  on A  i s l i n e a r on A  for a l l  iji .  Let a e S  )) = y a 4>o s (T  A))  =  ;J(  a  S-invariant positive  byty(A) = y(d> (T A ) ) n  and |^(A)j  for a l l  0  *o  A  e  A  = | y (<J)Q ( T A ) ) | g  .  Since  , then  0 <_ i n f {ty (T E)  <! Q(E) = 0  .  Hence  ty  ty  y e LIM  , then ty (T A) = ( T  s  A ) )  =  *  (  A  )  f  o  ra  f u n c t i o n a l on A  the normal p a r t ty^ o f ty i s S - i n v a r i a n t .  u )  c  = 0  g  P , <!>Q( ) >  E  n  Then t h e r e e x i s t s an S - i n v a r i a n t  <JU I| ||A||  so i s  0  ijj(E) = 0  we have  )  < j > can be chosen c e n t r a l .  ( T A ) | <.||d>|| ||T A||  0  I  • Moreover, i f  A e A  cf>Q  (  Thus  <b„(E) = 0 (J  for a l l  and  .  < j > on A  L e t y e LIM be f i x e d .  0  K  (*) I f E  Proof:  <U  1 *0  holds.  p o s i t i v e normal f u n c t i o n a l  s e S  )  n  So (2a) =  {<t„(T E ) : s e S} > 0 0 s  e P  )  L e t ty^ be a p o s i t i v e normal f u n c t i o n a l on  amenable.  addiiton  E  f o r a l l u e LIM  $ (E) = 0  Lemma 2-5: left  =  Q  u(4> (T E)) = 0  E  (<f> (T E ) ) =  : s  £  1  1  By  If E e P  S} <_ty(E)= 0  .  From (*)  s .  By C o r o l l a r y  2-2,  the normal  ijj^  part (*)  we  of  ip  have  (j, CE) = 0  has  <p (E) = 0  implies  n  implies  0  the p r o p e r t y  ^(E)  = 0  §Q<  M  ^  •  Now,  (<j> (T E)) = 0  for  E e P  ,  i f i n a d d i t i o n to  for  hence  9  E e P ~ ^  Q  , then .  Thus,  * by  C o r o l l a r y 2-2,  then  \p  9g  " ^1  is central.  <p = ij;^  ,  9  Let  <|>Q e A  Then the f o l l o w i n g are  (2)  E  E  9Q  P  T  s^0  i S  c  e  n  t  r  Theorem 2-A,  A  e  A  f u n c t i o n a l on  ~ 9  A  implies  Moreover, i f  , 9 (A) A  = 0 =>  Q  T <t>_ s 0  i n f {9  1  1  s  e  '  S  is central.  .  Let  #  S  left  amenable.  9 e A  such  (T E)  : s e S}  > 0  and  .  s  > 0  implies  i n f (9 (T A) : s 0  M C9 CT A)) = 0 L  0  S  is central for a l l  Y  a  US 0  9 (A)  r  .  = 0 => \ ( 9 C T E ) ) = 0 +  o  ij>  U  C3)  f  S - i n v a r i a n t p o s i t i v e normal  n  Q  l  be p o s i t i v e normal and  , (f> (E) > 0  9 (E)  a  equivalent:  There e x i s t s an that  I f  Hence, by  i s the r e q u i r e d  Theorem 2-6:  Cl)  '  g  S}  e  > 0  and  . s e S  A>  then  can be  chosen  central.  Proof: If  (3) =>  T 9 s*0  is central for a l l  n  chosen c e n t r a l . then s  e  9CT  S  <pCA) > 0 inf  Now,  A) = 9(A)  , hence .  M  Since  u  for a l l  (T A)) = 0 s  the s e t  (T A)  S  : s e S}  > 0  s .  .  .  (2) =>  then, by  E  S  Let  .  A e A  So  A e A  +  +  with  9  with  9 (T A)  = 0  (3).  //  be  9Q(A)  =  for a l l  n  we  2-5.  can  <f>(A) > 0 u  i s uniformly  , from C o r o l l a r y 1-N'  Hence (1) =>  (1) by Lemma  Lemma 2-5,  Let  {T A : s e S} s  g  U  +  suppose (1) h o l d s . =0  (9  i_i  s e S  {9(T A) : s e S} = <pCA) > 0  i n f {9  P C A  (2) i s c l e a r s i n c e  bounded have  , then and  Corollary 2-7: S  be l e f t  Let  <j>  be a p o s i t i v e normal f u n c t i o n a l on  amenable and  T d)-.-^ d)  S U are  for a l l s e S  .  A  .  Let  Then the f o l l o w i n g  \J  equivalent: (1)  There e x i s t s an  (2)  E e P  (3)  A e A  Moreover,  implies  i n f {d, ( T E ) : s e S} > 0  .  , 4> (A) > 0  implies  i n f {d> (T A) : s e S} > 0  .  Q  *  T i> s 0  if  Y  with  , (J> (E) > 0 0  +  .* d> e A  S - i n v a r i a n t p o s i t i v e normal  g  0  i s central for a l l s e S  n  g  then  d)  can be  chosen  central.  Proof:  Since  <j>(T A ) = 0  for a l l s e S  n  Invoke now  T d>„-< <b„ f o r a l l s E S s 0 U  Theorem 2 - 6 .  Corollary 2 - 8 : and  S  If  is left  A  0  (3)  0 Moreover, central.  if  +  .  <J)Q E A  ,  c l o s u r e of  (T  c l o s u r e of  {T A  E  :  S  s  £ S}  <J> e A  for  P .  E  i s not i n the s t r o n g  j  A e A  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  i s n o t i n the s t r o n g  0  for  n  implies  amenable then the f o l l o w i n g a r e e q u i v a l e n t :  (2)  E  (<j) (T A ) ) = 0  admits a f a i t h f u l p o s i t i v e normal  There e x i s t s an  f  M  d>„(A) = 0 0  #  (1)  0  , thus  , we have  E  A •k  T 6 s 0 r n  A  +  g  : s e S} f o r  .  i s central for a l l s E S  then  A  can be  chosen  Proof: A E A  Since .  +  9  is faithful,  S i n c e {T A : s e S}  9 (A)  =0  A  i f and o n l y i f  A = 0  i s u n i f o r m l y bounded by ||A|| f o r  for •  A e A  ,  +  s Theorem 1-1  and C o l l a r y 1-M we have i n f { 9 (T A ) : s e S} = 0  u only i f  0  i s i n the s t r o n g c l o s u r e of  s  { A  : s e S}  T  g  for  + Since  c|>(A) = 0  s E S  .  Apply now  Lemma 2-9: T  *  S  U  g  ^  \)  o r  a  C o r o l l a r y 2-7  <|>Q  Let  9n -< 9n  {T E  i f and o n l y i f  0  ^  A = 0  for  e  •  s  : s e S} C {E}"  AeA  A e  A  +  * , ^ i>Q<  9  s  to get C o r o l l a r y 2-8.  be a p o s i t i v e normal  s  i f and  f  o  r  a  1  1  Q  #  f u n c t i o n a l on  A  and  _  Suppose t h a t f o r any  E E P ,  then (1) i m p l i e s ( 2 ) : 00  (1)  If  AeA  then (2)  If  Proof:  +  and  9 (A) Q  E e P  Let  = 0  {s } , C n n=l  S  such t h a t  E T A e A 1 s n=l n  .  then  E e P  00  there are  9 (E) > 0 U  with  implies  inf{9 0  d> _ (E) > 6 > 0 0  but  (T E) s  : s e S} > 0  inf{ct> (T E) 0 s A  : s e S} = 0  00  {u } _ C S such t h a t 9„(T E) converges n n=l U u , then t h e r e i s u such t h a t <j>_(T E) < 6. • n, 0 u 1 1 n  then t h e r e are 6, = 8/2 k  k  n  ±  u  ,...,u n  l  n  have been chosen,  let  k  u (k+l)  be such  that  n  *0  ( T  u  E )  n  +  (k+l)  .  Z  1  =  +0 u  T  (T  1  n  u  i  T  n  u  E )  6  k l +  k  n  (k+l)  x. e. j, (T 0 u K  E) +  n  n  (k+l)  Z 9_(T 0 u k  - _ i 1  -  1  n  T  . . . u n  i  n  u  (k+l)  E)  < <S  k+1  to z e r o . .  If  Let  .  ,  3 2 .  T h i s can be done s i n c e  ft  we have  T m (T E) s 0 u n y  converges  T s  9g-<  <I>Q  f  o  r  a  1  s e S  1  converges t o zero f o r a l l s e S  rt  to zero f o r any  s E S  .  Let  u  k }. _ n. ,. x=0 k(i)  6 _ (T  -such t h a t  0  N  k+l k T T  u n  ^k+1 E) < - — • — k + 1  .  So  cj>-(T T E) 0 s u n  be the maximal o f  n  {u  (note: By P r o p o s i t i o n 1-N  (k+l)  f o r a l l n > n. , —  and  n N  k(0)  6  *0  ( T  u  ... u u n. T  n  k  E  <  8 E  e  {E}'  1  n > n  1  and  i = 1 , ...,k)  k  s E S  T E E {E}" s .  . Let  hence  Now  { I , E , T E,T I : s s  fora l l s E S  , we have  {T E : s E S} C { E } ' s  So  i s commutative.  {T I : s E S} C {I}" s B  r3  be the von Neumann a l g e b r a g e n e r a t e d by  forall  { I , E , T E : s e S} s  So  °  f  n  s. = u . Let k n, k s E S} . Since T E-E = E-T E s s  )  {E,TE: s  S E S } C { I } '  { I , E , T E,T I : s E S} s s  i s a commutative von Neumann a l g e b r a .  , thus and  i s commutative,  We a l s o have  T 8 C B  s for a l l  s E S  since a l l T  a r e p o s i t i v e normal s  1-A and Theorem  tation of  S  8  i s the u l t r a -  r  weak c l o s u r e o f the a l g e b r a g e n e r a t e d by  Theorem  and  1-J).  So  {T  { I , E , T E,T I : s £ S} s s  : s E S}  g  (see  i s a l s o an a n t i r e p r e s e n -  as p o s i t i v e normal c o n t r a c t i o n o p e r a t o r s on  8  .  By  it Theorem  1-P,  measure  space  f u n c t i o n a l on  8  i s isometrically (X,S,p)  .  L (X,S,p) oo  So  -isomorphic =  ^0^8  of be  Y  T  g  .  , a g a i n denoted by  Let  ) be the element i n  the measure  Y e  L (Y,S,p) oo  which c o r r e s p o n d s t o  L (X,S,p)  of a  oo  induces a p o s i t i v e  p o s i t i v e normal c o n t r a c t i o n o p e r a t o r on a g a i n denoted by  to  ip  A  L (X,S,p) oo  and  Xy  , and  T  g  induces a  fora l l s E S  (the c h a r a c t e r i s t i c  which c o r r e s p o n d s to I/J g i v e n by [ 8 ] Theorem a  normal  E  , function  , and IV-8-16.  v  33.  Then we have  T|) ( f ) = / f d v  = ^ ( X ) = / X dv Q  Y  and  y  ^(TgE) = / T  h = (x " Y  and  let  B e 8  for a l l  oo  n  ^  ^  dv  g X y  for a l l  ... X )  T s  s  n=l i = l  i  be the c o r r e s p o n d i n g element o f  Q  .  Q  , so  ^(E) = ^(E)  s e S  .  Let  e L>,S,p)  +  Y  n  d> (B) = ip (B) = 4> (h) = / hdv  Then  f e L^CX.S.p)  h  in  8 O  A  We want to show t h a t  <f>(B) > 0 Q  00  and  sufficient  oo  {a  that there e x i s t s  } , C n n=l  to show t h a t  S  f o r which  [ hdv > 0  E T B e A T a n=l n  i+k E T  and  J  J=l j >_ i >^ 1  and f o r a l l  k >_ 0  .  „ h < 1 s .. • . s . — J !  S i n c e , then, we  can l e t  h e L (X,S,p)  , hence  , we w i l l  get  E  I t xs  for a l l  i = 1  and  oo  00  k -> °°  •  T S « . . S  n=l  °°  1  n  E  T .  1  n=l  B e 8 C A. S  • • - o,  n  I  Since  oo  /  (  x  - h)dv < /  n  E n=l  E T i=l n g  >  s  X y  dv  i  N \ ! n=l  ±  , . > n=l i = l  <  E n=l  co  and  d> (E) = / Q  X y  dv  > S  , so  . \ s ...s X d v i=l n i  <  T  Y  n  E  E  6  (  T  s  = 6  / hdv > 0  .  ...s. > i E  n  .  Let  k = 0  , then  i+k E T  g  ...s/h  34.  i+k = T  h<T Y <T 1 < 1 s. — s. Y— s — i l l  f o r any  A  i  .  Assume t h a t  T h <_ 1 s ...s J --  E i  i+k +l  =  J  i x  1  Q  f o r any  i >_ 1  i > 1  any  and  .  k = k»  .  0  We need to show  Since  i+k +l  i  +  k  Q  S  T  j=i  s ...s. = j 1  0  +  ( T  s. s....s 1 J 1+1 T  h )  1 + 1  i l k +  = T  (h +  i+l k +  it  h(x)  i s enough to show  =  0}  and  N  2  h +  E j=i+l  h +  0  T  T  0}  h = -'- i+l S  j  ,  S  i + l  .  Let  N  -{ x  e  X  1+1  j  S  .  i  s  j "  h < 1  T  +  1  N  On  +  Q  E  h) S  0  = {x e X : h ( x ) >  i+l+k  +  E j=i+l  i  s  for  '  1  }. J=l  h  T, „ h <_ 1 s .. . . s. 3 i  E . . j=i  k  ±  0  E j=i+l  T S  j  < 1 n by  assumption.  On  N_  , we have  h > 0  h - x  , so  2  E n=l  _ Y  1  z  s  £=1  and  i + 1 + k  h +  0  E j=i+l  T S  j ' *  1  +  1  h < h + ~ n  < h +  n  E l  =  E *  = 1  oo  n  E  E  T n * T  n=l 1=1 n" l S  1 Xy £  1  S  h A X Y  T  Y  ...s n  X  Y  I  35.  So we have  -f  Be  A  , {a  }  . C S n n=l  which c o n t r a d i c t s to ( 1 ) .  Remark:  CO  So (1) =>  The i d e a o f the p r o o f  commutative von Neumann a l g e b r a L  (X,S,p)  and  in  E n=l  f o r some  T  such that  b  0 0  but  ff>„(B) > 0 0  //  o f Lemma 2-9 came from the f a c t i s isometrically  measure space  h e L (X,S,p) s • • • i n i  (2).  E T B e A , a n=l n  (X,S,p)  .  that a  -isomorphic to  The p r o o f  f o l l o w s Sachdeva's p r o o f  of  / hdv > 0  o f Theorem 3-3  [18] which i n t u r n i s i n s p i r e d by the i d e a i n the p r o o f  o f Theorem 2  in Granirer [9].  The  f o l l o w i n g theorem i s a g e n e r a l i z a t i o n o f H a j i a n - I t o ' 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  T V _ < rf>_ f o r a l l  s 0  0  <|>Q e A s e S  be p o s i t i v e normal, .  I f f o r any  E  e  S  left  amenable and  , {T E : s e S} C {E}" s  P  then t h e f o l l o w i n g a r e e q u i v a l e n t : (1)  There e x i s t s an that  (2a)  S - i n v a r i a n t p o s i t i v e normal  A e P  S  S} > 0  ((2b) A e A  ) then  +  < j > (A) > 0 => i n f {$ (T A) :  u If  A e P  ((3b) A e A  (4a)  If  A e V  ((4b) A e A  CO  I  U S  .  (3a)  E  such  <j>Q ~ < j >  If E  < f >e A  T A e A s n  then  +  ) then  <|>(A) > 0 => M (ty (T A)) > 0 U JJ U S ) and t h e r e e x i s t s {s } _ C S with n n=l  <f>(A) = 0 u n  .  n  36.  Moreover, i f T <b_ s0  i s central f o r a l l  K  s e S  <f> can be chosen  then  central.  Proof:  By Lemma 2-9, C o r o l l a r y 2-7 and P r o p o s i t i o n 2-4.  Remark:  I n case  S  amenable semigroup o f p o s i t i v e l i n e a r  a left  L (X,S,p)  and  1  A = L (X,S,p)  {T  g e L (X,S,p)}  [12]  of  : s e S  with  (consider  1  operator  g  s  f o r a p r o b a b i l i t y measure space  I f ( g ) = f(sg)  f o r f e L^X.S.p)  L (X,5,p) = L ( X , S , p ) ro  and  1  T  ) , our Theorem 2-10 reduces to H a j i a n - I t o ' s  and U. Sachdeva's theorem i n [18] (Theorem 3-3).  [ 7 ] , p. 31, Theorem 1, t h a t a von Neumann a l g e b r a  of  the Banach space  Since Neumann a l g e b r a  A^  , called  (see Theorem 1-Q, Theorem  A  Let A  transformation  on a f i n i t e von  {T  g  : s e S}  S  on a von Neumann a l g e b r a  A  be a f i n i t e von Neumann a l g e b r a w i t h  on  a p o s i t i v e c o n t r a c t i o n operator  is  central.  T  ^  .  We s t i l l l e t  an a n t i r e p r e s e n t a t i o n o f  function  and  function  1-R, Theorem 1-S and P r o p o s i t i o n 1-T), we w i l l  c e n t r a l valued A  results i n  i s the d u a l space  the c a n o n i c a l c e n t r a l v a l u e d  p o s i t i v e normal c o n t r a c t i o n o p e r a t o r s  Lemma 2-11:  the a d j o i n t  g  (Note: D i x m i e r proved  d i s c u s s our r e s u l t s above f o r f i n i t e von Neumann a l g e b r a s . be a semigroup and  ,  )  there i s a p a r t i c u l a r A  (X,5,p) ,  c o n t r a c t i o n o p e r a t o r s on  in  S  #  L e t <J>Q  as  the c a n o n i c a l  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  Then we have <j> ((TA)^) = cj> (T(A^))  on  A  such t h a t  fora l l  A e A  T ty^ .  37.  Proof:  Since both  <p  : e n t r a l f u n c t i o n a l on  and  n  A  T  .  By P r o p o s i t i o n  -- 9 ( T A ) = T * ( A ) = T % ( A ^ ) = 0  9 Q  Theorem 2-12: c e n t r a l valued Let is  S  be  Q  -  Let  A  be  (T(A^))  , and  we  for a l l  s e S  let  <j> e A  is a positive  n  have  ^((TA)^)  A e A  .  #  and  Z  is  canonical  be p o s i t i v e normal c e n t r a l ,  A  * T cb„ -< d>_ s U 0  l e f t amenable w i t h  central for a l l  9 ( )  1-T,  T cp  a f i n i t e von'Neumann a l g e b r a w i t h the  H  function  are p o s i t i v e , so  s e S  for a l l  S-stable,  .  Suppose  then the f o l l o w i n g  T  *  m s (J A  are  equivalent: (la)  There e x i s t s an such that  (lb)  There  (2a)  If  cj> ~  S - i n v a r i a n t p o s i t i v e normal  A z V C\ Z (  (A)  then  ((2b)  > 0  9 (A)  Z  e  , (2c)  +  inf {  A e Z  ((3b)  > 0  Q  A  implies  A e V C\ 1  If  <j> e A  9Q  e x i s t s an  then (3a)  S - i n v a r i a n t p o s i t i v e normal c e n t r a l  implies  9 o  (T A) : s g  , (3c)  +  A e P  A c ?  \p z I  with  , (2d)  A  S}  E  > 0  A e V HI  If  ((4b)  A  )  +  . A e A  , (3d)  )  +  M^(<j> ( T A ) ) > 0 n  g  4-  (4a)  A  e  CO  Z  £  ) , and  there  exists  {s  } n  . <_ S n=l  CO  with  Proof:  E  0  So  =  (2b)  9 ( ^) A  0  <=>  (4b).  //  0  and  then  9-(A) u  n  and g  (3b)  we  have (2c)  Lemma 2-11  . 9 CT A) =  (2d)  commutative, by <=>  AeA S  From P r o p o s i t i o n 2-4  From P r o p o s i t i o n 1-T  9 (A)  T  I  9 ( )  <=>  Theorem 2-10  = 0  <=>  .  (2d)  and  \<re have ( l a ) <=>  C C r A ) ^ ) = 9 (T A^) G  (3d). ( l b ) <=>  Q  If  Z  (2a)  g  is <=>  (3c)  S-stable, <=>  (3d).  ( l b ) , and  for a l l  (2b)  <=>  s  e  S , A z A  then, s i n c e  (3a)  <=>  (3b)  . Z  <=>  is (4a)  38.  Remark:  If  {T  : s e S}  g  i s any a n t i r e p r e s e n t a t i o n  of  S  as c e n t r a l  * A  o p e r a t o r s on s e S  then  Tg^g  I  §Q e A  c e n t r a l f o r any c e n t r a l  s  and  .  Corollary left  ,  A*  2-13:  A  If  amenable w i t h  {T  is a  : s E S}  g  normal c e n t r a l c o n t r a c t i o n following  are  (la)  a-finite  f i n i t e von Neumann a l g e b r a ,  an a n t i r e p r e s e n t a t i o n A  o p e r a t o r s on  and i f  Z  S  of  is  S  is  as p o s i t i v e  S - s t a b l e then the  equivalent:  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  central  * <j> e A (lb)  There e x i s t s an  (2b)  0 + A  V CM  E  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 ((2b)  0 =(= A E A  (2d)  +  A : s E S} . s A e P H Z ((3b)  0  =}= A  e Z  , (2c) 0 =f=  +  ) then  0  A E Z  ) , i f there e x i s t s  A  e P  ty  e 2  ,  i s n o t i n the s t r o n g c l o s u r e  of  {T  (3a)  +  CO  with  Proof:  Since  central  ty  inf  e A  E n=l A  a-finite  Hence  {<j) (T A ) : s e S} = 0 (J s {T A : s e S } for A E A s  Corollary  A = 0  .  f i n i t e , t h e r e i s a p o s i t i v e normal  i>g(A) = 0 <=> A = 0 i f and o n l y i f  n  bounded by ||A||  then  n  is .  A E A  T  {s } C S n n=l  +  (since  0  the s e t  for  from Theorem 2-12.  {T A : s E S} s  //  .  Hence  i s i n the s t r o n g c l o s u r e  , and a p p l y Theorem 1-1 and C o r o l l a r y  2-13 f o l l o w s  A E A  faithful  1-M) .  of  i s uniformly Thus our  39.  Proposition  2-14:  Let  A  be a von Neumann a l g e b r a ,  semigroup w i t h an a n t i r e p r e s e n t a t i o n operators < t >  an  on  A  for  all  ty ~  ty  Proof:  for  Clearly  ty  i s an  s E S  E E P  we have let  a  and  are  ty  .  ~ <j>  i s left  a  Let  0  CO  - G  Let  A = G - ? E, n . . . k k=l  converges to zero. 1-N' we have by  , then  Since =  <  C o r o l l a r y 1-N' a g a i n ,  Thus  iJ>(A ) n  ty^  ty  f ^ ^ )  n  A  .  S  e  U  a  .  ^(F) = 0  ty(A  n  ty (T E) = 0 n  u s  : s  g  S} >  £  So  .  Since  Now  both  {E : ty(E ) > 0}  : ty(.E ) > 0}  a. Let  , CO  G =  Z E i n=l  n  i . e . ty(Y. E ) = ^(G)  strongly.  uniformly  converges to zero  Since  such  ty (A ) O n  So  n  and ty ~ ty^ , by C o r o l l a r y  converges to zero  ^o^s^rP  E e P  If So  and  A  S-invariant  since  for E E P  ty(E)  CO  converges to zero  n  with  p o s i t i v e normal.  Z ty (E ) = Z d> (E ) . . U a ^ 0 n a n=l hence  A  0  a  n  A  i n f {cj) (T E)  : A (E ) > 0} U {E  is  converges to z e r o .  s  ~ ty  i s central.  m(S)  : ty (E ) > 0}  {E  < j > (F) = 0  , then  'K^g^)  , and  = u(<j>g(T A))  orthogonal p r o j e c t i o n s .  a  and  T  F = Z E  ty  S-invariant  s  {E } °°_ = {E  Z ty(E ) = Z ty(E ) a T n a n=l "  and  ty(A)  implies  i n f { ( J ) ( T E ) : s e S} <_  n n=l then  A  such t h a t  then  fora l l  AQ(E) > 0  A l s o we have  a r e p o s i t i v e , so b o t h  countable.  s E S  i n v a r i a n t mean on  be a f a m i l y o f p a i r w i s e  n  A  d e f i n e d by  <fc(T E) = <fc(E) = 0 s  , since  Q  of p o s i t i v e contraction  S - i n v a r i a n t p o s i t i v e f u n c t i o n a l on  , s i n c e ty ~ ty^ and ty i s  ty  {E )  y  then  T  all  for  ty  i s central fora l l  *.(£) = 0 u  amenable  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l on  ty  If  i s p o s i t i v e and  that  ty  , the f u n c t i o n  i s an  a left  be a p o s i t i v e normal f u n c t i o n a l on  T  .  : s E S}  g  p o s i t i v e normal f u n c t i o n a l on  y E LIM  A e A  ty^  Let  S-invariant  Then f o r each  <J>Q  .  (T  S  ) = ty(G) -  on  s  uniformly i  ty(E  Z T  k=l  . on )  Hence, s  we have  KG)  =  E if>(E ) , n n=l  • Hence  i>(Z E ) = i|>(G) = E i|>(E ) = Z f ( E ) a .. n a a n=l a  \p i s p o s i t i v e and completely a d d i t i v e , hence, by Theorem 1-B,  Thus  i s p o s i t i v e normal.  * I f T cp s u  i s central for a l l s e S  <j> (T (AB)) = T* (AB) = T*cp (BA) = 0  .  s  A, B e A  9()  .  ip(AB) = i^(BA)  So  ( T (BA))  0  (p (T (AB)) = 0  g  f o r a l l A, B  ^  Q  ( T  Q  g  ( B A ) )  in A  .  for a l l s e S  as elements i n m(S) i . e . \p i s c e n t r a l .  //  , then and . Hence  CHAPTER 3  SEMIGROUPS OF NORMAL  *-HOMOMORPHISMS  The main purpose o f t h e f i r s t that the connection  chapter  found by H a j i a n - K a k u t a n i  i n [11]  o f i n v a r i a n t measures,  s t i l l holds  first  t r u e f o r von Neumann a l g e -  Weakly-wandering s e t s w i l l be r e p l a c e d by weakly-wandering  p r o j e c t i o n s i n t h i s case.  The r e s u l t s o f H a j i a n - K a k u t a n i  g e n e r a l i z e d by L. Sucheston  [19] to any n o n - s i n g u l a r  [18])  to l e f t  amenable semigroups o f n o n s i n g u l a r  [11] were  t r a n s f o r m a t i o n and  then i n t u r n by E. G r a n i r e r i n [9] ( H a j i a n - I t o i n [12] in  i s to show  between the n o n e x i s t e n c e o f weakly-wandering s e t s  of p o s i t i v e measure and the e x i s t e n c e  bras.  part of this  and U. Sachdeva  transformations  (contractions).  In the second p a r t o f t h i s s u f f i c i e n t f o r the e x i s t e n c e  chapter  o f a complete s e t o f  normal f u n c t i o n a l s on a von Neumann a l g e b r a  In t h i s  chapter  we. w i l l  a n t i r e p r e s e n t a t i o n of a semigroup von  Neumann a l g e b r a .  The  we g i v e c o n d i t i o n s w h i c h a r e  A  assume t h a t S  S-invariant positive  as normal  {T : s e S} s  i s an  *-homomorphisms on a  The r e s u l t s i n Chapter 2 remain t r u e i n t h i s  main r e s u l t s o f t h e chapter  case.  a r e Theorem 3-4, Theorem 3-14,  C o r o l l a r y 3-7,. Theorem 3-15 and C o r o l l a r y 3-.17.  42.  Let  A  be a von Neumann algebra on a H i l b e r t space  H  . The  following Proposition 3-1 i s w e l l known and we bring i t s proof f o r the sake of completeness.  Proposition 3-1: T(P) C ?  Proof: Hence  Let T  , and  A e A  +  be a  ||TA|| <. ||A||  +  +  .  f o r some  If E e P  T ( E ) = T(E) = T(E*) = T(E) *  .  2  ||A|| <_ 1  , then  A*A <_ I  || TA ] <J|A|| f o r a l l A e A  -homomorphism on  for a l l A e A  <=> A = B*B  T(A ) C A  "k  .  So .  then So  .  A  If  .  TI = I  E  2  = E = E*  T(P) C V  .  , hence  Suppose  I f TI = I  T(U) C  U  Let A  TI = I  T(Z)d Z  Proof:  .  Then  *y • • Let  ^ » T = T o tj i f  7  . Hence  Z  i s T-stable.  For A e A  convex h u l l of  T a n c  be a  .  Hence  U*U = UU* = I  . So  -homomophism on  j only i f Z  we have  {UAU ^ : U e U}  A e Z  Suppose  o  T(Z) C Z .  and  i s T-stable  ).  I f tj o T = T ° kf , then, l e t t i n g  = (TA)^ e Z  .  be a f i n i t e von Neumann algebra with the  canonical c e n t r a l valued function  (i.e.  T(U) C U .  #  Proposition 3-2:  with  AeA  U e U then  and  ,  T(E.)T(E) =  T(A)*T(A) <_ TI <_ I => J| TA|| <_ 1  T(U) T(U) = T(U U) = T(I) = I = T<UU ) = T(U)T(U)  A  then  B e A => T(A) = T(B)*T(B) >_ 0  hence  .  4 - 4 -  T(A ) C_ A  Then  Since  o  , where A^  T(A) = T(A^)  i s T-stable, then  o  T(K ) C ^ .  Z  , we have  i s the  i s an uniform l i m i t of  h elements i n K  (see Theorem 1-R), so  T(A )  i s an uniform l i m i t of  ,  fl  elements i n for  K  AeA  .  K^H  T o tj =  Remark:  = 1  <j>(TA)I =  ([5],  Z tq  0  T  If  , then by  t r a c e on  ty  i s a m u l t i p l e of  T  A  H  But we  Z  , where  (TA)^ have  A  i s the unique f i n i t e  AeA  A  is a  of  A^ =  = T(<j)(A)I) = A (A) I  K  i s the  T(A^)  =  only  (TA)^  i.e.  d>  by  is  T-stable, A  t r a c e on  ty(A)I  for a l l  for a l l  that  AeA  AeA  and  hence  such  i s i n v a r i a n t under  .  ty  Hence  .  Hence  T  .  If  ty  is  1-0  Proposition  a p o s i t i v e number, hence  t r a c e s on a f i n i t e  A  -homomorphism on  , then by P r o p o s i t i o n 1-F  ty  Hence a l l f i n i t e  .  T  Z  C o r o l l a r y 1-T',  for a l l  finite  .  ty  ( T A ) ^ = T(A^)  any  K  K  //  , then the c e n t e r  .  T(Z) C  2 5 1 Theorem 1 2 ) ,  TI = I  ty(TA) = 4>(A)  T  .  p.  i s a f i n i t e f a c t o r and  o tj = tj o T  4(1)  f\  e K  V  A  If  such that T  T(A )  , i s the u n i f o r m c l o s u r e of  element i n Hence  Hence  i s i n v a r i a n t under  f a c t o r are i n v a r i a n t under  any  * -homomorphism which l e a v e s  I  fixed.  * Remark 3 - 3 : A  Proposition 3-1  i s a p o s i t i v e contraction operator  f u n c t i o n a l on =  implies  <j, (T(AB))  for and  0  and  A  =  T  is a  t h a t any on  A  -homomorphism .  If  -homomorphism on  < j ) ( T A T B ) = ty^ ( T B T A )  =  0  ^(TCBA))  i.e.  ty^  on  is a central  , then  A  T  T* (AB) Q  AQ(AB)  T  TJ (BA)  =  Q  a l l A,B in A . Hence T <j>^ i s c e n t r a l . Let S be a semigroup { T : s e S} be an a n t i r e p r e s e n t a t i o n of S as normal -homomors  phism on replace  A  .  Then a l l r e s u l t s i n Chapter 2 remain t r u e and we  the c o n d i t i o n  that  "T  ty  n  is central for a l l  s e S"  r  s e S" by  the  condition  "T  is s  s 0  central for a l l  or  can  "ty^  is central".  44.  From now on we l e t antirepresentation algebra  A  of  S  S  be a semigroup w i t h  as normal  an  H  ).  A projection  i f t h e r e e x i s t s a sequence  E  ^  s n  in  A  ^ - i ^ ~ -  ^  n  CO  •  such t h a t  : s e S}  g  -homomorphisms on the von Neumann  ( a c t i n g on the H i l b e r t space  i s s a i d to be weakly-wandering  {T  {T  ®' -±  g  a  n  family  of pairwise  orthogonal projections,  n We w i l l  f i n d a r e l a t i o n between the n o n e x i s t e n c e o f weakly-wandering  projections  and  S-invariant  The f o l l o w i n g  Proposition  Let and  3-A:  f e m(S) 6 > 0  Let  S  be r e a l and  Let  S be amenable and (1) <=>  i s Proposition  there i s some  Theorem 3-4:  p o s i t i v e normal f u n c t i o n a l s  on  A  1 of [9].  be a r i g h t amenable semigroup and  u(f) = 0 a e S  .  Then f o r any  such t h a t  a^,...,a  n f(cr) + E i=l  f(aa.)  f o r any  E e P  e S < 6  1  (p^ be a p o s i t i v e normal f u n c t i o n a l on E'T E = T E-E s s  u e RIM.  , s e S  A  . Let  .  Then  (2) and (1') <=> ( 2 ' ) :  * (1)  There e x i s t s an 9 -< Q  (2)  9  S-invariant  p o s i t i v e normal  < J >e A  such  that  •  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 e x i s t s an 9Q - 9  (2*) Moreover,  If  p o s i t i v e normal  <p e A  such  that  •  E E P  i f (J>  S-invariant  then  9 ( E ) = 0 => M(<J> ( T E ) ) = 0 Q  i s c e n t r a l then  g  < j >  , and (2) h o l d s .  can be chosen c e n t r a l .  45.  Proof:  Suppose ( 1 ) h o l d s .  and  {s } . C n n=l  let CO  Then  E T E _ s n=l n  <_ d> ( E  S  OT  T  I  s  A  E  be such t h a t  <f>(T E) = <|>(E) = 0  for a l l  = 0  .  that  ( 2 ) holds.  ty  Since  is  Then  ty  E}°° n-l  are pairwise  orthogonal, „ N N<j)(E) = <> j ( E T E) * s n=l n  S-invariant,  N > 0 => <j>(E) = 0 => <f> (E) = 0 u Let  E e P  s e S => <Ji (T E) = 0 0  with  <|>Q(E) = 0  for a l l  s e S =>  ( 1 ) and ( 2 ) , ( 1 ' ) i m p l i e s  p £ IM  i s an  ( 2 ' ) . Now  ij,(A) = u(d> (T A ) ) (J s  and d e f i n e  We prove now that  then M(<})Q(T E)) g  suppose  for a l l  n  A  S - i n v a r i a n t p o s i t i v e f u n c t i o n a l on  c e n t r a l i f ty^ i s c e n t r a l .  .  N  ( 1 ' ) holds.  ( 1 ' ) implies Let  be a weakly-wandering p r o j e c t i o n ,  s n  is  f o r a l l integers  Suppose  .  {T  n  Hence ( 1 ) = > ( 2 ) .  A e A  <> j  and, s i n c e  E) < co  E e P  Let  *Q-< ^  , and •  Suppose  E P such that ty(E) = 0 b u t <f> (E) > <5 > 0 . L e t <$ = 6 / 2 . Since the f u n c t i o n d> (T E) i s r e a l and u(<j> (T E)) = ty(E) = 0 , then, by 0 s U s E  Q  n  P r o p o s i t i o n 3 - A , there P , = E-T 1  E s^  k  rt  T  and  s^ e S  is  F, = { P . } 1  such t h a t  , then  1  means the sum o f a l l elements i n  E - E©  F^  ( ^ ( P ^ <. * Q ( T  <_ y (<j) ( T E ) )  = 0  sition  S  g  3-A,  * 0  Let  P  =  e S  C  <j) (T (E  and  T  s  (E - E ©  such  ( 2  0  E  -  E  ®  F  E) < 6  s  E S  ) < 6^  < E —  1  . Let  , where  of pairwise .  1  - I@  g  T  T  (from now on we l e t  the sum o f a l l elements o f a f a m i l y We a l s o have  <J>Q(  E €B F  to mean  orthogonal p r o j e c t i o n s ) . g  i s real,  there  ±  i s , by Propo-  that  F  l  )  }  +  •0  F-^'T, (E - E $  C  T  s s 2  Fj)  C  E  1  and  -  i  0 <_ y (<J>Q ( T (E - E (D V )))  Since  F^)  E <B F  i  F  e  2  F  l  }  = ¥  ±  )  <  6  2  U{P > 2  Let  46.  P  =  2 1  (E -  Then, by  F ^ ' T ^ E  ( i ) , we  - E  F )  is  ^0  real,  ( T  s  Let P  P  32  "  Let  = F  ( E 3  ( E  P  2  1  i  3-A we  "  ( T  ®  21  } )  *0  +  21  "  Z  ®  F  3 ' s s )  T  3  {  p  » 32' 32i P  3  P  *0  If  s ,s„,...,s ± z n 1  "  ( E  ( P  T  3  h  )  "  ( E 2  ®  E  3  F  ^  e  n  '  Z  }  ®  a  E  21  )  <  6  U  fP >  = F ^ (  2 1  ^ - ^ 1  b  F  0  n  1  21»  )  P  (1  d  F  +  w  = 0  *0  +  (  and F  21  ^3^'  y  2  32  =  F  2  3  3 ^  { P  h  a  v  *0 321 ( P  K  )  S  ^  n  t  ^'  e  a b o v e  a n c  S  " ®  hl»  Z  E  l  = F , L/ { P,} 21  32  }  3  ^'V  = 21 U F  = F  6  . Let  3  3 2 1  <  3 2  U  ^  3  ( ±  F , a family n(.n— I; . . . Z l l N  -' >  a  that  S  (where  F  s  ( 2  F^))  9n(T (E - E ©  e  a l l o f them l e s s  ^i» 2i' 321  s s  F ^ ) and F  e  have been chosen w i t h  and such  T  pairwise orthogonal projections, F  2  s^ E S  - E©  W  +  ( P  a g a i n have  (E-E©F  ^ *  }  s^  2  F  (E - E ©  s s 3  = (E - E © F . J ' T  3 2 1  U  F  - F  i P ^ ^ ) )  2 1  s  by P r o p o s i t i o n  Z  +0  +  F >))  0 <_ y(<|. (T (E - E © 0  F ^  have  2  W  Since  and  2  3  )  of  than o r e q u a l to  E  '  k-1 W  +  .f  1  •o k(k-l)...i ( P  )  <  k  6  ^  47.  where  k=  k(k-1)  P  1,2,...n , F £^  with  9 n  (T  <i>_(T (E - I © 0 s  (E - E e n+1  _  1  }  >  <  ^  = F _D (k  n ( n  _  ( k  _ ) . . . 21  0 <. y (<j> ( T (E - E © F 0  1 ) - f  _  2 1  ))  { P  2  , then we can choose g  i s real,  l N  F  +  k  F , n(n-l;...21  T  s  (  £ = l,2,...k  P r o p o s i t i o n 3-A, s i n c e and  k  +  E , (T £=1 Q  n  P  1  e  ) ) )i  ^  (E - Z © £  F  ( _ ) n  k ' k ( k - l ) , ' " ''  1  2  s n  1  +  > Y  s  D  ^  "  ^  =  such t h a t  _ n+1 n  s  n  < n+l 6  Let  F , = F , n+1  E©  a  n  d  „, U {P ^ }  1 X  n(n-l)...21 ^  F  F  (  (n+l)n  E  "  (n+l)n...£~  where  E  F  ®  F  F, , ~, (n+l;n... z l N  them a r e l e s s finite  than  4.^  +  4  •0  9  ,)  U C w  p  . / ,,x } n+1, ( n + l ) n  •  p  n  n + 1  s ...s/ n  P  (i ]_) n +  ( P  E  "  E  ®  L  -  »  w  to  E  .  F = ( , /i I N oo k ( k - l ) ... £  }  +  i  (E -  s n  et  F  (n+l)n...(£+1)>  }  have  e  Cn+l)n...i  P  s n  )  < n+l 6  I n t h i s way we g e t a c o u n t a b l e i n -  family p  n l  i s a family of pairwise orthogonal p r o j e c t i o n s , a l l of or equal  )  and  n+1  n+l' (n+l)n" " (n+l)n...£  Then, by  )  ,N  n ( n - l ; . . . 21  (n+l)n  ,^  { P  F,  (E - I ©  P, .... = (E - Z © F ,-, ) "T  n ( n - l ) . . .0 21 1  U  i  )  +  T  .  C P  n  Let  n(n-l)...21  £ = l,2,...,n  s  (  (n+l)n... ( £ + ! ) ) ' s  ^0 n+l  and  •  n+1  1  F, ,, > . = F ,  and  n+1  (n+l)n...£ =  P  -.x 9 n ) ' T n(n-l)...21  F ,  P , = (E - E © n+1  0  i = 1,2,...,k  and  k = 1,2,...}  ^  of  p a i r w i s e o r t h o g o n a l p r o j e c t i o n s , a l l o f them are l e s s CO  , and * (E<S F J  E  E  0  than o r equal t o  CO  [E  K. X  V  P  ( k - l ) . . .£>  < J;  ]  k  X/ — X  «  = «  k  •  Also,  K. X oo  from our c o n s t r u c t i o n o f F  , we have a sequence n  co t  E  h  ®  3  t  P  F  k(k-1)...£  k(k-1)  (£+!)'*  \  where  F  =  _  n )  k = 1,2,... in  A  .  ...  T  £  ±  t  h  .  Let j > i G = T  F  1  k ( u  _  1 )  ...  t t + 1 )  ' > • • •» ^ ~ ) 2  k  ^  k  { P  a  S k S k  n  E  i > >  8 A  d  F  '  k' k(k-l)''"'' k(k-1)..' P  P  ,  then  G <_ E  and <j> (£© F J < 6 ,4, 0  CO  {T G} . s . ..s.s, n=l n 21  and T  S . . . . S  j  2  i >_ 1  is a projection  (G) = <j> (E) 0  are p a i r w i s e  , then,  G = T .••T T . . . T G , S^ S . S . .. S, 1 1 1 l+l k  S  4  = T 1  So we need o n l y to show t h a t  ortho-  since  G-T G = T ...T (G'T ...T G) S ««»S_SSS, S*,-> s, 2 1 j 2 1 1 1 l+l j  S.*«*S_S_.  such  n  - ® Ck-i)...2i>  be two i n t e g e r s w i t h  ..•T G S . 1 1  )-T »  1  G = E - E Q F^  We c l a i m t h a t  S ,  1  =  4, (E) > 6  Since  T  F  " ( -i)...21  . Now l e t  S . - - - S . S . 1 2 1  £  n  n  ^k-D.-.z^'X^  E  <{>„(E© F ) > 0 ^0 °° gonal.  W  " ®  ( E  k ( k  (E - E ©  =  {s } C S =l _ . (E "  (G-T 2 1 j  G-T G =0 s , • ••s . _ J 1+1  G) i+l  Case 1:  If j = i + 1  , then  OiG'T S  *  Q  ha-i)...2i>  a  n  d  0  l  G  '  s  T  G i  +  1  -  G  i+1 -  E  G <_ T G^T (E i+1 i+l S  "  E  ®  F  i(i-D.-.2i  •  So  0 < G-T G i+1 S  < (E - Effi F . ^ ^ . ^ - T ^ C E  P..  F  <  1+1 —  0<G-T  But G-T S  _  1 ( 1  1 ) > > > 2 1  )  o  which i s o r t h o g o n a l to  E©  , so  i+1  G = 0 i+1 .  Case 2:  F  °  G<_G = E - E © F S  - Ee  .  If j > i + 2  , then  0 <_ G'T G s. • •• s.3 i+l <_ T  G i+l  3  <T  —  a  n  d  0  1  G  -  T  S . . . . S . . T  3  s . . s . ^ ^ 3 i+l G  0 < G'T  l+l  G  J  0  "  E  E 8 F  3(3-D...(i+2)  *  }  S  °  i+l  < (E - I S F . ^ j ,  P  (  G 3  -  ^  (E - E © F.j ( j - 1 ) . . . ( in+iO +2)  -l)...(i U +  1  ... E  ®  ( 1 + 2 )  F  -  )'V... . ( B  •  + 1  E  *®  F ](j  -l)...Ci42)'  5 0 .  0 < G-T  But  —  G < G = E - E © F  S .... S  1+1 G-T G = 0 S . • . • S , _. j i+l J  So  f o r any i n t e g e r s  pairwise  If  .  orthogonal.  (2') h o l d s ,  Hence with  G 2 1  T i , j >_ 1  Thus we" o b t a i n  , i.e.  {T  then  (2) h o l d s and i f  9 (E) =0  for  So  9  2-2 and Theorem 2-3,  .  So we have  (2') =>  the normal p a r t  p o s i t i v e normal f u n c t i o n a l on the case o f (2') h o l d s ) .  A  such t h a t  Moreover, i f  c e n t r a l , hence by Theorem 2-A, (1) and (2') =>  Remark:  (a)  by the p r o o f (b) S  o f Theorem  Let  A  f u n c t i o n a l on  A  ~ ip  .  ^  we  have  Now  apply  9  A  with  <f> A  ^  ^  {T .  ; s e S}  g  If  9  d>(E)  i s an E  for a l l  9  (resp.  Hence  9  s e S  .  S-invariant  tp^ ~  i s c e n t r a l then Let  ty^  9 =  3-4  ,  ^ is  then  i s i n part inspired  s e S  = 9(I)  H  an a n t i r e p r e s e n t a t i o n o f  . S  Let as  S - i n v a r i a n t p o s i t i v e normal .  9  on a H i l b e r t space  Then .  Since  , we have  <p(E,) = 9 (I)  9  E  and  i s the s m a l l e s t  ip  T E  > E  s 9 — s E S  1 E ' E , = E •T E, = E , f o r a l l 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 t h a t T E -E .  i s an  2 of G r a n i r e r [ 9 ] .  with support  = d)(E ) = 9(1)  9Q-^  o f Theorem  be a von Neumann a l g e b r a  A  projection i n  all  9Q~<  E z P  ij> o f  i s central.  The i d e a o f the p r o o f  -homomorphisms on  s e S  G e P  (1'). #  be a semigroup w i t h  d>(T E J s 9  are in j.  U  Corollary  (2) =>  G 2 1  S....S-S-  j  a weakly-wandering p r o j e c t i o n  U S  for  CO  to ( 2 ) .  tJj(E) = 0  T  G} s • • . s _ sn 2 1  That c o n t r a d i c t s  , hence  oo  i s o r t h o g o n a l to  .  M((p (T E ) ) = 0  E <£ F  °  i  i 4 j  <PQ(G) > 0  with  which i s o r t h o g o n a l to oo  —  . . .  S  for a l l  J  9 .  So, at l e a s t ,  = E "T E 9  9  9  S  for 9  51.  (c)  Let  be  as  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l on  A  such t h a t If  there  then  E  A,S  T E  and  = I  G  {T  , then  < E  .  9 — -9  I  ;  If  for a l l  g e n e r a t e d by  s  E > E  —  S  E  9  then  Thus  a l l projections  <J>  E e P  , hence  on  s e S is  N  9  9  Let  Let  A  9 ( E ) = 9(I)  for a l l  .  .  G  9 ( T E ) = 9 ( E ) = 9(1)  9Q(E)  i n (b).  9 ( E ) = 9 ( T E ) = <j>(I)  i s a p o s i t i v e normal f u n c t i o n a l  Q  Also  : s e S}  g  be  an  and  s  E >  .  such t h a t  hence  , hence  <f>Q-< 9  »  <p (E) = <p (I)  .  U  9 (T on  N  n  U  E ) = 9 (I) =  n  S-invariant  S  E  the  subalgebra  G > E  — 9 Remark 3-5: replace case,  In Theorem 3-4,  M  by  and  (1) <^>  (2)  <=>  S  if  T <j)_ 9s 0 ^ 0 Y  needs to be  (1')  <=>  (2').  for a l l  Y  only  If  left  9  s  E  S  then we  amenable.  In  i s c e n t r a l then  can  this 9  can  be  chosen c e n t r a l .  Proof:  I t i s c l e a r that  (1')  =>  (1).  The  proof  f o r CD  =>  C2)  is  the  * same as i n Theorem 3-4.  we  have t h a t  coincide. by  9 CE) = 0  Therefore  Lemma 3-6:  and  Let  s E S  f o l l o w i n g are  9^ and  we  T 4. 7 s 0  implies  Q  C o r o l l a r y 2-7  for a l l  Since  L  for a l l  M^^O^E)) = 0  need o n l y  to show C2)  the f o l l o w i n g lemma.  e A  be  =>  p o s i t i v e normal.  E'T E = T E - E s s  , i f  E e P  f o r any  .  Hence (2*)  (1').  and  T h i s w i l l be  (2) clear  #  E  Assume t h a t E  P  , s  E  S  T (j) _< g  .  n  Then  9^ the  equivalent:  If  C.2)  There i s no weakly-wandering p r o j e c t i o n  E  P  then  9 CE) > 0  (1)  E  SeS  0  Q  implies  inf' U n C ^ E E  E  P  : s with  E  S}  > 0  .  9QCE) > 0  Proof:  If E £ P  i s a weakly-wandering p r o j e c t i o n , then t h e r e  CO  sequence  {s } . C~ S n n=l  CO  So  CO  such t h a t  {T E} _ s n=l n  00  Z n=l  4>Q(  t  E S  ) = ^  £ n=l  n  to zero, hence holds.  Let E  E) < co n  T  s  i n f {<j) (T E) : s n  u  e  P  , which i m p l i e s  S} = 0  e  s  be such t h a t  are pairwise  orthogonal,  4>Q(T E) n  S} = 0  e  .  Let 6  th ere i s {u } _, C n n=l there i s u  k  S  = 6/2  and (1) => ( 2 ) . Suppose (2)  A (E) > 6 > 0  .  b u t i n f {<j>(T E) : n  such that 1  u s  Since  such t h a t  Y  converges  s  U s  i sa  i n f{^(l^E)  (b„(T E) 0 u n  <{>„(T E) < 6 -. U u 1 n^ T  : s  S} = 0  e  converges to z e r o . •  F o l l o w the p r o o f  ,  Hence f o r Theorem  3-4 s t e p by step w i t h the replacement o f P r o p o s i t i o n 3-A by the p r o p e r t y &  T A>rx-< s 0  <t>^ f ° a l l 0 r  Y  Y  the same p r o p e r t y  s e S  .  as t h e sequence  A l s o we g e t the same f a m i l y which we g o t i n the p r o o f by  u n  here,  CO  We g e t a sequence ^  F^  s n  ^n-l  ^  of pairwise  n  {u } , .. which has n^ k = l t  *  i e  P  r o  ° f °^ Theorem 3-4.  orthogonal  projections  o f Theorem 3-4 ( r e p l a c e t h e s ^  k = 1,2,...  ).  Let G = E -  F  i n Theorem 3-4  , we have, as i n  k  the proof  o f Theorem 3-4, t h a t  A  4> (G) > 0  with  6 (E) > 0  U  Q  implies  G  i s a weakly-wandering p r o j e c t i o n i n  which c o n t r a d i c t s t o ( 2 ) . i n f { 4 ( T E) : s  U S  n  e  S} > 0  Thus, f o r E £ P i . e . (2) => ( 1 ) .  , #  Remark:  (a) Lemma 3-6 i s a g e n e r a l i z a t i o n o f Theorem 6 i n [2j (p. 305).  The  i n [2] i s i n c o r r e c t .  proof  of s e c t i o n 1 ) . [9].  For a c o r r e c t proof  Remark 3-5 g e n e r a l i z e s  The i d e a o f o u r p r o o f s  the remark o f G r a n i r e r section 1 of [9].  s e e J9J ( l a s t  theorem  the "remark" a f t e r Theorem 2 i n  f o r Remark 3-5 and Lemma 3-6 a r e i n s p i r e d by  ([9]) a f t e r Theorem 2, and the l a s t  theorem o f  (b)  In the p r o o f  o f (2) => (1) i n Theorem 3-4 ( r e s p . i n Lemma 3-6)  we proved t h a t i f there i s a p r o j e c t i o n (p (E) > 6 > 0  but where  (A) = u(<f> (T A)) n  mean on  m(S)  G  9 ( )  (E) > 6 > 0  for A  g  in A  in A but  and  such that  <KE) = 0  i n f (cj> (T E) n  u  : s e S} = 0  g  .  i f we r e p l a c e  G  such  I t w i l l be much e a s i e r to prove t h e e x i s t e n c e  o f such  the c o n d i t i o n t h a t  E e P  " E - I E = I E'E  s " by a s t r o n g  c o n d i t i o n that  E e P  '.  ( i n s p i r e d by the p r o o f  The p r o o f  f o r any  ,  s  ' { T E : s e S} C {E}"  s e S  ),  i s a fixed invariant  , then there i s a weakly-wandering p r o j e c t i o n  <PQ(G) > 0  that a  (resp.  0  E  f o r any  g  o f Theorem 2 i n [9]) i s as  follows: k  Let  6  k  = 6/2  .  <P (T E)  Since  P r o p o s i t i o n 3-A t h e r e  0  is s  y(9 (T E)) = 0  i s r e a l and  S  e S  such t h a t  0  9 ( n  E) < &  T  ^  u n  ^ -]_ ^ ^ n  to zero, hence there i s s. = u 1 n^ s.. , s„,. . . ,s, (s. = u , s. = u 1 2 k 1 n, 2 n„ 1 2  ( i n case o f  1  CO  Lemma 3-6, there i s a sequence  , by  g  such t h a t  9Q^  such t h a t  u  E) converges n ). I f  6 (T E) < 6. 0 s^ 1 s, = u i n case o f Lemma 3-6). k n, k n  *  have been chosen, then by P r o p o s i t i o n 3-A (by T 9Q-< f° and P r o p o s i t i o n 1-N i n case o f Lemma 3-6) there i s ( 4 - ^ ) ^ ra  ±  s e S  ±  s  s  e  k  case o f Lemma 3-6,  s  fc  cp (T U  (k+1) co  F =  V  9 (F) n  U  V [(T  1 x=l., <_ E n=l  k+  N  ( k +  E) + E <p (T i=l (k+l) k''  j i  oo  ,.. = u n  ( D S  S  E ) - E ] , then  E  n  that  E) < 6 - S  S • • •s. n x  n  ) such  i)  ^  . Let  i F  i s a p r o j e c t i o n and  co  c p (T n  1=1  because o f the f a c t  n that  E) < E 6 l n=l (T  =5  .  The i n e q u a l i t y i s t r u e  E ; i = 1,2,...n and • • •s* n l commutative (the commutativity o f t h i s f a m i l y comes from S  n = 1,2,...} a r e { T E ; s e S} G ( E } " ) g  54.  Let  G = E - F  The f a c t  YQ( )  that  s....s s 2  i s a weakly-wandering p r o j e c t i o n w i t h i s clear since  j > i > 1  (G-T ^ ...s  1  G  > 0  G  C o n s i d e r any T  , then  S  <_ T  j  E <_ F  (  i  +  1  , we have G ) 1  )  and  F  4>Q(E) > <5  (  J Q( ) ,  G  <j> (F) < <5  and  •  Q  G-T G = s....s„s s . • • •s_s i 2 1 j 2 1 Since T G = T (E-F) j-" (i+l) j-" (i+l)  .  T  S  S  S  i s o r t h o g o n a l to  G  , we  S  have  j'" (i+l)  S  S  G = 0 (i+l)  G'T J f o r any i.e.  .  T  G x  j > i > 1  G  Thus  , hence  {T  S  n  o r t h o g o n a l to  Let  ^  e A  E-T E = T E-E  s  G}  • • • S „S  oo  2 1  and  T A  s  //  There e x i s t s an Y  (2a)  Let  be  E E P  f o r any  left ,  s e S  AeA  such  If  A  P  e  ((2b)  A e P  If  s  T  A  )  A e P  If  A e A  ) then  +  ((3b)  >  )  T Q  (A)  > 0  implies  . d, (A) > 0  A e A  +  )  A e A  +  ) and t h e r e e x i s t s  then  implies  0  ((4b)  {s }°° . C S n n=l  CO  that  (5)  E T A e A n=l n g  then  A (A) Q  = 0  .  Tliere i s no weakly-wandering p r o j e c t i o n 4> (E) > 0 0  Moreover,  that  ~ YQ  W (4a)  S  u  S - i n v a r i a n t p o s i t i v e normal  i n f {i>AT A) : s e S} > 0 0 s (3a)  -< A  s u  2 1  are p a i r w i s e o r t h o g o n a l  XI—X  be p o s i t i v e normal.  Then the f o l l o w i n g are e q u i v a l e n t : (1)  G J  * amenable w i t h  T  2 1  i s a weakly-wandering p r o j e c t i o n .  C o r o l l a r y 3-7;  if  A^  >  E z P  such t h a t  . i s c e n t r a l then  A  can be chosen  central.  such  U  55  Proof: in  A  Suppose (4a) h o l d s .  Let  {s  oo  } -, C n n=l  S  and  let  E  be  be  such  that  a weakly-wandering {T s  CO  gonal,  Now  then  £ T E n=l n g  our C o r o l l a r y  3-7  P r o p o s i t i o n 2-4.  Theorem 3-8: and  E-T  s  A  e  .  oo  E} , n=l  n  are p a i r w i s e o r t h o -  (4a), 9 ( E ) = 0  So, by  .  Q  f o l l o w s from Lemma 3-6,  projection  Hence  Remark 3-3,  (4a) =>  Corollary  (5)  2-7  an  //  Let  z A  <p  n  E = T E-E s  be p o s i t i v e normal.  f o r any  E e P  , s e S  .  Let  S  be  left  amenable  Then the f o l l o w i n g are  equivalent:  * (1)  There e x i s t s an 9  (2a)  ~ <j)  0  If  Proof: a  e S  ^  u e LIM  .  A e A then  with  9 (E) = 0  Now,  for a l l  u e LIM  Q  y(£ 9 (T E)) = y(9 (T E)) = 0 0  A  9 (T E) 0  O  S  n  so  S  9 ( A ) = 0 <=>  ) then  +  <j)  PC  A  0  for  all  a  z  S  .  So  chosen .  +  Suppose (2a) h o l d s . 0  G  = 0  n  (  L  9()  .  g  T 9 -< g  E)) =  n  (T (T^E)) ) = 0 9  0  for  Q  all  s  Let  for a l l  , u (cf> (T ( I E ) ) ) = u(cp (T M  = 0  g  central.  y(9 (T E))  , then  *  =  with  ^ ( ^ Q (T A))  Q  can be  (2a) i s c l e a r s i n c e  E z P  and  ((2b)  i s central  (2b) =>  <J> e A  .  A e P  Moreover, i f  S - i n v a r i a n t p o s i t i v e normal  Hence 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  00  such  that  {T  E} S  Tl * J .  E  s i n c e (2a) h o l d s ,  (2a) =>  (1).  E e P  9Q( )  with  Suppose (1) h o l d s .  E  =  u  •  9Q(E) > 0  .  S  n=l  weakly-wandering p r o j e c t i o n have  E z A  T -  -  By P r o p o s i t i o n 2-4,  ^  00  are p a i r w i s e o r t h o g o n a l , hence  n  we  ^ n ' ^ i ^  n  Hence t h e r e i s no .  From Remark  From Theorem 2-6  we  have  3-5,  4> (A) > 0  u  implies  n  u  s  and  u s  for A e A  M (<}> (T A ) ) = 0 J_I  i n f (<j> (T A) : s e S} > 0 .  +  d>„(A) = 0 <=> M (<J>„(T A ) ) = 0 U L 0 s  * (A) = 0  u  n  Apply P r o p o s i t i o n 2-4 a g a i n , we have  for A e A  .  +  Hence (1) =>  by Remark 3-3 and Theorem 2-6, i f Q i s c e n t r a l then  dp  T  central.  #  Remark:  I n case  and  S  for  A = L (X,S,p)  to Theorem 3 o f E. G r a n i r e r  c  a  (2b).  and  [9].  x e X  X  X  and  Moreover,  °e chosen  n  o f some f i n i t e measure space  i s a semigroup of measurable maps on  a l l s e S , f e L (X,S,p)  implies  (X,5,p)  ,  T f (x) = f (sx)  , our Theorem 3-8 reduces  The above p r o o f f o l l o w s  the idea i n  [9].  Theorem 3-9:  Let  A  be a f i n i t e von Neumann a l g e b r a  normal c e n t r a l f u n c t i o n a l on Then  (1) <=> (1)  (2a) <=>  (2a)  T  Q  T  S  $Q  be amenable and  (2*a) <=>  Z  a positive S-stable.  (2'b):  S - i n v a r i a n t p o s i t i v e normal c e n t r a l  | e A  <j)  with  There e x i s t s an such t h a t  (2'a)  Let  There i s no weakly-wandering p r o j e c t i o n E e P )  (1 )  .  (2b) and ( I ' ) <=>  There e x i s t s an with  A  and  T Q  (E)> 0  E e PA  Z ((2b)  .  S - i n v a r i a n t p o s i t i v e normal c e n t r a l  <j) e A  <j>Q ~ <j>  If  E e P n Z  and  (2a) ( r e s p .  ((2'b)  E  E  P  (2b)) h o l d .  ) then  d, (E) = 0 => M ^ O ^ E ) ) = 0  57.  Proof: A  Assume (1) h o l d s .  Let  E  be a weakly-wandering p r o j e c t i o n i n  {s } , C n n=l  S  such t h a t  .  So  CO  then t h e r e e x i s t s  .  {T  CO  orthogonal,  hence  £ T E e A _ -1 s n=l n  N = l i m E <j>(T E) = l i m N<j>(E) => N-*» n=l n N-*» s  (2b) =>  (2a).  I f ( l ) holds,  9 (T E) = 0 0  for a l l  g  u £ IM => M(<j> (T E)) = 0 u s Suppose  (2a) ( r e s p .  .  hence  00  E <p(T E) = s n=l n .  Q  9 (E) => <p(T E) = 9(E)  (1') => Since  So (1) =>  for a l l  S n  (2'a)) h o l d s .  are pairwise  9 (E) = 0  s e S => u (<j> ( T E ) )  Also  0  . n=l  00  vj  s e S =>  E}  « > (p( Z T E ) = - s n=l n  9(E) = 0  then  f  0  s n  (1). Z  = 0  g  for a l l  So ( l ) =>  (2'b) =>  1  i s commutative and  (2'a).  S-stable,  * t h e r e e x i s t s an  (resp. then  <pQ — ^ 9  S - i n v a r i a n t p o s i t i v e normal  ) on  Z  for a l l  A E A  9(E)  <=>  0  (resp. \p  _ 9  and  .  if)(E^)  on  n  Z  ).  9  =  VQCE^) =  0 =>  Hence  s E S  S  case.  9(A) = ty(A^) f o r A E A  cp -< 9 n  <=>  9 (E) Q  a E S  E E P  , then  =  since  $ -<  Q  <p  Let  Let  9 (E^) = 0 <=> (resp.  .  = \p (T (A^)) = ip(A^) =  S-invariant. 0  A  n  _ 9  0  9 (E) = 0 Q  ).  4>  Q  Thus  , ,  9(A)  on  Z  since (2a) =>  (1)  ( 1 ' ) . //  In Theorem  C l ) <=>  is  i>(E^) = 0 <=>  Remark 3-10: , then  ^(T^A) = ^ ( ( T ^ A ) ^ )  Hence  9(E) = 0 <=>  (2'a) =>  1  Let  9Q-N <|>  such t h a t  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  from Lemma 2-11 we have  =  by Theorem 3-4.  i> £ Z  (2'a) <=>  3-9, i f we assume t h a t  T <!>-•-< d>„ s 0 0 y  need o n l y be l e f t amenable and (1) <=> (2'b).  M  r  for a l l  (2a) <=>  would have to be r e p l a c e d by  M^  (2b) <=>  i n this  58.  Proof;  Replace Theorem 3-4 by Remark 3-5 i n the above p r o o f , and u s i n g  the f a c t  (2a) <=>  Corollary S  (2'a) i n Remark 3-5.  3-11:  Let  A  be l e f t  amenable and  (1)  There e x i s t s an  be a Z  //  a-finite  f i n i t e von Neumann a l g e b r a .  S-stable.  Let  Then the f o l l o w i n g a r e e q u i v a l e n t :  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  central  AeA (2a)  There i s no non-zero weakly-wandering  A  in  Proof: ful  Since  central  A  ^  Proposition  ((2b) c e n t r a l ) p r o j e c t i o n  is  e A  a-finite .  3-12:  Let  finite,  there i s a p o s i t i v e normal  The C o r o l l a r y f o l l o w s from Remark 3-10.  A  A  .  Let  T I = I  "  and  Z  be  S-stable,  then  ?/  be a f i n i t e von Neumann a l g e b r a and  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  faith-  <J>Q  for a l l  a s e S  s  the f o l l o w i n g a r e e q u i v a l e n t : oo  co  4-  (1)  If  A e A  A (A) 0  and t h e r e i s  = 0  {s } , C n n=l  If 0  Proof: be  (1) =>  and t h e r e i s  {s } , C S n n=l  with  .  (2) i s c l e a r s i n c e  the c a n o n i c a l c e n t r a l v a l u e d  Z  +  C  £  . n=l  T  A e A  then  T A e Z s n  then  s n  CO  CO  A e Z  4> (A) = 0  with  .  +  (2)  S  A  +  .  f u n c t i o n on  E  . n=l  Suppose (2) h o l d s . A  .  Let  A e A  CO  suppose t h e r e i s  {s }°°_ C S n I T — J.  with.  E T A e A s n=l n n  Then, from  +  Let and  59  P r o p o s i t i o n 3-2  = ( E  T  I  A)  S  and Theorem 1-S, we have  e Z  .  By  (2) we have  E T (A^) = E (T A / n=l n n=l n  A (A) = A (A^) = 0 u u  .  7  Hence  n ( 1 ) . //  (2) =>  Let  A  be a von Neumann a l g e b r a  on a H i l b e r t space  . 1  H  ft subset there  J C A is  i s s a i d to be complete i f f o r any non-zero  A e J  such t h a t  A (A) 4 0  .  Let  S  A e A  be a semigroup  , and  ft {T  : s e S}  g  on  an a n t i r e p r e s e n t a t i o n o f  Then  J  there  is  -homomorphisms  3-13:  A e J  Since  +  0 4 E e P that  be a s e t o f p o s i t i v e f u n c t i o n a l s on  P (2. A  A  0 4 E e P  we have  cE <_ A  and hence  Let  E e A  S and  .  Since  such t h a t  E 4 0  A  A(E) > 0 c > 0  , there  be amenable and such t h a t .  necessary.  o f Lemma 1-G we have  is  .  S-invariant  J  Let  and  A e J  So  Now .  such  i s complete  E'T E = T E'E s s  Then the f o l l o w i n g a r e  There i s a complete s e t o f f u n c t i o n a l s on  A e J  A(A) >_A(cE) = cA(E) > 0  s e S  E e A  .  , the c o n d i t i o n i s o b v i o u s l y  +  such t h a t  Theorem 3-14: projection  <j>(E) > 0  , then from the p r o o f  A(E) > 0  (1)  J  such t h a t  suppose f o r any A e A  Let  i s complete i f and o n l y i f f o r any non-zero p r o j e c t i o n  Proof:  T  as normal  A  Proposition  0  S  for-any  equivalent:  p o s i t i v e normal  60.  (2)  There i s no non-zero weakly-wandering  (3a)  0 4 A e P  If  Proof:  (resp.  strong  c l o s u r e of  {T A  (3b). =>  (3a) s i n c e  P C A  be a weakly-wandering 3  0 4 A E A  (3b)  A  projection i n ),  +  0  i s not i n the  : s E S}  g  .  +  Suppose  (3a) h o l d s .  CO  {s } C S n n=l  p r o j e c t i o n and  such t h a t  n  E E P  Let {T  E}  s n  co  n=l  oo  are pairwise  orthogonal p r o j e c t i o n s .  00  || E  , and  00  I  T  Ex || =  E  2  s n  to zero by  E T E E A n=l n  Then  f o r any  (3a),  || T Ex || s n  _ n=l  x s H  E = 0  .  for a l l x e H  (3a) =>  then t h e r e  a r e p o s i t i v e number  cE <_ A  Since  Let  .  E 4 0  <j> (A) = ( A x | x ) 0  Q  n  ,  A  c l e a r that  T  E  (2). c  there  Suppose  H  is  x  E H  A  ,  Then  9^  F s A  normal f u n c t i o n a l  9  on  9Q(A)  .  Hence t h e r e  >_ ctp(E) > 0  such t h a t  p o s i t i v e normal f u n c t i o n a l s on A E A  +  and  {s } C_ S a a  A  = x  A  4 0  n  i s the i n n e r  ^Q^)  there  9Q9  i s an •  be a n e t such t h a t  It i s F  and  9(E) > 0  , and  S-invariant  Suppose T  •  u  S-invariant positive  Thus  (2) =>(1).  >  ^QO?) > 0  i s a complete s e t of and  Ex  i s a p o s i t i v e normal  A  (1) h o l d s .  converges to  s  a Then f o r any p o s i t i v e normal f u n c t i o n a l  zero  strongly.  have  I(J(T A) converges t o z e r o . In p a r t i c u l a r ( T a f o r any S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l  zero  such t h a t  ( | )  such t h a t  So, by Theorem 3-4,  Hence  0 4 A E A  Let  such t h a t  where  i s weakly-wandering.  Let  E E P  and non-zero  .  i s no p r o j e c t i o n  A  EX|| converges  strongly.  (2) h o l d s .  (see [ 4 ] , p. 54, Theorem 1) w i t h  there  )|T s  converges to zero  for a l l A E A  p r o d u c t o f the H i l b e r t space f u n c t i o n a l on  Thus  n  i.e.  So  .  9  f  on  A  we  A) converges to a 9 on A But  61.  <j>(T  A) = <j>(A)  fora 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 (1) =>  (3b).  Remark: A  s  A  any  S  x e H  cE < A  .  .  .  By (1) we have  A = 0  .  Hence  A 4 0  For i f 0 4 A e A T  So  s  A  be such  +  converges t o zero  that  s t r o n g l y , then  a  (T Ax|x) s  converges t o zero  for  a  , there a r e  c > 0  and  0 4 E E P  such t h a t  Hence, f o r each  _< (T Ax|x) s a x E H  Since  .  such t h a t  converges to zero weakly. "  a  .  o f (3a) and (3b) i s t r u e f o r any von Neumann  and any semigroup  t h e r e i s a n e t {s } C S a T  A  //  The e q u i v a l e n c e  algebra  < j > on  a, c|JT Ex [j = c ( T E X | T E X ) = c ( T Ex|x) s s s s a a a a x E H . Thus || T Ex jj converges t o zero f o r any s • a  fora l l  Hence  T  E  converges to zero s t r o n g l y b u t  E 4 0  a  Theorem 3-15:  Let  A  be a von Neumann a l g e b r a and  Then (*) o r (**) i m p l i e s (*) (**)  Cl)  Z  is  E"T E G  = T E-E G  in (3a)  f o r any  A  o f ( 1 ) , ( 2 ) , (3a) and ( 3 b ) .  E £ P  There i s a complete s e t o f  A  be amenable.  S-stable.  f u n c t i o n a l s on (2)  the e q u i v a l e n c e  S  i s finite,  , s E S  .  S - i n v a r i a n t p o s i t i v e normal c e n t r a l  A and t h e r e i s no non-zero weakly-wandering p r o j e c t i o n  A is finite,  and i f 0 4 A E P  i s n o t i n the s t r o n g  closure of  ((3b) 0 4 A £ A ( T A : s E S} g  +  ) then  0  Proof: proof (3b)  The  proof  (3a) =>  o f Theorem 3-14 =>  (3a) by  P C  (2) and  (see A  (3b)  Suppose (2) h o l d s .  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 (note  t h a t , by P r o p o s i t i o n 1-C  of f i n i t e  normal t r a c e s on  f u n c t i o n a l s on F e A  A  and  A  By  the same as i n  Q  c p  Let  on  n  Theorem 1-B,  •  Q  So  cp(A) > 0  .  we  can  9  •  So  <J>(A) > 0  .  I f (*)  I f (**)  .  holds,  holds,  Thus we  T  be  A  e A  ±  a  A  Let  be  -homomorphism such t h a t  = T ( B A ) = T(A 1  T-stable.  Let  a von on  T(A )  = A  1  A .  A  cj> on  .  (p on  A  Hence (2) =>  >  U  set  which i s onto.  For  A e A  B e Z  Hence, i f  a semigroup.  If  {T  that  there i s  such  that  S-invariant #  i t s center.  then  Hence  there  such  (1).  Z  .  A  then from Theorem 3-4  and  1  be  the  then from Theorem 3-9  Neumann a l g e b r a  B) = T ( A ) T ( B ) = A*T(B) S  A  identify  have a complete s e t o f  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 s on  Remark 3-16:  9Q( )  there  i s no weakly-wandering p r o j e c t i o n  an S - i n v a r i a n t 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 9Q-<  then  +  such t h a t  S - i n v a r i a n t 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  9 ..< 9  Neumann a l g e b r a ) .  0 4 A e A  A  the  w i t h the s e t of p o s i t i v e normal c e n t r a l  ( 2 ) , there  9 (F) > 0  such t h a t  is a  ).  are  the d e f i n i t i o n of a f i n i t e von  .  +  (1) =>  Let  , there i s  T(B)-A = T(B)T(A^)  T(Z) C  : s e S}  Z  i.e.  i s an  Z  is  antirepresen-  * t a t i o n of  S  as normal  S-stable.  Hence the  -homomorphisms on  condition " Z  hypotheses i n b o t h Chapter 2 and  onto  A  (note  to  I  so  the c o n d i t i o n "  ,  that a  A  onto  S-stable"  S  as normal  -homomorphism on T I = I  A  for a l l  , then  can be  this chapter (chapter  an a n t i r e p r e s e n t a t i o n of a semigroup A  is  A  onto s e S  3}  Z  o m i t t e d from i f we  must c a r r y  " can be  the  consider  -homomorphisms A  is  on I  omitted).  C o r o l l a r y 3-17: space {T  g  H  If  , and  S  : s e S}  A  is a finite  an amenable semigroup w i t h an  of  S  as normal  Then the f o l l o w i n g a r e (1)  (2a)  (3a)  S-invariant  If  0 4 A e P  (resp.  (3b)  closure of  By Theorem 3-15,  central projection) 0 + A e A  (a)  I f we  and r e p r e s e n t a t i o n and  S  LIM  : s e S}  Remark 3-16,  S  S  M^  Theorem 3-15,  be l e f t  Then f o r any  a ,a„,...,a  on  as o p e r a t o r s and  RIM  (note:  amenability A  on  by r i g h t A  of  . S  If  and  S and  amenability  r e s p e c t i v e l y (also  Hence Theorem 3-4,  S  Theorem  remain t r u e i f we  3-9,  consider  as the c o r r e s p o n d i n g o p e r a t o r s  an analogue o f p r o p o s i t i o n 3-A  y e LIM in  of  1-T,  r e s p e c t i v e l y ) then a l l our r e s u l t s  and C o r o l l a r y 3-17  as a r e p r e s e n t a t i o n  amenable and  i s not i n  //  as o p e r a t o r s  of  0  A  Theorem 3-9, P r o p o s i t i o n  r e p l a c e b o t h the l e f t  by  i n these theorems  ) then  +  (2b)  {T A : s e S} s  Chapter 2 and Chapter 3 remain t r u e .  A  traces  in  Theorem 3-14,  on  normal  no non-zero weakly-wandering  replace  g  finite  (resp.  the a n t i r e p r e s e n t a t i o n o f  {T  A  There i s no non-zero weakly-wandering p r o j e c t i o n  Remark 3-18:  in  onto  equivalent:  and the p a r a g r a p h a f t e r P r o p o s i t i o n 1-C.  S  A  A  the s t r o n g  of  on a H i l b e r t  antirepresentation  -homomorphisms on  There i s a complete s e t o f on  Proof:  von Neumann a l g e b r a  f e m(S) 6 > 0  is:  i s r e a l and  , there  is  a e S  Let  y(f) = 0 such  .  that  f(o") +  n E. f ( a . a ) < <S i=l  ).  Lemma 3-6 and P r o p o s i t i o n 3-12 a l s o  1  remain t r u e i f we r e p l a c e the " a n t i r e p r e s e n t a t i o n o f  S  tation of  of  S  ", s i n c e we do n o t need the a m e n a b i l i t y  " by  "represen-  S  i n Lemma 3-6  A  i s normal,  and P r o p o s i t i o n 3-12.  ft (b)  Since  -automorphism on a von Neumann a l g e b r a  all  r e s u l t s i n Chapter 2 and Chapter 3 remain t r u e  is  S - s t a b l e " and "  (the conditions "  Z  T I = I f o r a l l s e S " can be omitted) i f we l e t s be an amenable group w i t h an a n t i r e p r e s e n t a t i o n ( o r r e p r e s e n t a t i o n )  S  *  {T  : s e S}  g  of  Let lim  sup i n case  lim  S  be a l e f t  ps  p  -automorphisms on  be a l e f t  j| £  (X,5,p)  A  amenable semigroup.  f(st)  {1,2,...}  S =  sup f ( s ) = i n f s  Let  L  as  sup f ( s ) = i n f sup s t  S  lim  S  S  .  Define,  f e m(S)  T h i s d e f i n i t i o n reduces to the u s u a l  with  addition.  If  f  i s positive  then  measure  . Let  f||  be a measure space w i t h  finite  amenable semigroup o f measurable maps on  fora l l  forreal  s e S  , where  (X,S,p)  i s a von Neumann a l g e b r a  (X,5,p)  .  ps ^"(E) = p (s ^E)  X  p  such t h a t  for E e 5  (commutative) on the H i l b e r t  .  Then  space  CO  L  L (X,S,p) oo  then  T  l(x)  = 1  of  S  s  by  For  f  E  L (X,S,p) , s e S co  T f(x) = f(sx)  . i s a normal fora l l  as normal  ps -< _1  ft  x e x  ft  (since  -homomorphism on .  Thus  {T  , we d e f i n e  g  A  p with  so  T  g  : L^CXS.p) ->  T f e L (X,5,p) g  T 1 = 1 s  oo  ),  , where  : s e S} i s an a n t i r e p r e s e n t a t i o n  -homomorphisms on the von Neumann a l g e b r a  L (X,S,p)  65.  with  T 1 = 1 s  YQC^)  d e f i n e d by L^CXjSjp) on  A  fora l l  .  s e S  / fdp  =  The f u n c t i o n a l  o f an  a f i n i t e i n v a r i a n t measure  S-invariant  <j> ~  such t h a t  (X,S,p)  map on  with  X  X  on  L (X,S,p) 0 0  <J>Q  p o s i t i v e normal f u n c t i o n a l  i s equivalent  to the e x i s t e n c e o f  u ~ p  I t has been shown by H a j i a n - K a k u t a n i measure space  4> 0  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  The e x i s t e n c e  L^CXjS^)  .  and  S = {s  ps  p  .  [11] t h a t : G i v e n t h e f i n i t e  : n _> 1}  11  , where  The c o n d i t i o n  s  i s a measurable  "p(E)-> 0 => l i m sup(s E ) > 0 n  n->oo  for  E e S "  measure  i s not s u f f i c i e n t  u ~ p  .  f o r the e x i s t e n c e  of a f i n i t e i n v a r i a n t  Hence i n the von Neumann a l g e b r a  L (X,S>p)  w i t h the  antirepresentation { T : s e S where T f(x) = f(sx) fora l l n * of S = {s : n >_ 1} as normal -homomorphisms on L (X,S,p)  x e X}  g  T  1 = 1 n s l i m sup  fora l l  r  n  >  w  \)  S  o f an  (T x) S  n > 1 —  , the c o n d i t i o n "  i°  u  r  E e S "  Y  duCXrJ > 0 0 E  implies  A  i s not s u f f i c i e n t  with  f o r the e x i s t e n c e  IJ  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> ~ Q  (The p r o j e c t i o n s  T  functions  Y  a-finite A "(*)  in  hi  in  ^(XjSjp)  L (X,5,p)  a r e j u s t the c h a r a c t e r i s t i c  for E e S  co  ).  (a commutative von Neumann a l g e b r a  Thus, even f o r a commutative  i s f i n i t e ) von Neumann  w i t h 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 T Q  (E)> 0  sufficient <j>  on  and  {T  phisms on  lim sup g  f o r the e x i s t e n c e  A g  implies  such t h a t  : s e S} A  .  r Q  o f an  <J>Q ~ <J>  (T E) > 0 g  on  A  , the c o n d i t i o n  for E e P"  i s not  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l  , where  S  is a left  i s an a n t i r e p r e s e n t a t i o n o f  But there i s a s u b c l a s s  which the c o n d i t i o n  AQ  algebra  (*) i s s u f f i c i e n t .  of l e f t  S  amenable semigroup  as normal  -homomor-  amenable semigroup f o r  That i s the c l a s s o f e x t r e m e l y  66.  left  amenable (ELA)  semigroups  multiplicative left for  f , g e m(S)  Theorem 3-19: .  tion  {T  E'T E  (1)  n  : If  s £ S  Proof:  S  be  m(S)  i.e.  i f there i s a  u(fg) =  S  a p o s i t i v e normal  n  A  an  antirepresenta-  .  Suppose  , then the f o l l o w i n g are  real  A e A  )  +  <j> can be  (if  then  by  (A)  9  = 0 <=>  chosen c e n t r a l .  .  9Q  i s c e n t r a l then  lim  I f one  4>  c p ( T AT B) = u s s n  [10] p. 68, we  have  g  sup  of  can be 9  M  f i r s t p a r t of our  theorem f o l l o w s from Theorem 2-8.  (2b) h o l d s ,  functional left  <p  on  i n v a r i a n t mean on  iJ>(A) = u(<j> (T A)) n  of Theorem 3-8  then there i s an  A  such t h a t m(S)  for a l l  9 (TgA) = 0 . 0  ( 1 ) , (2a)  and  chosen c e n t r a l ) which  (T A ) c p (T B) u s u s  (f) = l i m  and  (2a)  and  (2b)  (2a)  to the c o n d i t i o n s  (2a)  on  (2b)  are  respectively. Now  s  sup  f(s)  equivalent Hence  i f one  of  the (1),  S - i n v a r i a n t p o s i t i v e normal  <pQ „ 9  .  Let  u  be a m u l t i p l i c a t i v e  , then P r o p o s i t i o n 2-14 A E A  for  .  Hence the c o n d i t i o n s and  <p  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l  are such t h a t  i s ELA,  =  <j>  i|)(AB) = ^(A)iJi(B)  f E m(S)  T E*E s  equivalent:  XJ  for  u(f)u(g)  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l  <J>Q ~  e A  A,B  <j>  semigroup w i t h  -homomorphisms on  , s E S  <f ~ 4>Q  , then  Since  on  an ELA  then there e x i s t s an with  satisfies all  E e P  i s c e n t r a l then  A  u  i s c a l l e d ELA  a von Neumann a l g e b r a and  as normal  such t h a t  (2b) h o l d s , on  Let  I f A e P ((2b)  (2a) <p  .  be  There e x i s t s an A  If  A  Let  : s e S}  s  f o r any  g  i n v a r i a n t mean  S  ).  A  f u n c t i o n a l on  (a semigroup  i s an  implies  that  S - i n v a r i a n t p o s i t i v e normal  f u n c t i o n a l on  A  such t h a t 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 t h a t  cj)- (T AT B) = ty (T A)ty (T B)  u then ty (AB) = y  Q  , since  g  Remark:  s  u  s  u Q  s  y  0  p -1  on  (Y)  Let  (X,S,p)  fora l l  (i-e. y(s  _ 1  Y ) = y(Y)  ps ^ ).  S  be a f i n i t e  s e S  and  Y  X( i . e .  d e f i n e d by  i n v a r i a n t measure  S  e  a left  into  i s a measure on  Let y  fora l l  X  S  ) such  that  •  space  on  , where  for Y e S  The a l g e b r a  all  0  be a f i n i t e measure space, and  s e S  = p ( s "hr)  S  y ~ P  g  i s m u l t i p l i c a t i v e . //  amenable semigroup o f n o n - s i n g u l a r measurable maps on  ps  s e S  Theorem 3-19 i s a g e n e r a l i z a t i o n o f Theorem 7 i n G r a n i r e r [9]  Example 1:  ps  fora l l  s  ( T A B ) )) = y (_ty ( T A T B ) ) = y (ty ( T A ) d > (T B) ) = y(<fr (TA))  y ( Q ( T B ) ) = i|i(A)ij;(B) Y  s  L (X,S,p)  .  2  .  oo  algebra.  So  ty  on  Q  ty^  , then  oo  i s a von Neumann a l g e b r a  Qo  Define  f e L (X,S,p) L (X,S,p)  L (X,5,p)  by  = /  ty (£) Q  i s a p o s i t i v e normal f a i t h f u l  L (X,S,p) oo  The p r o j e c t i o n s  L^X.S.p)  on t h e H i l b e r t  i s a commutative  i n L (X,5,p)  fdp f o r functional  a - f i n i t e von Neumann  a r e the c h a r a c t e r i s t i c f u n c t i o n s  OO  X  for Y E S  Y  on  X  .  F o r each  f Cx) = f(sx)  by  s e S  xe  for a l l  g  f e L (X,S,p)  and  , we d e f i n e  oQ  X  .  Then t h e f a c t  that  f  ps \<  p  g  implies  that  e L (X,S,p)  f  °o  S  L  (X,S,p)  into  L (X,S,p)  co  S  t  T f = f  for a l l  S  f  T  L (X,S,p)  e  on ,  oo  on  -homomorphism on  L (X,5,p) 00  S  L (X,S,p) 00  i s the ' c o n j u g a t i o n ' ) .  S  = T T  tion of  , we d e f i n e  *  i s a normal t  s e S  S  •  T  involution  T  by  F o r each  co  then  S  .  (note t h a t t h e It i s clear  that  S  for s,t i n as normal  S  .  Hence  {T  -homomorphisms on  : s e S} L (X,S,p)  i s an a n t i r e p r e s e n t a .  I t i s c l e a r that  68.  *0 .X ) = POO  and  (  Y  u  i s equivalent  ^(Ay) to  = p(s Y)  p  and i n v a r i a n t under  i n f {p(s *hf) : s £ S} > 0  implies  f o r any  1  s E S , Y E S  S  , we have  for Y E S  .  Since  p(Y) > 0  i.e. ^(Xy)  >  implies  u  g  inf {^Q(T x ) By  s E S} > 0  :  y  x  f o r any p r o j e c t i o n  i  y  L^CX.S.p)  n  Lemma 3-6, .there i s no weakly-wandering p r o j e c t i o n -1 l^Cxy) > 0  such t h a t s £ S  (note t h a t  x  L^CXjSjp)  n  s * -<  ( T ) ip -< ip  p =>  Q  for a l l  Q  ). Let  space  H  .  M  be a non-commutative von Neumann a l g e b r a  , the p r o d u c t o f  Qo  a non-commutative von Neumann a l g e b r a the H i l b e r t sum o f A  is  on  A  o f form by  on a H i l b e r t  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  Let  A = M x L (X,S,p)  Let  ps  Xy  H  and  A E M  cp ((A,f)) = ^ ( A ) + ^ ( f ) 0  A  p o s i t i v e normal f u n c t i o n a l on  and  L (X,S,p)  on the H i l b e r t . space  L2(X,S,p)  ( A , f ) where  M  .  M Ai s  , then  H © L2(X,S,p) ,  (see [4], p . 2 1 ) . An element i n and  f e L (X,S,p)  for  (A,f)  oo  F o r each  e A  .  Define  , then  s £ S  cf>  9  n  isa  0  , we d e f i n e  T s  s on  A  into A  phism on normal that,  by  .  Hence  since  _x  A  g  {T  t S  Y  (T )^ -< ^  ? Q " <  S  Q  $Q  )•  A  are projections  is a projection in  T ((A,f)) g  -homomorphisms on  we have and  A  (since  in M  Q  = (A,T f )  : s £ S} A and  in M  in  and  s  n  (0,x ) v  X y  )) in  n  ( (A,T f))  i s o f form  A  oo  fora l l S  9 ( )  L (X,S,p)  (E,Xy)  X y  i s a normal  g  T <p -< <p  n  projection  T  i s an a n t i r e p r e s e n t a t i o n  T*<j> ((A, f ) ) =  , then  t,s e S ) . A p r o j e c t i o n s  , then  such t h a t  T ( (E, ) ) ( ( E , g  *  of  A  S X  as (note  = ^ (A) + ^ ( T f ) ,  (E,Xy)  respectively.  y -T^)  S  s E S  S  Q  where If  i s n o t a weakly-wandering  = (E, T  -homomor-  E  0 4 E  projection  4 (0,0) f o r any  i s weakly-wandering i f and o n l y  69.  if  Xy  1  S  weakly-wandering p r o j e c t i o n i n  a  L^CXjSjp)  Hence t h e r e i s no weakly-wandering p r o j e c t i o n <J)Q((E,X )) Y  with  (E,Xy)  Corollary A  >  0  . i°  r  Since a n  Y  L^CXjSjp)  projection  In t h i s example A  Moreover, s i n c e zero i f and o n l y  if  i|>  0  =  wandering p r o j e c t i o n  in  L (X,S,p)  wandering p r o j e c t i o n  in  A  of  S-invariant  Example  2:  .  (X,S,p)  : s e S}  and  Neumann a l g e b r a  For  each  s e S  {T  a projection  .  ,  we  define  3-14,  n  l^CXjSjp)  on  t h e r e i s a complete s e t A  space, and l e t  be as i n Example matrices  T  ^  Xy  T h e r e f o r e t h e r e i s no non-zero weakly-  be a f i n i t e measure  of a l l 2 x 2  on  » hence t h e r e i s no non-zero weakly-  By Theorem  : s E S}  A  S-invariant  s {f  By  i s non-commutative.  normal p o s i t i v e f u n c t i o n a l s  Let  .  that commutes  S  s e S  : s e S) ) .  such  T ((E,Xy))  t h e r e e x i s t s an  is faithful,  ij; (xy)  and  A  in  {T  p o s i t i v e normal f u n c t i o n a l  A  and  A  in  <f> ~ <J)Q  p o s i t i v e normal f u n c t i o n a l on  is  (E,Xy)  S-invariant  .  X  i s commutative,  3-7,- t h e r e i s an  such t h a t  (E, y)  (under  A  on  f  h  e  g  into  1.  A  Let  S , ^  be the von  with entries i n  A  ,  L (X,S,p)  by  s  f  then  T  s  f  h  Vf  T h"  e  g  T e  T g_  i s a normal  an a n t i r e p r e s e n t a t i o n  S  S  S  -homomorphism on of  S  as normal  f e  A  —  s  h s  s  g °s_  .  Hence  {T  s  -homomorphisms on  : s £ S} A  is  70.  Let matrices.  t  be the t r a c e on the a l g e b r a o f a l l 2 4. = t ill. 0 r 0  Define  f  on  if, (f)  of  a p o s i t i v e normal f u n c t i o n a l  under  {T  g  : s e S}  functional {T  g  if) on  : s e S}  .  Q  g  i s a p o s i t i v e normal f a i t h f u l  A  if) (h)  A  = t e  <p  9  on  f u n c t i o n a l on A  A  such t h a t  .  existence  and  n  i s e q u i v a l e n t to the e x i s t e n c e o f a p o s i t i v e L^CXjSjp)  such t h a t  if) ~ if)  A  and i n v a r i a n t  S i n c e the e x i s t e n c e o f such an  g {T  : s e S}  p o s i t i v e normal f u n c t i o n a l to  The  9 ~ <f>  ) , the e x i s t e n c e o f an  9  on  A  such t h a t  invariant normal  under  if) i s e q u i v a l e n t  n o n - e x i s t e n c e o f a non-zero weakly-wandering p r o j e c t i o n i n (with r e s p e c t to  complex  by  h  Vo  then  A  2  x  9  to the  L (X,S,p)  S-invariant is  "0  equivalent  the n o n - e x i s t e n c e o f a non-zero weakly- wandering p r o j e c t i o n i n s  L  (X,S,p)  (with r e s p e c t to  : s e S}  {T  S - i n v a r i a n t p o s i t i v e normal f u n c t i o n a l this r e a d i l y implies projection i n if  A  9  ). on  (see the p r o o f o f Theorem 3-4  i f there e x i s t s  such t h a t  the n o n - e x i s t e n c e o f a non-zero  {T  : s e S}  (For i f  x  A  , then  ((1) =>  L (X,S,p)  (2))). A  Conversely,  , then t h e r e with respect  L (X,S,p) w i t h r e s p e c t to {T weakly-wandering p r o j e c t i o n i n  i s a weakly-wandering p r o j e c t i o n i n i s a non-zero TXv 0" : s e S} , then 0 0 A ) . So the n o n - e x i s t e n c e o f a non-zero  weakly-wandering p r o j e c t i o n i n  A  •  ^  9 ~. <p  an  weakly-wandering  t h e r e i s no non-zero weakly-wandering p r o j e c t i o n i n  i s no non-zero weakly-wandering p r o j e c t i o n i n to  A  Now,  y  8  u  implies  the e x i s t e n c e o f an  S-invariant  p o s i t i v e normal  We  functional  show now  T E-E ^ E'T E s s L  '  on  A  T E ^ E s  .  Let  g  , then  h  h  1-f  Let  E  Now  T E = E s  in  A  such  and  0 < f  —  1 2  where  —  1  to  f  s  S  S  = — — r < 1 g +1  .  S  h = f (1-f)  i s equivalent  that  be any p o s i t i v e element i n  0 < f < 1  g+1  r  E  g  , l e t f =-fr f  d> ,. 6  such t h a t  t h a t there are p r o j e c t i o n s  2  (X,S,p) c°  if  A  2  , then  = f  E  is a projection in  (hence e q u i v a l e n t  g  to  s  = g  Since  f E'T  s  s  fh  s  + h(l-f ) s  s  E hf f  s  s  hh  + h (1-f) s  f + hh  hf  s  +  s  (1-f)(1-f ) s  + h (1-f) s  s  T E-E = s  and  fh  it  f + hh  i s c l e a r that  fh  s  1  ff (l-f ) s s  2  f (l-f  [f f 2  +  s  (1-f)(1-f ) s  i f and o n l y i f  1  1  + f (l-f) (l-f 2  2  1 1 1  2  ) [f f 2  s  11 i.e.  T E-E = E'T E s s  1  2  1 i.e.  hh  + h(l-f ) s  + h ( l - f ) = hf + h (1-f) s s s  1 i.e.  s  2  s  +  (l-f) (l-f 2  s  +  (l-f) (l-f  1  2  s  1  1  s  1  2  I  1 2  1  s  )  2  I  2  2  2  i 2  I (l-f) (l-f  2  1  2  I  - f (l-f) ] = 0 s 2  1  + f (l-f ) (l-f) s s 2  ) ] = f (l-f) [f f + s s s 2  ) ]-[f (l-f 2  2  I  1 2  1 2  s  1  ) = f (l-f) f 2  (a)  s  )  2  72.  1  1 1 Since  (a)  0 <_ f < 1  and  h o l d s i f and o n l y  i f f (l-f 2  s  i.e.  )  2  2  f (l-f  Y  Z  s  2  - f (l-f) s 2  I  1  I  = f (l-f) s  f(1-f ) = f (1-f) s s  i.e.  f - f f=f s s  i.e.  f = f  i.e.  T E = E s  T E-E 4 E*T E s s  g  i f T E 4 E s  f  2  2  .  Hence  = 0  I  /  i.e.  Hence  1  0 <_ f < 1 , [ f f + (l-f) (l-f ) ] > 0 s s s 1 1 1 1 2  /  f s  (i.e. g  g  = g  )  . g  Now i f on Let E  T  g  i s n o t the i d e n t i t y on  L^XjSjp) f, E in  A  A  then  , hence there i s p o s i t i v e  as above then such t h a t  T E»E 4 E*T E s s  T E-E 4 E*T E s s  T  i s n o t the i d e n t i t y  g e L^CXjSjp) .  f o r some  Hence t h e r e s e S  s z S  and  g  are projections  T E-E = E-T E s s  E e P " i n the h y p o t h e s i s o f our main theorems i n  Chapter 3 can be r e l a x e d . prove a s t r o n g e r  g  .  T h i s example shows t h a t the c o n d i t i o n t h a t " f o r any  such t h a t  We do though n o t know a t t h i s  v e r s i o n of i t .  time how t o  4 g  73.  Example 3:  Let  H  L (H)  be a H i l b e r t space and  bounded l i n e a r o p e r a t o r s on  H  Let  H. = H  be the a l g e b r a o f a l l  for a l l positive  integers  CO  i  .  if  C o n s i d e r the p r o d u c t  A =  and o n l y i f A = { A ^ } ^ ^  II i=l  L(H.)  (see [ 4 ] , p. 21), then  such t h a t  A_^ e  L(H./)  and  sup ||A || < i  CO  A  = max is  S = {1,2,...}  {i,j}  S  , then  i f i < n —  A 2>*'-) n+  •  : n e S}  and  Then  be the s e t o f p o s i t i v e  i s a commutative  , the H i l b e r t  1  semigroup  i o j  integers with o  under  .  Hence  S  CO  n e S , l e t T (A) = {B.}. -. such t h a t n i i—l B. = A. i f i > n i . e . TA=(0,0,...,0,A,,, x i n n+1  an amenable semigroup.  B. = 0 l  {T n  H.  00  ^ j ^ i - i  Let  S  E © 1=1  i s a von Neumann a l g e b r a on the H i l b e r t space  sum o f  A e A  1  For  i s a normal  i s a s e t o f normal  *  -homomorphism on  -homomorphisms on  A A  . .  Hence  Let  m, n  in  , then  Case 1:  m < n  T T (A) = T ({B. ; B. = 0 mn m x i  i f i < n , B. = A. — x x  i f i > n})  = {C. ; C. = 0  i f i < m , C. = B.  i f i > m}  = {C. ; C. = 0 x x  i f i < n , C. = A. — x x  i f i > n}  X  X  —  T T (A) = T '({B. ; B. = 0 nm n x x  X  X  i f i < m , B. = A. — x x  i f  i > m})  = {C. ; C. = 0 x x  i f i < n , C. = B. — x x  i f  = {C. ; C. = 0  i f i < n , C. = A.  i f i > n}  X  X  —  1  T (A) = T (A) = T (A) = {C. ; C. = 0 m°n n°m n x x  i > n}  X  i f i < n , C. = A. — x x  i f i >  Hence  T T m  Case 2:  = T T n  n  = T m  = T n°.Ti  m°n  m > n  T T (A) = T ( { B . mn - m i  ; B. = 0 x  if  i < m —  , C. = B . l l  if  i > n}  = {C. i  ; C. = 0 i  if  i < m —  , C. = A. l l  if  i >  0  if  , B . = A. l l  i < m —  if  m} i >  m})  = {C. i  ; C. = 0 i  if  i < n , C. = B .  if  i > n}  = {C. i  ; C. = 0 i  if  i < m —  i f  i >  T T = T T = T m n n m m°n : n e S}  n  i > n} )  if  I  —  =T  I  , C. = A. i i  ; C. = 0 l  1  {T  if  1  ; C. = 0 i  T (A) = T (A) = T (A) = {C. m°n n°m m l  Thus  1  = {C. i  T T (A) = T ( { B . ; B . = n m n l l  Hence  , B . = A.  i < n —  if  i < m —  m} , C. = A. l i  if  i >  nom  i s a representation  of  S  ( a l s o an  antirepresentation  * of  S  since  S  i s commutative) as normal  -homomorphisms on  A  .  A  CO  E e A  projection  i s o f form  E_^  where  are p r o j e c t i o n s i n CO  .. ,|2 E ||x.|| < ~ i=l  00  L(H.)  for a l l i  .  Let  & > 0  x  and  e  1  00  There i s  N > 0  E © i=l  H.  , then  1  1  2 E ] | x . || < 6 . For n > N , we have i=N CO i < n , B. = E. i f i > n} T Ex = { B . x . } . ^ — ' i i ' n i i i=l  such that  1  T E = {B. : B . = 0 n i l  if  00  0  so  ]|T E X | |  J  CO  n  E ||B.x.|| i=l  =  E ||E.x.|| i=n  .  Since  E.  <_ I  , hence oo  00  llE^ill  2  1 jl^li  2  •  E ||E.x.|| < E || X . f < S i=n ^ i s a r b i t r a r y , j| T^Ex|| converges to z e r o  Thus we have  j|T Ex|| = 2  2  n  1  since  n > N  .  Since  5 > 0  ,  2  9  =  L  =  n  ,  f o r any  {T^E  x e  Z 0 i=l  : n E S}  H.  i.e.  that  In p a r t i c u l a r i f  <> j  a l l projections S-invariant.  E  ACT^E)  is in  E = {E.}. , 1 i=l  converges to zero,  S-invariant A  Hence  then <f>  T h i s proves t h a t there  p o s i t i v e normal f u n c t i o n a l on CO  converges to zero  strongly.  Since  i s a bounded s e t , a l l normal p o s i t i v e f u n c t i o n a l s  have the p r o p e r t y  is  T E  such t h a t  wandering p r o j e c t i o n .  E  J  i  = 0  A  .  4(E)  i s the i s no  = zero  for a l l ty(T^E)  finite  i s zero  i  for <> j  S-invariant  I t i s c l e a r t h a t any  f o r a l l but  E E P  functional i f  non-zero  <J>Q  projection  i s a weakly-  76.  Bibliography  [1]  Bachman, G. and N a r i c i ,  L., " F u n c t i o n a l A n a l y s i s " , Academic P r e s s  1966. [2]  Blum, J^R. and Friedman, N., transformations",  "On i n v a r i a n t measures f o r c l a s s e s of Z. W a h r s c h e i n l i c h k e i t s t h e o r i e  verw.  Geb. 8 (1967) 301-305. [3]  C a l d e r o n , M., Sci.  [4]  A l b e r t o , P., Paris  "Sur l e s mesures i n v a r i a n t e s " , C.R.  (1955) V. 240,  Acad.  1960-1962.  D i x m i e r , J . , "Les A l g e b r e s D ' o p e r a t e u r s Dans L'espace H i l b e r t i e n " , P a r i s 1957, 1 s t e d i t i o n .  [5]  D i x m i e r , J . , "Les anneaux d ' o p e r a t e u r s de c l a s s e f i n i e " ,  Ann. Ec.  Norm. Sup., t.66 (1949) 209-261. [6]  Dixmier, J . , " A p p l i c a t i o n s  tf  dans l e s anneaux d ' o p e r a t e u r s " ,  Compos. Math., t.10 (1952) 1-55.. [7]  D i x m i e r , J . , "Forms l i n e a i r e s Soc.  [8]  Math. F r . , t . 81 (1953) 9-39.  York, 1958.  G r a n i r e r , E.E., "On f i n i t e  equivalent  groups o f t r a n s f o r m a t i o n s " , [10]  i n v a r i a n t measures f o r semiTo appear i n Duke Math. J .  G r a n i r e r , E.E., " F u n c t i o n a l a n a l y t i c p r o p e r t i e s o f e x t r e m e l y amenable semigroups", T r a n s . AMS  [11]  137 (1969) 53-75.  H a j i a n , A.B. and K a k u t a n i , S., "Weakly wandering s e t s and i n v a r i a n t measures", T r a n s . AMS  [12]  Bull.  Dunford, N. and Schwartz, J.T., " L i n e a r o p e r a t o r s I " , I n t e r s c i e n c e , New  [9]  s u r un anneaux d ' o p e r a t e u r s " ,  Hajian,  A. and I t o , Y., for  110  (1964) 136-151.  "Weakly wandering s e t s and i n v a r i a n t measures  a group o f T r a n s f o r m a t i o n s " , J o u r n a l of Math, and  Mechanics, V o l . 18,.No. 12 (1969) 1203-1216.  77.  [13]  Hewitt and  Yosida., 72  [14]  Nakamura, M.,  Natarajan,  10  S.,  " C o n t r i b u t i o n s to E r g o d i c Theory", T h e s i s ,  Neveu, J . , "Sur  R i c k a r t , C.E.,  Sachdeva, U.,  Institute, Calcutta,  The  1968.  l ' e x i s t e n c e de mesures i n v a r i a n t e s en t h e o r i e CR.  "General  Acad. S c i e . P a r i s ,  260  (1965) 393-396.  theory of Banach a l g e b r a " , Van  P r i n c e t o n , N.J., [18]  Sem.  (1958) 189-190.  ergodique", [17]  AMS  "A p r o o f of a theorem of T a k e s a k i " , K6dai Math.  " Indian S t a t i s t i c a l [16]  a d d i t i v e measures", T r a n s .  (1952) 46-66.  Rep. [15]  "Finitely  Nostrand.  1960.  "Research F u n d a t i o n " ,  T h e s i s , The  Ohio S t a t e U n i v e r s i t y  1970. [19]  Sucheston, L., Z.,  [20]  T a k e s a k i , M.,  "On  e x i s t e n c e of f i n i t e i n v a r i a n t measures", Math.  86  (1964) 327-336.  "On  the conjugate  Math. J . 10 [21]  T a k e s a k i , M.,  "On  space of o p e r a t o r a l g e b r a " , Tohoku  (1958) 194-203.  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  functional  on o p e r a t o r a l g e b r a " , P r o c . Japan Acad., t . 35 365-366.  (1959)  

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