UBC Theses and Dissertations

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UBC Theses and Dissertations

Topological invariant means on locally compact groups Wong , James Chin Sze 1969

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T O P O L O G I C A L I N V A R I A N T M E A N S O N L O C A L L Y C O M P A C T G R O U P S b y J A M E S . C H I N S Z E WONG B . A . ' (1st H o n o u r s ) U n i v e r s i t y o f H o n g K o n g , 1963 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F j D O C T O R O F P H I L O S O P H Y i n t h e D e p a r t m e n t o f M A T H E M A T I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s , i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a i i T h e s i s S u p e r v i s o r : D r . E . E . G r a n i r e r A B S T R A C T T h e s t u d y o f i n v a r i a n t m e a n s o n s p a c e s o f f u n c t i o n s a s s o c i a t e d w i t h a g r o u p o r s e m i g r o u p h a s b e e n t h e i n t e r e s t o f m a n y m a t h e m a t i c i a n s s i n c e v o n N e u m a n n ' s w o r k o n i n v a r i a n t m e a s u r e s a p p e a r e d i n 1929* I n r e c e n t y e a r s , m a n y i m p o r t a n t p r o p e r t i e s o f l o c a l l y c o m p a c t g r o u p s h a v e b e e n f o u n d t o d e p e n d o n t h e e x i s t e n c e o f a n i n v a r i a n t m e a n o n a s u i t a b l e t r a n s l a t i o n -i n v a r i a n t s p a c e o f f u n c t i o n s o n t h e g r o u p . I n t h i s t h e s i s , w e d e a l m o s t l y w i t h i n v a r i a n t m e a n s o n t h e s p a c e L ^ G ) o f b o u n d e d m e a s u r a b l e f u n c t i o n s o n a l o c a l l y c o m p a c t g r o u p G . S e v e r a l c h a r a c t e r i s a t i o n s o f t h e e x i s t e n c e o f a n i n v a r i a n t m e a n o n L ^ G ) a r e g i v e n . A m o n g o t h e r r e s u l t s , we p r o v e t h e r e m a r k a b l e t h e o r e m t h a t L o o ( G ) h a s a l e f t i n v a r i a n t m e a n i f a n d o n l y i f G i s t o p o l o g i c a l l y r i g h t s t a t i o n a r y , a n a n a l o g u e o f a r e c e n t r e s u l t f o r s e m i -g r o u p s b y T . M i t c h e l l . H o w e v e r o u r a p p r o a c h i s e n t i r e l y d i f f e r e n t . i i i A C K N O W L E D G E M E N T S I a m g r e a t l y I n d e b t e d t o P r o f e s s o r E . E . G r a n i r e r f o r h i s v a l u a b l e h e l p a n d e n c o u r a g e m e n t d u r i n g t h e p r e p a r a -t i o n o f t h i s t h e s i s . T h e g e n e r o u s f i n a n c i a l s u p p o r t o f t h e N a t i o n a l R e s e a r c h C o u n c i l o f C a n a d a a n d t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a w h i c h m a d e i t p o s s i b l e t o c a r r y o u t t h i s r e s e a r c h i s g r a t e f u l l y a c k n o w l e d g e d . T A B L E O F C O N T E N T S P a g e I N T R O D U C T I O N 1 C H A P T E R I I N V A R I A N T M E A N S O N L O C A L L Y C O M P A C T G R O U P S 1.1. N o t a t i o n s a n d T e r m i n o l o g i e s 3 1.2. F u n d a m e n t a l L e m m a s 5 1.3• A r e n s P r o d u c t 10 1.4. L o c a l i s a t i o n T h e o r e m o n I n v a r i a n t M e a n s 16 1.5. T o p o l o g i c a l S t a t i o n a r y L o c a l l y C o m p a c t G r o u p s . . .18 1.6. I n v a r i a n t a n d M u l t i p l i c a t i v e I n v a r i a n t M e a n s . . . 21 C H A P T E R I l \ T O P O L O G I C A L A L M O S T C O N V E R G E N C E 2 . 1 . T o p o l o g i c a l L e f t A l m o s t C o n v e r g e n c e 25 2 .2 . C r i t e r i o n ' f o r T o p o l o g i c a l L e f t A l m o s t C o n v e r g e n c e 25 C H A P T E R I I I T O P O L O G I C A L I N T R O V E R T E D S P A C E S 3 . 1 . E q u i v a l e n t D e f i n i t i o n s 31 3 .2 . P r o p e r t i e s o f T o p o l o g i c a l L e f t I n t r o v e r t e d S p a c e s 34 3.3* E x a m p l e s o f T o p o l o g i c a l L e f t I n t r o v e r t e d S p a c e s 36' C H A P T E R I V A P P L I C A T I O N S 4 . 1 . I n v a r i a n t M e a n s o n W ( G ) 4 l 4 .2. I n v a r i a n t M e a n s o n U C B r ( G ) 44 C H A P T E R V I N V A R I A N T M E A N S A N D F I X E D P O I N T S 5 .1 . A c t i o n s o f L ^ ( G ) a n d M ( G ) 47" 5 .2 . S i l v e r m a n I n v a r i a n t E x t e n s i o n P r o p e r t y 53 B I B L I O G R A P H Y 57 1 I N T R O D U C T I O N A s e m i g r o u p S i s c a l l e d r i g h t s t a t i o n a r y i f f o r e a c h f e m ( S ) , t h e w e a k * c l o s e d c o n v e x h u l l o f r i g h t t r a n s l a t e s o f f i n m ( S ) = ^ ( S ) * c o n t a i n s a c o n s t a n t f u n c t i o n . I n [23], T . M i t c h e l l p r o v e d t h a t a s e m i g r o u p S i s r i g h t s t a t i o n a r y i f a n d o n l y i f m ( S ) h a s a , l e f t i n -v a r i a n t m e a n . I n t h i s c a s e , t h e s e t o f v a l u e s n ( f ) w h e r e H r u n s o v e r a l l l e f t i n v a r i a n t m e a n s o n m ( S ) c o i n c i d e s w i t h t h e s e t o f c o n s t a n t s i n t h e w e a k * c l o s e d c o n v e x h u l l o f r i g h t t r a n s l a t e s o f f . T h e m a i n p u r p o s e o f t h i s t h e s i s i s t o p r o v e a t o p o l o g i c a l a n a l o g u e ( w h i c h i s a l s o a g e n e r a l i s a -t i o n ) o f t h i s t h e o r e m f o r l o c a l l y c o m p a c t g r o u p s . I t t u r n s o u t t h a t o u r a n a l o g u e o f M i t c h e l l ' s t h e o r e m i s a c o n s e q u e n c e o f a m o r e g e n e r a l " L o c a l i s a t i o n T h e o r e m " o n i n v a r i a n t m e a n s w h i c h we a r e g o i n g t o p r e s e n t i n C h a p t e r 1. I n C h a p t e r 2, we s t u d y t h e c o n c e p t o f t o p o l o g i c a l l e f t a l m o s t " c o n v e r g e n c e . C h a p t e r 3 i s d e v o t e d t o t h e s t u d y o f t o p o l o g i c a l l e f t i n t r o -v e r t e d s p a c e s o f . b o u n d e d m e a s u r a b l e f u n c t i o n s o n a l o c a l l y c o m p a c t g r o u p . T h i s c o n c e p t i s a g e n e r a l i s a t i o n o f l e f t i n t r o v e r t e d s p a c e o f b o u n d e d f u n c t i o n s o n a s e m i g r o u p i n t r o -d u c e d b y M . M . D a y i n [4] a n d s t u d i e d i n s o m e d e t a i l b y R a o i n [29]. I n C h a p t e r 4, we g i v e s o m e a p p l i c a t i o n s o f t h e " L o c a l i s a t i o n T h e o r e m " a n d i t s i m m e d i a t e c o n s e q u e n c e s . I n t h e l a s t C h a p t e r , c e r t a i n f i x e d p o i n t p r o p e r t i e s o f l o c a l l y c o m p a c t g r o u p s a r e i n v e s t i g a t e d . I t w i l l b e p r o v e d t h a t a 2 l o c a l l y c o m p a c t g r o u p G i s a m e n a b l e ( o r e q u i v a l e n t l y L o c ) ( G ) h a s a t o p o l o g i c a l l e f t i n v a r i a n t m e a n ) i f a n d o n l y i f G h a s a n y o n e o f t h e s e p r o p e r t i e s . A n i m m e d i a t e c o n s e q u e n c e o f t h i s i s t h a t e v e r y l o c a l l y c o m p a c t g r o u p h a s a n " i n -v a r i a n t e x t e n s i o n p r o p e r t y " s i m i l a r t o t h a t s t u d i e d b y R . J . S i l v e r m a n i n [32]. 3 C H A P T E R I I N V A R I A N T M E A N S O N L O C A L L Y C O M P A C T G R O U P S 1 . 1 . N o t a t i o n s a n d T e r m i n o l o g i e s . F o r g e n e r a l t e r m s i n h a r m o n i c a n a l y s i s , we s h a l l f o l l o w H e w i t t a n d R o s s [19] ( u n l e s s o t h e r w i s e s t a t e d e x -p l i c i t l y ) . L e t G b e a l o c a l l y c o m p a c t g r o u p w i t h a f i x e d l e f t H a a r m e a s u r e X a n d m o d u l a r f u n c t i o n A . S y m b o l s l i k e . . . d x a n d . . . d y w i l l a l w a y s d e n o t e i n t e -g r a t i o n w i t h r e s p e c t t o X . L e t B M ( G ) b e t h e a l g e b r a ( p o i n t w i s e o p e r a t i o n s ) o f a l l r e a l - v a l u e d b o u n d e d X -m e a s u r a b l e f u n c t i o n s o n G w i t h s u p r e m u m n o r m || || a n d N t h e c l o s e d i d e a l o f a l l l o c a l l y n u l l f u n c t i o n s i n B M ( G ) ( [ 1 9 , D e f i n i t i o n 1 1 . 2 6 ] ) . L e t L w ( G ) = B M ( G ) / N b e t h e q u o t i e n t B a n a c h a l g e b r a w i t h q u o t i e n t n o r m - || ( i . e . e s s e n t i a l s u p r e m u m n o r m ) . F o r f e L r o ( G ) , g e L 1 ( G ) , l e t ( f , g ) d e n o t e t h e d e f i n i n g b i l i n e a r f u n c t i o n a l o f t h e p a i r L ( G ) a n d L 1 ( G ) . I f g e L 1 ( G ) , s e G , t h e f o l l o w i n g m a p s ^s 3 r s 3 l g 3 r g : L ~ ( G ) "* L » ( G ) a r e d e n n e d b y y ( t ) = f ( s t ) , r g f ( t ) = f ( t s ) , t&(f) = | g ~ * f a n d r _ ( f ) = f * g ~ f o r a n y f e L ( G ) , t e G . ( i f i s s o m e -g 0 0 s t i m e s w r i t t e n a s f a n d . r f a s f• ) . I t i s c l e a r t h a t S S S t h e m a p s i, , r Q , I , r a r e n o r m c o n t i n u o u s a n d l i n e a r a n d s a t i s f y \\lj = \\rj = 1 . , | k g | | , | | r g | | ' < i | g | | -A l i n e a r f u n c t i o n a l m i n L ( G ) * i s c a l l e d a m e a n i f e s s i n f f <_ m ( f ) <_ e s s s u p f f o r a n y f e L _ ( G ) ( s e e [ 1 8 , § 2 . 1 . ] ) . A m e a n o n L ( G ) i s l e f t ( r i g h t ) i n v a r i a n t i f f o r e v e r y f e L _ ( G ) , x e G , we h a v e m(l f ) = m ( f ) ( m ( r f ) = m ( f ) ) . I t i s t o p o l o g i c a l l e f t ( r i g h t ) i n v a r i a n t i f f o r a n y f e L r o ( G ) , a n d a n y cp e - P ( G ) = (cp € L - ( G ) : cp >_ 0 , ||cp||. = 1 } , we h a v e m ( c p * f ) = m ( f ) ( m ( f * c p ~ ) = m ( f ) ) . H e r e cp~ i s t h e f u n c t i o n d e f i n e d b y c p ~ ( x ) = c p ( x _ 1 ) , x e G . m i s ( t o p o l o g i -c a l ) i n v a r i a n t i f i t i s b o t h ( t o p o l o g i c a l ) l e f t a n d ( t o p o -l o g i c a l ) r i g h t i n v a r i a n t . I f X i s a l i n e a r s u b s p a c e o f L _ ( G ) c o n t a i n i n g t h e c o n s t a n t s , X i s c a l l e d l e f t ( r i g h t ) i n v a r i a n t i f la(X) c X ( r (X) c X) f o r a n y s e G . I t i s t o p o l o g i c a l l e f t ( r i g h t ) i n v a r i a n t i f P ( G ) •*.. X c X (X * P ( G ) ~ c x) . I n v a r i a n t m e a n s ( l e f t , r i g h t , t o p o l o g i -c a l l e f t o r t o p o l o g i c a l r i g h t ) o n s u c h l i n e a r s u b s p a c e s X c a n b e d e f i n e d s i m i l a r l y . L e t C B ( G ) b e t h e B a n a c h s p a c e o f a l l r e a l v a l u e d ^ N o t e t h a t i n H e w i t t a n d R o s s [ 1 9 ] , t h e n o t a t i o n f * i s u s e d i n s t e a d o f f ~ . T h e l a t t e r i s t a k e n f r o m H u l a n i c k i [20] a n d G r e e n l e a f [ l 8 ] i bounded continuous functions on G and l e t UCBr(G) (UCB (G)) consist of a l l f e CB(G) such that the map s -• JL f (s -* r _ f ) from G into CB(G) i s continuous when CB(G) has the usual supremum norm. Functions i n UCBr(G) (UCB (G)) are ca l l e d r i g h t ( l e f t ) uniformly continuous. Notice that some authors c a l l functions i n UCBr(G) (UCB (G)) l e f t (right) uniformly continuous and denote them by LUC(G) (RUC(G)). See f o r example [14] and [ 2 7 ] . I t i s known that the spaces ^ ( G ) s CB(G) ,' UCB (G) and UCB(G) = UCB (G) fl UCB. (G) are l e f t invariant as well as topological l e f t i n v a r i a n t . Moreover, any topo-l o g i c a l l e f t invariant mean on X = L (G), CB(G), UCB^(G) or UCB(G) i s always l e f t i n v a r i a n t . I t i s also known that the existence of a l e f t invariant mean on any of the spaces L (G) , CB(G) , UCB (G) or UCB(G) implies the existence CO 1/ of a topological l e f t invariant mean on any of them. In this case, we c a l l G amenable (as a l o c a l l y compact group). The reader should consult Greenleaf [ 1 8 , § § 2 . 1 . and.2 . 2 . ] f o r a detailed description of these f a c t s . 1 . 2 . Fundamental Lemmas. In this section, we state a number of lemmas which are needed i n proving the main theorems. Several of these are quite well-known: For completeness, we s h a l l bring i n the proofs. Lemma 1.2.1. Let G be a l o c a l l y compact group, then, (a) . The mapping g - j g~ i s a l i n e a r isometry of L^G) onto i t s e l f . I t i s also a homeomorphism when L-j_(G) has Nthe weak topology. (b) The mapping f -» f ~ i s a l i n e a r isometry of L_(G) onto i t s e l f . I t i s also a homeomorphism when L^G) has the weak* topology. (c) For any f e L_(G) , g,cp e L-^G) , we have (f*g~,cp) = (f ,<P*g) = (j <P~*f ,g) • 1 1 1 " (d) For any g,cp € L-^G) , ^(cp*g)~ = - g~*-cp~ . (e) For any f e L_(G) , cp e L.(G) , f~*cp~ = (cp*f)~ . (f) For any f e L_(G) , cp,g e L-jG), , cp*(f*g~) = (cp*f )*g (g) For any f e L_(G)' , cp,* e L^(G) , (ep*ijr)*f = <p*(i|r*f) "". Proof: (a) By [19, Theorem 20.2], we see that g •-• •— g~ i s a li n e a r isometry of L-^G) onto i t s e l f . Since i t i s 7 self-inversed, the second part of (a) i s immediate. (b) Let T : L 1(G) - L 1(G) be the map i n (a). Consider i t s adjoint T* : L j G ) - L w(G) . I f f e L ^ ) * g € L X(G) , then (T*f)g = {t,j s~) = (f~,g) by [19, Theorem 20.2] . Hence T*f = f ~ . . Since inversion i n G preserves l o c a l l y n u l l sets, ||f~||e0 = ||f !!„ and the map f f ~ i s a l i n e a r isometry which i s s e l f -inversed. Hence being the adjoint of T , i t i s also a homeomorphism when L r o(G) has the weak* topology. (c) Let f e ^ ( G ) , g,cp e L-^G) , we have (f*g~,cp) = (f*g~) (x)cp(x)dx = f (y)g~(y - 1x.)dycp(x)dx g(x _ 1y)cp(x)dxf (y)dy = f (cp*g) (y)f (y )dy = (f,«p*g) , by [19, Corollary 2 0 . 1 4 . ( i i i ) ] , Pubini's Theorem and [19, Theorem 2 0 . 1 0 .(i)] respectively. On the other hand, cp~*f ,g) = J*(^  cp~*f) (x)g(x)dx ^(y~ 1)cp(y" 1)f-(y~ 1x)dyg(x)dx = r[cp(y)f (yx)dyg(x)dx f(yx)g(x)dx cp(y)dy = (f*g~) (y)cp(y)dy = (f*g~,«p) by [19, Corollary 20.14.(i), Theorem 20.2.], Fubini's Theorem and [19, Corollary 2 0 . 1 4 . ( i i i ) ] respectively 8 i n the second through f i f t h e q u a l i t i e s . (d) The map g - ^ g~ i s an involution of the group algebra L - j J g ) (with convolution as m u l t i p l i c a t i o n ) as i s w e l l -known . (e) Let f e L r o(G) , cp e L 1(G) , then (f~*cp~)(x) = f~(y)cp~(y _ 1x)dy = ff (y' 1)cp(x" 1y)dy = ff (y" 1x" 1)cp(y)dy 0 0 0 = (cp#f)~(x) . (f) L e t \ f e LjG) , cpj,g e L]_(G) , then ((cp*f )*g~, <|r) = =(f ,\ cp~*(t*g)) = (f, {\ cp~*i|/)*g) = (cp*(f*g"),t) by applying (c), repeatedly. Hence (cp*f)#g~ = cp#(f#g~) . (g) Let f e L j G ) , cp,i|i,g e L^G) , then ((<Mr)*f,g) = (f,^(<P*.*)"*g) = (f ,\ <T*(| «P~*g) = , | <P~*g) = (cp*(t*f ) , g ) by applying (c) repeatedly. Therefore (cp*t)*f = cp*(\|r*f) . Lemma 1.2.2. Let G be a l o c a l l y compact group, then (a) For f i x e d cp e L^CG) , the map f - cp*f o f . L o o(G) into i t s e l f i s w*-w* continuous. (b) For f i x e d f € L t t(G) , the map cp - cp*f of L^G) into L (G) i s w-w* continuous. 9 Similar assertions hold f o r the mappings f -* f#cp~ and cp f*cp~ . Proof: (a) Let cp e L 1(G) be f i x e d . Define a map T : L-^G) - L.(G) by Tg = ± cp~*g , g e L]_(G) . Clearly T i s bounded l i n e a r . Consider i t s adjoint T* : LjG) - L_(G) . We have (T*f)g = (f,|q)"*g) = (cp*f^g) by lemma 1.2 . 1 . ( c ) . Hence T*f = cp*f . Thus the map f -• cp*f being the adjoint of T i s w*-w* continuous. (b) Let X, Y be Banach spaces and T : X - Y* be bounded l i n e a r . Then T •: X -» Y* i s w-w continuous by [ 8 , Theorem 1 5 , p.-422] and a f o r t i o r i w-w* continuous. Take X = L-^G) , Y = L w(G) and Tcp = cp*f- , the r e s u l t follows. The assertions about the maps f -» f*cp~ and cp -• f*cp~ can be proved s i m i l a r l y . Lemma 1.2«3« Let cp be an approximate i d e n t i t y in.. L, (G) (see [ 1 9 , D e f i n i t i o n 20.26]), then (a) — cp~ i s also an approximate i d e n t i t y i n L^(G) . 10 (b) I f f e L (G) , then w* lim cp * f = f = w* lim f*cp~ . Proof: (a) I f cp e L^G) , * = | cp~ e L-(G) . Hence j cp~ *cp = A v l *~ = _"(***a^~ "* A" *~ = v l n n o r m t o P o l ° S y o f , by Lemma 1.2.1.(d) and (a). S i m i l a r l y cp*y cp~ -* cp i n norm topology of L- (G) . La C£ JL (b) Let, v cp be an approximate i d e n t i t y i n L, (G) . By (a), so i s j cp~ . I f f e L^G) , then f o r any g e L 1(G) , j cp~*g - g i n norm topology of L.(G) . In p a r t i c u l a r , ^ c^*_ "* S weakly i n L-^(G) . Hence (cp a*f,g) = ( f ^ 9 ~ * g ) - (f,g) be Lemma 1.2.1. (c). Consequently cp * f -• f i n w* topology of L (G) . Si m i l a r l y f*cp~ - f i n w*. topology of X (G) . 1.3* Arens Product. In [1], R. Arens has shown how to define an associative m u l t i p l i c a t i o n i n the second conjugate space B** of a Banach algebra B . This m u l t i p l i c a t i o n renders B** a Banach algebra and extends the m u l t i p l i c a t i o n i n B. We s h a l l describe the procedure i n the sp e c i a l case when B i s the group algebra L,(G) , with convolution as mul-11 t i p l i c a t i o n . This Arens product plays an important role i n subsequent discussions. For f e L ^ G ) , g e L - ^ G ) , we define f © g e L W ( G ) by (f © g)cp =. (f ,g*cp) f o r each cp e L - ^ G ) . For v e L m ( G ) * , f e Lv(G) , we define v © f e L r o(G) by (v © f )g = v(f © g) f o r each g e L-^G) . For |i,v € L ^ G ) * , we define |i © v e L j G ) * by (u © v)f = n(v © f ) f o r each f e L W ( G ) . \ \ The following lemma consists of a multitude of res u l t s analogous to those i n M. M. Day's c l a s s i c a l paper on amenable semigroups [4, §6]. The proofs are also "to-po l o g i c a l variants" of those i n his paper. Lemma 1.3.1. • ' (a) Let v e L W ( G ) * - , f e LjG ) . , cp,g e L - ^ G ) Then f © g = A"S~*f , cp © f = f*cp~ and cp © g = cp*g . (b) If \x, v are means on ^ ( G ) , so i s n 0 v . (c) I f cp e L 1(G) , n,v e L m ( G ) * . Then cp © v = I* v , _ • n © <P = r * n . (d) For f i x e d cp e D - L ( G ) , the maps n — | i © cp and v -• cp © v are w*-w* continuous. (e) For f i x e d v e L r o(G)* , the map |-i -• yx © v i s w*-w* 12 continuous. (f) M- © v i s topological l e f t invariant i f n i s topolo-g i c a l l e f t i n v a r i a n t . I t i s topological right i n -variant i f v i s topological r i g h t i n v a r i a n t . (g) I f v i s a topological l e f t invariant and \i i s a mean on .^(G) , then n © v = v.. Proof: (a) For any i|r e L.(G) , (f © g)i|r = (f_g**) = (jg*f, *) by \ • Lemma 1.2.1. (c). Hence f © g = ^-g~*f .. Also (cp © f H = cp(f 0 if) = (|-t~*f,cp) = (f*cp~,t) by Lemma 1.2.1. (c) again. Hence cp © f = f*cp~ . ' F i n a l l y i f f e L r o(G) , (cp © g)f = cp(g © f ) = cp(f*g~) = (f*g~,cp) = . (f ,cp*g) = (cp*g)f . Hence cp © g = cp#g . (b) Let v be means on L^G) . Then c l e a r l y ||n©v|| _. Ill-1 IIIIvI - 1 • Now i f 1 denotes the constant one function on G we have 1 © g = ^ g " * l = UgH^'l and (v © l ) g = v ( l © g) = ||g||- = (l,g) f o r any g e L-|_(G-) and g >_ 0 . Hence by l i n e a r i t y v © 1 = 1 and therefore [yi © v ) l = |i(v © 1)- = -M-(3.) = "1 . Con-sequently |i © v i s a mean on L (G) . (c) We have (cp © v)f = cp(v © f ) = (v © f )cp = v(f © cp) = v(|cp~*f) = v(£ f ) = U * v ) f f o r any f e LjG) . Hence cp © v = l*v . Also (|_ © cp)f = )_i(cp © f ) = n(f*cp~) = ^ ( r c p f ) = ( r tJ l i ) f f o r a n y f e LJG) ' S o - H © < P = r*n . (d) This i s immediate from (c). (e) Suppose n a - H i s w* topology of L O T ( G ) • Then fo r any f e L W ( G ) , 0 v)f = P a(v © f ) - n(v 0 f ) = (n © v)f . , ' -(f) Let \i be topological l e f t i n v a r i a n t / t h e n f o r any . cp e P ( G ) , © v) = ? 0 ^ 0 v) = (cp © |_) © v = (^l-1) © v = (j © v by (e) and topological l e f t i n -variance of i-i . I f v i s topological r i g h t invariant, then f o r any cp € P ( G ) , r*(n © v) = (n © v) © cp = p. © (v © cp) = p © (r*v) = p © v by s i m i l a r arguments. (g) Let n be a mean on L _ ( G ) . By weak* density of the set P ( G ) i n the set of a l l means on ^ ( G ) (see Hulanicki [20]), there i s a net cp i n P ( G ) such that cpa p i n weak* topology of L _ ( G ) * . Then cp„ © v -• p © v weak* by (e). Now cp © v = -t*v = v by (c) and topological l e f t invariance of v . Hence-p © v = v . . D e f i n i t i o n 1.3.2. For each m e L M ( G ) * , define a mapping 14 mL : LjG) - LjG) by m L(f)(g) = m(^g~*f) f o r any f e L w(G), g e L^(G) . The operator Is c a l l e d the topological l e f t introversion of m . S i m i l a r l y the topological r i g h t i n t r o -version mR : L r o(G) - L^G) i s defined by m^f )g = m(f*g~) f o r any f e LjG) , g e L-^G) . Lemma 1.3.3. (a) mL : L j G ) - L j G ) i s bounded l i n e a r , H m ^ f ) ^ < ||m|l |r and \ n L ( f ) = m © f f o r any f e L m(G) . (b) If v e LjG)* , f e LjG) , cp,g e L-jG) . Then . ^L = rcp ' v L ^ s * f ) = S*v L(f) . (c) I f m_, m i n norm topology of L m ( G ) * , then (m a)^ -* m^  i n uniform operator topology. (d) I f m -* m i n weak* topology of L (G)* , when f o r any f e L^CG) , ("^L^ "* ^ L ^ ) i n w e a k * topology of L„«0 . • i ' Proof: (a) |m L(f)g| == |m(|g~*f)| < UmllUgU-Jf ^  . Hence )IL 1 H m l l l l f L ' B u t mL(f)s = ™ ( i g~*f) = m(f © g) • ' = (m © f )g by Lemma 1.3.1. (a). Thus m^f) = m © f . (b) I f cp e L-(G) , cp L(f) = cp © f = r f by (a) and Lemma 1.3.1. (a). Hence cpL = r ^ . I f f e LjG) , g € L-(G) and v e L_(G)* , then v L(g*f)cp = V(^ CP~ * ( g * f ) ) = v ( ^ g ~ * c p ) ~ * f ) = v L(f)(^g~*cp) = ( v L ( f ) , i g % c p ) = (g*v L(f),cp) f o r any cp e L^G) , by Lemma 1.2.1.(g), (d) and (c). Hence v L ( g * f ) = g*v L ( f ) . -(c) Assume m - m i n norm topology of L (G)* . Since ( mA " = ( m a " m ^ L H ( ( m A " m L ) f l l » --||ma - m|I||f||_ by (a). Hence ||(ma)L - mL!| < ||ma - m|| - 0 . (d) Let ma -* m i n weak* topology of L r a(G)* . Then fo r any f e L_(G) , g e L. (G) , ( ( m a ) L f ) g - (m Tf)g = m a(^g~*f) - m(|g~*f) - 0 Remark 1.3»4. Lemma 1.3.3* has an analogue f o r topological r i g h t introversion (with some modifications). In f a c t we have = lO ' v R ( f * s ~ ) = v R ^ f ) * s ~ a n d mR f = m © f where © R R i s the Arens product a r i s i n g from the Banach algebra L^(G) where m u l t i p l i c a t i o n i s convolution with the order of factors interchanged. The rest of Lemma 1.3>3» remain v a l i d i f we replace L by R . We s h a l l not need these 16 r e s u l t s i n what follows. 1 . 4 . L o c a l i s a t i o n Theorem on Invariant Means. D e f i n i t i o n 1 . 4 . 1 . A l i n e a r subspace X of L e j(G) i s said to be topological l e f t introverted i f f o r any mean . m on L c o(G) , mL(X) c X .(Thus by d e f i n i t i o n of mL , L r o(G) i s always topological l e f t introverted). Theorem 1 . 4 . 2 . (Localisation) Let X be a topological l e f t introverted and topological l e f t invariant l i n e a r subspace of D m(G) con-ta i n i n g the constants. Then X has a topological l e f t invariant mean i f f f o r any f e X , there i s a mean mf on X such that mf (cp*f) = mf (f) f o r every cp e P(G) . Proof: Necessity i s t r i v i a l . To prove s u f f i c i e n c y , define f o r each f e X , = (m : m i s a mean on L^G) ~ m(cp*f) = m(f) f o r any cp e P(G)} . By assumption, K f ^ (j) (since any mean on X can be extended to a mean-on L r o(G) by Hahn Banach Theorem). Clearly K f i s a weak* closed subset of the norm closed unit b a l l i n L (G)* which i s -CO * ' 17 weak* compact. We show that the family (K f : f e X} has the f i n i t e i n t e r s e c t i o n property. Let f ^ , _ 2 , f e X. n-1 When n = 1 , t h i s i s c l e a r . Assume Pi K„ ^ (J) and l e t i = l 1 i n-1 m e D K~ . Define f = ITL. (f ) € X (since X i s topo-i = l f i X n l o g i c a l l e f t introverted).' Let u e and consider the mean |i © m (Lemma 1 .3.1.(b)). We claim that n l_ © m e fl K- . •i=l x i For 1 < i < n-1 , i f cpe P(G) . mT (f. )cp = m(^cp~*fi) = m(f i) since m e Kf^ . Therefore (m^f-^cp) = (m(f1)«l,cp) f o r any cp e P(G) which spans L-^G) . Hence m L ( f 1 ) = m(f 1)«l and i t follows that i f cp e P(G) , (p © m)(cp * f ± ) = p(m © (cp*f ±)) = p(m L(cp*f ±)) = p(cp*m L(f i)) = u(cp*m(f1) «1) = p ( m ( f 1 ) ' l ) = . n(m L(f 1)) = (u © m)f ± , where we have made use of Lemma 1 .3 .3•(a) and (b) respectively i n the second and t h i r d equality. On the other hand, i f cp e P(G) (|i © m)(cp*fn) = ^(ir^(qp*_ n)) = n(cp*m L(f n)) = p(cp*f) = y(f) (since [X e K f ) = p(m L(f n)) = (\i © m)f n . . n Consequently \x © m € H K„ . By weak* compactness of the i = l i norm closed unit b a l l i n L„(G)* , D[K f : f e X} ^ (j) . Any mean i s t h i s i n t e r s e c t i o n i s necessarily a topological l e f t 18 invariant mean on X . Remark 1.4.3. For discrete groups G , this theorem i s due to E. Granirer and Anthony Lau [17] and the above proof follows the idea i n [17]• The corresponding theorem there i s true f o r semigroups. There i s also another version of the L o c a l i s a t i o n Theorem which i s formally stronger. In fa c t , X has a topological l e f t invariant mean i f f f o r each f e^X and cp e P(G) , there i s a mean (depending on f and cp) on X such that m(i|/*(f - cp#f)) = 0 f o r any t e P(G) . The proof i s almost the same. We consider the sets K(f,cp) = (m : m i s a mean on L r o(G) and m(i|i*(f - cp*f)) = 0 f o r any i|r e P(G)} instead of K f , use compactness to f i n d a mean m e H[K(f,cp):f e X, cp € P(G)}. Then m s a t i s f i e s the condition mL(cp#f) = m^(.f) f o r any cp e P(G) , f e X , and u = m © m i s a topological l e f t invariant mean on X . For. discrete groups, ..this stronger version can also be found i n [17]• 1.5' Topological Stationary L o c a l l y Compact Groups. D e f i n i t i o n 1.5.1. Let X be a l i n e a r subspace of L r o(G) , we say that X i s topological r i g h t stationary i f f f o r each f e X , there i s a net cp e P(G) such that f*cp~ con-verges weak* to a constant function i n L r o(G) . G i s ca l l e d topological r i g h t stationary i f f L r o(G) i s topolo-g i c a l r i g h t stationary i n analogy to the d e f i n i t i o n s i n Mi t c h e l l [ 2 3 , § 3 ] . For any f e L m(G) , define i n analogy to M i t c h e l l [ 2 3 ] , Z R ( f ) a s t h e weak* closure of -[r f : cp € P(G)} (or Z R ( f ) = w*CL(f*P(G)~)) and K R ( f ) = ( a : a r e a l , o>l e Z R ( f ) } . The set K R ( f ) may be empty. In these notations, X i s topological r i g h t stationary i f f K ^ f ) 4 § f o r any f e X . Theorem 1 , 5 . 2 . Let X be a topological l e f t introverted and topological l e f t invariant l i n e a r subspace of -L^G) con-taining the constants. Then X has a topological l e f t i n -variant mean i f and only i f X i s topological r i g h t stationary. In this case f o r any f e X , /3 e K R ( f ) i f f there i s a topological l e f t invariant mean n on X such that |i(f) = jB . Proof: Assume that X i s topological r i g h t stationary, f e X . Then there Is a net cp i n P(G) such that f*cp~ converges weak* to a constant function jS.l i n L (G) . Passing to a subnet i f necessary, we can assume CO that cp^  converges weak* to some mean m i n L r a(G)* by weak* compactness of the set of means i n L M ( G ) * . Consider the mean m © m on L (G) . We show that m © m e K„ CO x ' j_ (defined as i n the proof of Theorem 1.4.2.)'. Observe that f o r any ,g e L-^G) , m L(f )g = m(js~*f) = lim cpa(-^g~*f) = lim (vg~*f,cp ) = lim (f*cp~,g) = (j3»l,g) by Lemma 1.2.1. a A a a , (c). Hence m L(f) = j3 • 1 . Now (m©m)(cp*f) = m(mL(cp*f)) = m(cp*mL(f)) = m(cp*j8'l) = m(j3.l) = m(m L(f)) = (m © m)f f o r any cp e P(G) . Consequently m © m e K^ , . By theorem 1.4.2., X has a topological l e f t invariant mean v e n[Kf : f e X) . Let |i = v © m .., then |_i(f) = (v © m)f = v(m L(f)) = v(j3*l) = 0 . Since v i s a topological l e f t invariant mean, so i s p. . For i f f e X , cp e P(G) , n(cp*f.) = (v © m) (cp*f) = v(m L(cp*f)) = v(cp*m L(f)) = v(m L(f))"" = (v © m)f = n(f) , since m L(f) e X . (Note that we cannot invoke Lemma 1.3.1.(f) d i r e c t l y because the mean v i s topological l e f t invariant on X only, but not necessa-r i l y on L (G)) . 21 Conversely", assume that X has a topological l e f t invariant mean n . Then there i s a net cp^  i n P(G) such that cp - \i i n weak* topology of X* . Let f e X , then (f*cp~,cp) = (|cp~*f,cpa) = cp a(icp~*f) - p(^cp~*f) = p(f) = (n(f)«l,cp) f o r any cp e P(G) which spans • Here we have made use of Lemma 1 . 2 . 1 .(c) and topological l e f t invariance of u . Hence converges weak* to the constant function n ( f ) « l i n L M ( G ) . In f a c t \i(f) e K - ^ f ) . \ • \ Remark 1 .5 .3 . I t i s possible to prove Theorem 1 .5 .2 . (at least f o r the case X = L r a(G)) by a method si m i l a r to that employed i n [Ik] or [23]• For discrete groups G , i t i s re a d i l y seen that w*CL(f*P(G)~) = w*CL CO[r f : a e G] c l = pointwise CL CO{r f : a e G] (see [23, Lemma 3]) . Thus 3. Theorem 1 .5 .2 . reduces to a r e s u l t of Mitchell's [23, Theorem 3] f o r groups. 1.6. Invariant and M u l t i p l i c a t i v e Invariant Means. Let G be a l o c a l l y compact group. X a topolo-g i c a l l e f t invariant l i n e a r subspace of" L r a(G) containing the constants. Consider the semigroup S = P(G) with convolution as m u l t i p l i c a t i o n . Define a map T : X -• m(S) 22 by Tf (cp) = (f,cp) cp e P(G) . Here m(S) i s the Banach space of a l l bounded r e a l valued functions on the semigroup S . Let W be the norm closed subalgebra of m(S) genera-ted by T ( X ) . In analogy to M i t c h e l l [25, §5], we have the following lemma and theorem. Lemma 1.6.1. W i s a l e f t invariant subalgebra of m(S) con-tai n i n g the constants. \ Proof: Since X contains the constants, i t i s clear that W also contains the constants. We now show that T ( X ) i s l e f t i n v a r i a n t . Let f e X , cp,t e P(G) . Then (Tf)(i|r) = Tf(cp*i|r) = (f,cp*$) = = T ( y ) ( i | r ) . Hence ^ ( T f ) = r ( i ^ f ) e T ( X ) since i f e X . Now the l e f t t r a n s l a t i o n operator i n m(S) i s norm continuous, l i n e a r and m u l t i p l i c a t i v e . I t follows that W i s l e f t i nvariant. Theorem 1.6 .2. X has a topological l e f t invariant mean i f and only i f W has a m u l t i p l i c a t i v e l e f t invariant mean (see [14] f o r d e f i n i t i o n ) . 23 Proof: I t i s clear that the map T i s norm continuous and l i n e a r . Moreover ||Tf|| <_ ||f and T(1) = 1 . Let T* : W* -• X* be the adjoint of T . I t i s r e a d i l y v e r i f i e d that T*p. i s a mean on X whenever )i i s a mean on . W . Now - ( T f ) = T ( A f ) by Lemma 1.6.1. Hence ( T * H ) (cp*f) = | i ( T ( ^ f ) ) = u ( ^ ( T f ) ) = | i(Tf) = (T*p)f i f H i S l e f t invariant on W and = •^•cp~ . Thus X has a topological l e f t invariant mean i f W has a l e f t invariant mean (this i s a stronger statement than the s u f f i c i e n c y part of the theorem). Conversely assume that X has a topological l e f t invariant mean 9 . Let cp^  be a net i n P(G) such that cpa -» 8 weak* i n X* . Consider the evaluation f u n c t i o n a l . 6 on m(S) defined by 6 (F) = p(cp ) f o r any F e m(S). pa • a Passing to a subnet i f necessary, we can assume that 6 converges weak* to some m u l t i p l i c a t i v e mean \i i n m(S)* since the set ( 6 ^ : cp- € S} i s weak* dense i n the set of -mu l t i p l i c a t i v e means on m(S) . Then p(Tf) = lim 6 (Tf) a. c^a = lim (Tf)(cp ) = lim (f,cp ) = lim cp (f) = 0(f) f o r any a a • a .. • • f e X . Hence H ^ T f ) ) = n ( t ( ^ f ) ) = Q(l^f) = 9(f) = u ( T f ) 24 f o r any f e X . Since the l e f t t r a n s l a t i o n operator i n m(S) i s continuous, l i n e a r and m u l t i p l i c a t i v e , i t follows that u i s a m u l t i p l i c a t i v e l e f t invariant mean on W . This proves the theorem. 25 CHAPTER II TOPOLOGICAL ALMOST CONVERGENCE In this chapter, we s h a l l prove a theorem on topological l e f t almost convergence. I t i s an analogue of a r e s u l t i n [15] on l e f t almost convergence of bounded functions on semigroups. 2 . 1 . Topological Left Almost Convergence. D e f i n i t i o n 2 . 1 . 1 . Let G be a l o c a l l y compact group such that L r a(G) has a topological l e f t invariant mean. A function . f e L c o(G) i s said to be topological l e f t almost convergent ( t . l . a . c . ) to j3 i f and only i f p(f) = /3 fo r every topological l e f t invariant mean (TLIM) \i on L_(G) . Let K denote the l i n e a r subspace of a l l functions i n n L (G) which can be represented as _ (f. - cp.#f. ) where 0 0 i — 1 1 1 f± e L_(G) , cp± e P(G) , 1 < i <_ n , n f i n i t e . 2.2. C r i t e r i o n f o r Topological Left Almost Convergence. Lemma 2 . 2 . 1 . Let G be a l o c a l l y compact group, m a net.of means on L (G) such that lim )\l m. - m 11 =0 f o r any a T cp e P ( G ) . Then for any f e K (closure taken i n norm topology of LjG)) , lim ||(ma)Rf||aj = 0 where (m a) R i s a the topological r i g h t introversion of m . Proof: Consider f i r s t functions i n K of the form f - cp*f , f e L o t ( G ) , cp e P ( G ) . We have, i f g e L . ( G ) , IK^R^ " = l m a ^ f " c P * f ) * S ~ ) l = |m a(f*g~ - f P * ( f * g ~ ) ) | = |ma(f*g~) - ma[lq(f*g~))\ (where * =^cp~ e P ( G ) ) = |(ma - ^ m a ) ( f * g ~ ) | < ||ma - •t|ma|| ||f !lra||g||^ . Here we have made use of Lemma 1.2. (f) i n the second equality. Hence l l ( m a ) R(-f ~ cP * f ) l l 0 0 _. l l ma " * J m c J N f L - 0 . I t follows that lim ll(m a) Rf||„ = 0 ct for any f e K . I f f e K~ , l e t e > 0 , there i s some f e K such that ||f - f || < e . Then ||(m )„f|| < e + ! ! ( m a ) R f 0 l l 0 0 • Consequently lim ||(ma)Rf ||„ = 0 f o r any Ct f e K~ . Theorem 2.2.2. Let G be a l o c a l l y compact group for which 2 7 L^CG) has a topological l e f t invariant mean and C the constants i n L (G) . Then: (a) C © K~ i s the space of a l l t . l . a . c . functions i n . L m ( G ) . f being t . l . a . c . to j8 i f f f e j B . l + K~ . (b) I f f i s topological l e f t almost convergent to jS and m^  i s any net of means on L r a(G) such that lim |k*m - m || = .0 f o r any cp e P(G) , then a * lim ||(ma)Rf - /3-lL = 0 . CL s (c) f i s topological l e f t almost convergence to j6 i f and only i f j8»l belongs to the norm closure of P(G)*f i n LjG) . Proof: (a) Suppose f i s t . l . a . c . to zero, we claim that f e K~. Otherwise there would e x i s t some m e L (G)* such that CO ' - m(K ) = 0 and m(f) ^ 0 by Hahn-Banach Theorem. Hence m i s topological l e f t i n v a r i a n t . Write m = c^m^ - Cgmg where m^ , m^  are means on L r a(G) and l e t 9 be any TLIM on L j G ) (which exists by assumption), we have m' = 9 © m = 0^(9 0 m^ ) - c 2(9 © m2) by Lemma 1.3.1.(g) where both 9 © m^  and -9 © m2 are TLIM on L j G ) by Lemma 1.3.1.(f). Hence m(f) = 0 which i s a contradiction. Therefore f e K~. 28 On the other hand, i f m i s . any TLIM on L (G) , c l e a r l y m annihilates K~ . Thus K~ = . f f e L (G) : f i s t . l . a . c . to zero} . Now f i s t . l . a . c . to j8 i f f f - j3»l i s t . l . a . c . to zero, i f f f e j3'l + K~ . To show that the sum C © K~ i s di r e c t , l e t j8»l e K and choose any TLIM m on L^G) . Then j3 = m(j3«l) = 0 . This proves (a). (b) Let f be t . l . a . c . to j3 and ma a net of means on L (Gx) such that lim |U*rn - m l| = 0 f o r any a r cp e P(G) . Then f - JB• 1 e K~ . By Lemma 2 . 2 . 1 . lim ||(m a) R(f - JB.I)!^ = 0 . But (m a) R(/3.l) = jS-1 . a Hence lim l|(m ) R f - j B - l ^ = 0 . a (c) F i r s t we observe that i f f,g e L (G)~ and g e P(G)*f, then m(f) = m(g) f o r any TLIM on L w(G) . There-fore i f ]8'1 belongs to the norm closure of P(G)#f , then j8 = m(/3.l) = m(f-) f o r any TLIM m on L w(G) , or f i s t . l . a . c . to j3 . Conversely, i f this i s the case, l e t cpQ be a net i n P(G) such that lim ||cp*cp - cp || '•= 0 f o r any cp e P(G) [18, -Theorem a a a ' 2 . 4 . 2 . and 2 .4.3. ] • Since. cp*cpQ = £*(<Pa) by Lemma 1.3 .1 . (c) ,we have l i m IU*(qpQ) - q>J| = Q f o r any a ^ 29 cp e P(G) . But f i s t . l . a . c . to j3 , hence lim l|(cp ) R f - = 0 by (b). Now (cp ) f = I f . a For i f g e L^G) , ((cp a) Rf)g = cp a(f*g~) = (f*g~,cp a) = (| cP~*f,g) or (cp a) Rf = I f e P(G)*f . Consequently a j3• 1 belongs to the norm closure of P(G)*f . This completes the proof of the theorem. Remark 2 .2 .3 . (a) When G i s discrete, t h i s i s p r e c i s e l y Theorem I I I . 7 . of Granirer [15] r e s t r i c t e d to groups (see also Witz [35, §4]). The above proof follows the same idea there. (b) Let W(G) be the space of a l l weakly almost periodic functions on G (see [18, §3.1.] f o r d e f i n i t i o n ) . It-i s well-known that W(G) has a unique invariant mean which i s also topological invariant, whether L m(G) has a TLIM or not. (For existence and uniqueness, consult Ryll-Nardzewski [31] and Deleeuw and Glicksberg [7]). Hence i f L w(G) has a TLIM , W(G) c C © K - -because every function i n W(G) i s t . l . a . c . I f L (G) does not have a TLIM , then K~ = L (G) and co 3 co ' t r i v a l l y W(G) cr L M(G) = C~+ K~ although the sum i s no longer d i r e c t . 30 (c) I f the group G i s compact, then L m(G) = C © K~ . For then L r o(G) has a unique topological l e f t i n -variant mean, namely the l e f t Haar i n t e g r a l X , and K" = (f € L w(G) : X(f) = 0} . Therefore K~ and (1) together span L w(G) and L r o(G) = C © K~ . \ \ 31 CHAPTER III TOPOLOGICAL INTROVERTED SPACES In chapter 1, we Introduced the concept of topo-l o g i c a l l e f t introverted l i n e a r subspace of L r o(G) and proved the "Localisation Theorem" f o r such spaces. In the present chapter, we study the properties of these spaces i n some d e t a i l . \ 3 ' ! ' Equivalent D e f i n i t i o n s . Lemma 3»1 -1• For any f e L w(G) , w*CL(f*P(G)**) = w*CL CO(r af : a e G} . Proof: Let a 6 G and l e t cpQ be an approximate i d e n t i t y i n L 1(G) . I f f e L w(G) , then f*<P~ - f weak* i n L_(G) by Lemma 1 .2 .3 . (b). Since r i s w*-w*_ c l continuous, we have (f*qO ~* £ = r f weak* i n L (G) . ' v era a a » v ' But (f*cp~) a = f*(cp~) a = f * ( _ 1 ^ a ) " € f*P(G)~ . Hence a r f e w*CL(f*P(G)~) which i s weak* closed and convex. Therefore w*CL C0|> f : a e G) c w*CL(f*P(G)~) . 32 Conversely, l e t cp e P(G) , then there i s a net n a a (2) of f i n i t e means co = S j3. 6 v ' such that cp - cp a i = l 1 a" a i n weak* topology of CB(G)* . I f g e (G) , then n Cfc (f*qp~,g) = (^ -g~*f,<p) = lim cp (^g~*f) = lim J ' p"(W**f) (a°) a a a A a i = l . a x• = lim 2 f(ya^)g(y)dy = lim (^  S 0^ r a f,g) . (Here a i = l a i a i = l we have made use of Lemma 1.2.1.(c) and [19, Corollary 20.14 and Theorem 20.2.1). Hence f*cp~ e w*CL C0(r f : a e G} which i s weak* closed and convex. Hence w*CL(f#P(G)~) c w*CL C0[r f : a e G)..'. 3. Remark 3.1.2. The i n c l u s i o n w*CL(f*P(G)~) c w*CL C 0 [ r f : a e G} was suggested by Professor M..M. Day (written communication). The above proof i s d i f f e r e n t from h i s . The author wishes to thank Professor Day f o r th i s important suggestion. ( 2) 6„ i s defined by 6 f = f (a) f o r any a a f e CB(G) . . 33 Theorem 3 . 1 . 3 . Let X be a l i n e a r subspace of L r o(G) . Then the following conditions on X are equivalent: (a) X i s topological l e f t introverted. (b) w*CL(f*P(G)~) c X f o r any f e X . (c) w*CL CO(r f : a e G] c X f o r any f e X . Proof: \ Suppose that X i s topological l e f t introverted and f e X . I f f 1 e w*CL(f*P(G) ~) , then f*cp~ -» f± weak* i n L^G) f ° r some net cpa i n P(G) . Passing to a subnet i f necessary, we can assume that cpa converges weak* to some mean m i n L m ( G ) * . Hence ^a^L^^ ~* m L ^ ^ * n weak* topology of L r o(G) by Lemma 1 . 3 . 3 .(d). But (cp a) L(f) = r ^ f = f*cp~ - f 1 i n weak* topology 'of LjG) . Therefore f ] L = m L(f) e X . Consequently w*CL(f*P(G)~) <= X fo r any f e X . Conversely i f this i s the case, l e t m be any "~ mean i n L (G)* , f e X . Then cp - m weak* i n L (G)* for some net cp i n P(G) . By same arguments as above, f*cp~ = («PQ)Lf - ^ ( f ) weak* i n L j G ) . Hence m L(f) e w*CL(f*P(G)~) c X and X i s topological l e f t introverted. Thus (a) and (b) are equivalent. I t follows from Lemma 3.1.1. that (b) and (c) are equivalent. Remark 3»1«4. The equivalence of (a) and (b) i s a topological analogue of a r e s u l t i n [17, Lemma 2]. Properties of Topological Introverted Spaces. Lemma 3«2\l. If X i s a topological l e f t introverted l i n e a r subspace of L (G) , then so i s X (norm closure). Proof: Since the map m^  : L a j(G) -• L r o(G) i s continuous, this lemma i s obvious. Lemma 3 • 2.2. If X i s a topological l e f t Introverted l i n e a r subspace of L (G) , then X i s topological r i g h t i n v a r i a n t . Proof: By Lemma 1.3»3«(b), f*<p~ = r ^  - «PL(f)- f o r any f e ^ ( G ) , ? e P(G) . Therefore X i s topological r i g h t invariant i f i t i s topological l e f t introverted. 35 D e f i n i t i o n 3.2.3* Let X be a norm closed topological l e f t invariant subspace of L^G) . Let B L = (f e X : m^(f) e X f o r any mean m on L r o(G)} ( i n analogy to the d e f i n i t i o n of i n Rao [29]). Theorem 3 .2 ,4 . B ^ i s a norm closed topological l e f t invariant l i n e a r subspace of X . It i s the unique maximal topolo-g i c a l l e f t introverted l i n e a r subspace of X . Proof: Clearly B ^ i s a l i n e a r subspace of X . Assume f e B L and l e t cp e P(G) . Then mL(cp*f) = cp*mL(f) e P(G)*X c x by Lemma 1 . 3 * 3 * ( b ) , d e f i n i t i o n of B L and topological l e f t invariant' of' X . Hence cp*f e B ^ or P ( G ) * B L C B l . Since the operator : L { o(G) -» L j G ) i s norm continuous and X i s norm closed, B ^ i s also norm closed. To show that B L i s topological l e f t introverted, we f i r s t observe that, i f H , v are means on L (G) , then * CO 1 ' •* CM- © V ) L = \ 0 v L • For i f f e L r o(G) , g e L^G) , we have (|i 0 v ) L ( f ) g = (n © v ) ( | g * f ) = |i(v L ( | g ~ * f ) ) = n(^g~*v L(f)) = P L ( v L ( f ) ) g = ( i ^ o v L ) ( f ) g . Hence 36 (u © v ) L = P L o v. L Consequently i f f e B. L > v are means on L (G) , then |X.(v T(f)) = (|i © v ) T f e X . There-topological l e f t introverted. F i n a l l y suppose B i s any topological l e f t introverted subspace of X then c l e a r l y B c B^ . Hence B^ i s unique and maximal. 3»3. Examples of Topological Left Introverted Spaces. \ Lemma 3 . 3 . 1 . fore v^(f) e B-^  f o r any mean v on ^ ( G ) , or B T i s For any m e L (G)* , f e L (G) , s e G We commute. Proof: If g e L^G) , we have U s ( m L ( f ) ) , g ) = ?' (m L(f), _ l g) = m(|( _ l g ) ~ * f ) = m(.A(s)(|g~) s*f) = {v^U^) ,&) s s by [19, Remark 20 .11. ] . Hence £ s ( m L ( f ) ) = m L ^ s f ^ * Theorem 3 .3 .2 . UCBr(G) i s topological l e f t introverted. Proof: 37 Let s a - s i n G , f e UCBr(G) . Then \\l f - I f || - 0 . Hence i f m i s any mean on L w(G) , s a s K ( m L ^ f ) ) " * s ( m L ( f » , l = K^s f ) - ^V^L a a lk-U_ f - V^L ^ l^ s f " V^L'"* 0 b y L e m m a 3 - 3 . 1 . and a a Lemma 1 . 3 . 3 .(a). Hence m^f) e UCBr(G) . Lemma 3»3«3» (Granirer) \ Let f e UCB (G) and m a mean on L (G) . \ ' • C O . ' Then m T(f) = m,(f) where m. i s the l e f t i n troversion L i l \, of m (see Day [K, §10] or Rao [29] f o r d e f i n i t i o n ) . Proof: I t i s well-known that UCBr(G). i s l e f t i n t r o -verted. In other words m,(f) e UCB (G) fo r any v r f e UCB r(G) and.any mean m on L r o(G) ( e s s e n t i a l l y because I and m commute). Consider f i r s t the case when S \, m = cp e P(G) . We have f o r any g e L 1(G) , (cp (f),g) = J ( s f , c p ) g ( s ) d s = J ( f , _ 1cp)g(s)ds = J J f(t ) c p(s' 1t)dtg(s)ds = s (f,g*cp) = (f*cp~,g) = (cp L(f),g) by Fubini's Theorem and Lemma 1 . 2 . 1.(c). Hence ^ ( f ) = W^?) • In .general, l e t cp be a net i n P(G) such that cp -* m in. weak* topology 38 of L w ( G ) * . Then (<P a) L f - , ' m r J ( f ) l n weak* topology of LjG) by Lemma 1.3.3- ( d ) . But ( c p a ) L f = (va)which converges to m f pointwise on G . Since f e UCB (G) , i t i s e a s i l y v e r i f i e d that (cp ) f i s a norm bounded equicontinuous family of functions on G . By [21, Theorems 7.14 and 7*15], ^a) £ c o n v e r S e s t o m i ^ ) uniformly on compacta and a f o r t i o r i , (cp ) f converges weak* to m-(f) i n L £ o(G)^ . Consequently . ^ ( f ) = m ^ ( f ) • Remark 3 > 3«4. Incidentally, this lemma gives another proof that UCB r(G) i s topological l e f t introverted. Theorem 3*3.5. The space W(G) of a l l weakly almost periodic functions on G i s topological l e f t introverted. Proof: I t i s well-known that W(G) <=. UCBr(G) (Eberlein -[9]) and that W(G) i s l e f t introverted (Deleeuw and Glicksberg [7])* The theorem i s therefore on immediate < consequence of the preceeding lemma. 39 Theorem 3*3.6. Let G be a l o c a l l y compact group such that L w(G) has a topological l e f t invariant mean, K the l i n e a r subspace of L W ( G ) defined i n §2.1.1. Then C © K~ i s a topological l e f t introverted and topological l e f t invariant l i n e a r subspace of ^ ( G ) containing the constants. Proof: By d e f i n i t i o n C © K~ contains constants. I t i s topological l e f t invariant because C © K~ i s p r e c i s e l y the topological l e f t almost convergent functions i n L ' ( G ) by Theorem 2.2.2.(a). To show that C © K~ i s topological •left introverted, consider f i r s t functions of the form f - cp*f with f e L E O ( G ) , cp e P ( G ) . We have i f m e LjG)*, m L(f - cp*f) = m L(f) - cp*mL(f) where m L(f) e L W ( G ) , by Lemma 1.3.3.(b). I t follows that mL(K) c K and hence m^(K ) c K by continuity. Consequently Tn^(C © K~) c C © K" . Remark 3«3»7. (a) The space C © K may contain functions which are not continuous. In f a c t the set (f - cp#f : f e L M ( G ) } i s not included i n CB ( G ) f o r any cp , provided that 40 L m ( G ) / CB(G) (since cp*f e CB(G)). (b) I f L M(G) has more than one topological l e f t invariant mean (for example, when G i s separable, non compact and amenable as a discrete group. See [16, Theorem 1, p. 124 and Remark l( a ) p. 118]). Then not every function i n L^G) can be topological l e f t almost convergent. Hence C © K i s a proper l i n e a r subspace of L (G) . C O (c) Consider G = R the additive reals with u s u a l topolo-gy. Let E = U [2n,2n+l) and F = U [2n-l,2n). n=-<» n=-°o Then X E,X p € L j G ) , x E + X p = 1 while ^(Xg) = X p ( i . e . Xp i s the l e f t translate of x E hy one). Hence both x E and x p are topological l e f t almost convergent to .1/2 . In thi s case C @ K~ even con-tains a c h a r a c t e r i s t i c function. 41 CHAPTER IV APPLICATIONS In this chapter, we indicate how we can obtain two known results from the "Localisation Theorem" and Its consequences. 4.1. Invariant Means on W(G).. Lemma 4.1.1. Let ¥(G) be the space of a l l weakly almost periodic functions on G and f e L (G) . Then the following statements are equivalent: (a) f e W(G) (b) [f#cp~ : cp e P(G)} i s weakly r e l a t i v e l y compact i n CB(G) (c) [cp*f : cp e P(G)} i s weakly r e l a t i v e l y compact, i n CB(G) . Proof: By Lemma 3.1.1., w*CL(f*P(G)~) = w*CL C0(r f : 3. a e G} . Using t h i s i d e n t i t y and the f a c t that the map f -» f ~ i s a w* homeomorphism of ^((J) onto i t s e l f , we 42 have w*CL(P(G)*f) = w*CL C O [ i f : a e G} . Also since the weak topology i n CB(G) i s stronger than the weak* topology of L M(G) r e s t r i c t e d to CB(G) , we clearly.have wCL(f*p(G)~) c w*CL(f*P(G)~) and wCL CO[r af : a e G} c w*CL CO[r f : a e G} . a Suppose now f e W(G) , then wCLfr f : a e G) i s weakly compact. By Mazur's Theorem [8, Theorem 6, p. 416], wCL CO(r f : a e G) i s weakly compact, hence w* compact and a f o r t i o r i weak* closed. I t follows that wCL(f*P(G)~) c w*CL(f*P(G)~) = w*CL C O ( r f : a e G} = wCL CO(r f : a e G} which i s weakly compact. Therefore... 3 wCL(f*P(G)~) i s also weakly compact and (a) implies (b). If we assume that wCL(f#P(G)~) i s weakly compact, the same arguments ensure that wCL CO(r f : a e G) and a 3 f o r t i o r i wCL(r f : a e G] i s weakly compact. Hence (b) 3 also implies (a). That (a) and (c) are equivalent can be proved s i m i l a r l y , using the i d e n t i t y w*CL(P(G)*f) = w*CL CO[lf ~i 3 a e G) . Lemma 4 .1 .2 . W(G) i s topological l e f t i n v a r i a n t . 43 Proof: We f i r s t observe the following f a c t i n general topology. Suppose T : X -* Y i s a continuous map between Hausdorff spaces A c X and A i s compact. Then T(A~) = T(A)~ . By continuity, i t i s clear T(A~) C T ( A ) ~ . Since T(A") => T(A) and T(A~) i s compact (a f o r t i o r i closed), we have T(A~) => T(A) . Hence equality. Now suppose f e W(G) , cp e P(G) . Let X = Y = CB(G) with weak topology. T = r , A = [r f : a e G} . 'Then wCL(r a(<t f ) : a e G} = wCLU ( r a f ) : a € G] = £„(wCL{r.f : a € G}) which i s weakly compact by the above observation. Hence ^ ( ^ ( G ) ) c W(G) f o r any cp e P(G) . Theorem 4.1.3. (Ryll-Nardzewski) For any l o c a l l y compact group G , W(G) has a topological l e f t Invariant mean. Proof: Take f e W(G) , then K = weak closure of (f*cp~ : cp e P(G)} i s weakly compact by Lemma 4.1.1. Since (f*cp~) = f*(cp~) = f * ( ncp)~ i t follows from the weak-a a _ — i -a weak continuity of r_ that r.(K) c K f o r any a e G . Now consider the semigroup (r : a e G] of weakly continuous 44 a f f i n e mappings acting on the weakly compact subset K of the Banach space CB(G). . This action i s d i s t a l with respect to the norm topology of CB(G-) (0 belongs to the norm closure of [r f- : a e G} I f f f = 0). By R y l l -3. Nardzewski's f i x e d point theorem (see [18, Theorem 3»3»1«] or [31]), there i s some jS e wCL[f*cp~ : cp e P(G)} C w*CL{f*cp~ .: cp e P(G)) such that r 8 = 6 f o r any a e G . Hence fl i s a constant function on G . Consequently G is topological r i g h t stationary. Now W(G) i s a topological l e f t introverted and topological l e f t invariant (Theorem 3»3»5« and Lemma 4 .1.2. ) l i n e a r subspace of L W(G) containing constants. Therefore W(G) has a topological l e f t invariant mean by Theorem 1-5.2.. 4.2 . Invariant Means on UCB (G) . Lemma 4 . 2 . 1 . Let T denote the topology of uniform convergence on compacta. For each f £ UCB r(G) , w*CL(f*P(.G)~) = T CL C0[r f : a € G]\. c a Proof: We observe that i f f e UCB (G) , then ^5 CO[r f : a e G} i s a norm bounded equicontinuous family of functions on G by a straight forward argument. The same i s true f o r the family 'P = T CL CO(r f : a e G) by [21, Theorem 7.14.]. Let cp e P(G) . Then there i s a net n a cp = S jS. 6 of f i n i t e means such that cp -» cp i n CX . n ~L A CX CC 1 = 1 a i weak* topology of CB(G)* . Now f-x-cp~(x) = f (xy)cp(y)dy = n J a a ( f,cp) = lim cp ( f ) = lim S fr! r n f(x) . Hence there i s \ a a i = l a. \ l a net f = S fi" r rt f e CO{r f : a e G) such that 1=1 a^ f Q f*cp~ pointwise and hence i n the TQ topology by [21, Theorem 7 .15. ] . Hence f*P(G)~ c F which i s T compact by Ascoli's Theorem [21, Theorem 7.-I7.]. Since the family F i s norm bounded, the T topology i s stronger than the w* topology of L r o(G) r e s t r i c t e d to F . Since they are both Hausdorff, they must coincide on F . In pa r t i c u l a r F i s weak* closed i n ^ ( G ) . Therefore w*CL(f*P(G)~) c T CL CO(r f : a e G] . On the other hand, by Lemma 3,1.1. w*CL(f*P(G.)~) = w*CL CO[r f : a e G] . Since the family CO[r„f : a e G} a a i s bounded i n norm, T CL COfr.f : a e G} c w*CL CO(r f : c a a a e G] = w*CL(f*P(G)~) . Consequently 46 T CL CO[r f : a € G} = w*CL(f*P(G)~) . c a \ \ / / Theorem 4 .2 .2 , (E. Granirer and Anthony Lau [17]) UCBr(G) has a l e f t invariant mean i f f f o r each f e UCB (G) , T CL CO[r f : a e G] contains a constant i ? C c t function. In th i s case 6 .1 e T CL CO{r f : a e G) i f f c s. there i s a l e f t invariant mean |i on 'UCB^G) such that H(f) = S . \ Proof: ; It was proved i n [18] that TLIM and LIM on UCBr(G) are the same. By preceeding lemma, T CL C0{r of : a e G] = w*CL(f*P(G)~) . The present theorem reduces to Theorem 1 .5 .2 . with X = UCBr(G) . 47 CHAPTER V INVARIANT MEANS AND FIXED POINTS Let G be a l o c a l l y compact group , G Is said to have the f i x e d point property (f.p.p.) i f whenever G acts a f f i n e l y on a compact convex subset S i n a separated l o c a l l y convex space ( l . c . s . ) E with the map G x S -» S (denoted by (x,s) -» T (s)) j o i n t l y continuous, there i s a point s^ e S such that T x ( s Q ) = s Q f o r any x e G . N. Rickert [30] has proved that G i s amenable i f f G has the above f i x e d point property. In t h i s chapter, we consider other (analogous) fi x e d point properties f o r actions of the group algebra L^(G)' and the measure algebra M(G) (of a l l bounded regu-l a r Borel meausres on G) . I t turns out that they are both characterisation of amenability of G . 5.1. Actions of L X(G) and M(G) . D e f i n i t i o n 5*1.1. Let G be a l o c a l l y compact group, k^(G) i t s group algebra (with convolution as m u l t i p l i c a t i o n ) , M(G) i t s measure algebra, E a separated l o c a l l y convex space. An action of L n(G) on E i s a homomorphism T of L n(G) 48 into the algebra (functional composition as mu l t i p l i c a t i o n ) of a l l l i n e a r operators i n E . Thus we have a map L-, (G) X E - E (denoted by (g,s) - T (s)) such that: . g (Al) For each g e L 1 (G) , the map s - T (s) i s l i n e a r (A2) For each s e E , the map g -* T (s) i s l i n e a r (A3) For any & 1 , S 2 e L^G) , T ^ = T ^ o T ^ . S i m i l a r l y we can define an action of M(G) on E . An action of L-^G) (M(G)> on E i s c a l l e d separately continuous i f the map L^G) x E - E (M(G) X E - E) i s separately continuous. Let S be a compact convex subset of E , we say that S i s P(G)-invariant under the action L X(G) _x E -» E i f T (S) c S f o r any cp e P(G) . Let M (G) be the p r o b a b i l i t y measures i n M(G) (|-i e M (G) i f f n > 0 , Hull = 1). We say that S i s M (G) -invariant under the action M(G) X E -• E i f T (S) c S f o r any |i e M (G) . In the f i r s t case S induces an action of the convolution semigroup P(G) on the compact convex subset S (as a f f i n e maps now), s t i l l denoted by T : P(G), x S - S. In the second case, S induces an action T : M (G) x S -• S of the convolution semigroup -M (G) on the :compact convex 49 subset S . We s h a l l r e f e r them as the induced actions. Theorem 5.1*2. Let G be a l o c a l l y compact group. Then the following conditions on G are equivalent: (a) G i s amenable. (b) For any separately continuous action T : L-^G) x E - E of L 1(G) on a separated l o c a l l y convex space E as l i n e a r operators i n E and any compact convex P(G)-invariant subset S of E , the induced action T : P(G) x S -• S has a f i x e d point. (c) For any separately continuous action T : M(G) X E -• E of M(G) on a separated l o c a l l y convex space E as l i n e a r operators i n E and any compact convex M (G)-invariant subset S of E , the induced action T : M (G) x S: - S has a-fixed point. Proof: Assume that G i s amenable. Then there i s a net cp i n P(G) such that j|cp*cp_, - cp |L -» 0 f o r any Ct Ct Ct J_ cp e P(G) , by [18, Theorems 2 .4 .2 . and 2 . 4 . 3 . ] . Let T : L^(G) x E -» E be a separately continuous action, S c E compact convex and P(G)-invariant. Consider T (s) where s e S i s a r b i t r a r y but f i x e d . By compactness of S , we can assume T (s) -• S q i n S , passing to a . a subnet i f necessary. We claim that s Q i s the required f i x e d point of T : P(G) x S -* S . For i f cp e P(G) T (s ) = T (lim T (s)) = lim T (T (s)) = lim T (s) = lim [T m (s) + T (s)} = lim T ( S ) = S , by cp*cp - cp v ' cp cp x ' o separate continuity of T :L-^(G) xE.-»E , l i n e a r i t y of • g -» T g ( s ) and t n e f a c t that cp*cpa - cpQ -> 0 i n norm topo-logy of ^L^G) . Hence (a) implies (b). Conversely, l e t E = L r o(G)* with weak* topo-logy and define T : L-, (G) x E -* E by T (u) = . J- g g Since I = I o I , i t i s clear that T defines an g l * g 2 S 2 S X J action of L^(G) on E ( l i n e a r i t y i n g or p i s obvious). We claim that T i s separately continuous. Suppose g n -* g i n L X(G) , then f o r fix e d u e' L ^ G ) * and f e L j G ) , we have |T (u)(f) - T ( u ) ( f ) | = f ) - < IMIIIg., ~ slUlfll - 0 • Hence g - T (u) i s continuous f o r n J. °° g fix e d n . Evidently the map \i .-* T (n) i s continuous f o r g f i x e d g since I* i s w*-w* continuous. Now take S S to be the set of a l l means i n L (G)* . Then S i n w*-CO ' compact convex and i s P(G)-invariant under the separately 51 continuous action T : L^G) x E -• E . Consequently, i f we assume (b), the induced action T : P(G) x S -• S must have a f i x e d point which i s necessarily a topological l e f t Invariant mean on L^G) .' Therefore (b) implies (a). To prove that (a) i s equivalent to ( c ) . Assume that G i s amenable and consider any separately continuous action T : M(G) x E - E of M(G) on E . Since L-^G) i s a subalgebra of M(G) , the r e s t r i c t i o n T : Ii-j^G") x E -» E i s a separately continuous action of L^(G) on E . If now S i s a compact convex M q ( G ) -invariant of E , then S i s P(G)-invariant since P(G) e M Q(G) . The equivalence of (a) and (b) ensures that the action T : P(G) x S - S has a f i x e d point which i s necessarily a f i x e d point of the action T : M Q(G) x S S since P(G) i s an i d e a l i n M Q(G) (as convolution semi-groups). Therefore (a) implies ( c ) . Conversely, assume ( c ) . For each p e M(G) , define a map l : LjG) - L j G ) by I (f) = u~*f [19, Theorem 20.23 :]. t i s bounded-linear [19, Theorem 20.12] ' u ; n~ i s defined by H~(E) = n(E ) f o r any Borel set E . and tlt = I o i ,K*> Let E = L (G)* with weak* topology. Let T : M(G) X E E be defined by T (m) = l*m \s e M(G) , m e L o o(G)* . Then T ' Is a separa-t e l y continuous action of M(G) on E . Let S be the set of a l l means on L (G) again. Then S i s weak* compact convex and M -(G)-invariant under T : M(G) x E -* E. Therefore the induced action T : M (G) x S - S has a o fix e d point which i s necessarily a topological l e f t i n -\ \ variant mean on L^G) . Hence (c) implies (a). This completes the proof of the theorem. Remark 5.1.3• In the proof of the preceeding theorem, we have never used the f a c t that the map T : L-^ G") x E -• E where (g,s) -• Tg(s) i s l i n e a r i n s (for f i x e d g)~. This i s not s u r p r i s i n g since the l i n e a r i t y of the map (g,s) -> T (s) S i n g (for f i x e d s) and the condition that T „ = g 1 * g 2 T o T (g.,gp e L (G)) together imply that T i s ^2 l i n e a r on the subspace E, = [T (s) : g e L-,(G)}- of E This i s a consequence of Pubini's Theorem. 53 (for each f i x e d s ) . This follows from continuity of T g and the f a c t that ^ ( ^ ( a ) + O g T ^ s ) ) = ^ ( T ^ ^ f . ) ) = T ( s ) = T ( s ) = g * ( a 1 g 1 + a 2 g 2 ) v ; a 1g*g 1 + a 2g#g 2 v ; a-,T „ (s) + a QT „ (s) = a,T (T (s)) + cuT (T (s)) f o r 1 g*g^ y 2 g*g 2 1 1 g g]_ 2 g v g 2 v / y any , a 2 r e a l any Z-]_>Z2 e L]_(G) • The space E^ i s c a l l e d the orb i t ' o f s . Silverman's Invariant Extension Property. D e f i n i t i o n 5.2.1. Let G be a l o c a l l y compact group, L^(G) i t s group algebra. Let M be an abstract M-space with unit e (see [22] f o r d e f i n i t i o n ) . A r i g h t action of L^G) on' M i s an antihomomorphism of L^(G) into the algebra of l i n e a r operators i n M (denoted by T : L-^(G) x M -• M where (g,s) -» T _ ( s ) ) ^ ) S U c h that T : M -• M i s a posit i v e l i n e a r operator and Tq,( e) = e each cp e P(G) . The-(5) any one variable i f the other i s f i x e d and • T = T oT J g-,*go go g This mean that (g,s) -» To.(s) i s l i n e a r i n o ^1*S2 ^2 . * i f o r any g- L Jg 2 € L-^G) . action i s called.separately continuous i f the map T : L-^G) x M -» M i s separately continuous (see [27] f o r d e t a i l s ) . I f A i s a l i n e a r subspace of M with e e A , we say that v e A* i s a mean on A i f ||v|| = v(e) = 1 . A i s c a l l e d P(G)-invariant under the right action T i f T (A) c A f o r any cp e P(G) . Let M* be the continuous dual of M and l e t E be the separated l o c a l l y convex space M* with w* topology* The r i g h t a c t i o n L^(G) x M => M i n d u c e s a n action L-^G) x E - E (i n the sense of §5.1.1.) such that (g,|i) - f o r any g e L (G) , [i e M* . It i s r e a d i l y seen that this induced action i s separately continuous whenever the ri g h t action i s . Theorem 5*2.2. Let G be a l o c a l l y compact group such that L £ o(G) has a topological l e f t invariant mean. Suppose T L 1(G) x M -• M i s a separately continuous right action of _ L^(G) on an abstract M-space M with unit e and A i s any P(G)-invariant subspace of M containing e . I f v e A* i s a mean on A such that v(T (s)) = v(s) f o r any cpeP(G)-, s e A . Then there i s a mean n e M* 55 extending v such that U-(T (s)) = n(s) f o r any cp e P(G), s e M . Conversely, any l o c a l l y compact group G with this "invariant extension property" i s amenable. Proof: Consider the induced action L-^G) x M* - M* which sends (g,p) into T*[i . As remarked, this action S i s separately continuous when M* has w* topology. Let S = e M* : u i s a mean on M and n extends v} . ' S 5 / (J) by Hahn-Banach Theorem. In fac t S i s a w* closed convex subset of the unit b a l l i n M* and i s therefore w* compact. We claim that T*(S) c S for any c p e P(G) . Let | i e S , cp e P(G) . Since T^ : M - M i s posit i v e l i n e a r and T^(e) = e , we have ||T (s)|| <_ ||s|| f o r any s e M . Therefore HT^ H < 1 " and ||T*u|| < ||T < Hu|| = 1. But T*n(e) = p(T (e)) = u(e) = 1 . Consequently ||T*u|| = T*p(e) = 1 or T*u i s a mean on M . Also T*p extends v since i f s e A , then T*n(s) = t-i(T (s)) = 1 cp cp v(T (s)) = v(s) ( r e c a l l that T (A) c A). Therefore S i s P(G)-invariant under the action L 1(G) X M* - M* . By Theorem 5.1.2., there i s some u e S such that T*u = (i ' o cp' o o 56 f o r any cp e P(G) . H Q i s the required extension of v . Conversely, i f G has this "invariant extension property", we can take M = L (G) and define a ri g h t action L-^G) X M - M by T g ( f ) = ^g~*f = l&f , f e LjG) , g e L-^G) • This action i s c l e a r l y separately continuous. Also choose A to be the constants i n L (G) and define C O v ( j 3*l) = j8 f o r any j8»l e A . Then A i s obviously P(G)-invariant under the ri g h t action- L-^G) x M -* M and v i s \ \ a mean on A s a t i s f y i n g v(T (f)) = v(f) , cp e P(G) , f € A . Any invariant extension si of v to M = L^G) is necessarily a topological l e f t Invariant mean on L W ( G ) . Remark 5»2.3» The above characterisation of amenability of l o c a l l y compact group i s an analogue of a theorem of R. J. Silverman [32, Theorem 15]. Right actions of M(G) on abstract M-space with unit can also be defined and a theorem of sim i l a r type can be proved with obvious - modi-f i c a t i o n s . 57 BIBLIOGRAPHY 1. R. Arens, The adjoint of a b i l i n e a r operator, Proc. Amer. Math. Soc. 2- (1951) 839-848. 2. N. Bourbaki, Integration, A c t u a l i t i e s S c i . Ind. 1175, Paris (1952). 3. N. Bourbaki, Espaces v e c t o r i e l s topologique, A c t u a l i t i e s S c i . Ind. 1189, Paris (1953). 4. . M. M. Day, Amenable semigroups, I l l i n o i s J. Math. 1 (1957s) 5 0 9 - 5 ^ -5. M. M. Day, Fixed-point theorems f o r compact convex sets, I l l i n o i s J. Math. 5 (1961) 585-590. 6. M. M. Day, Correction to my paper "Fixed-point theorems fo r compact convex sets", I l l i n o i s J. Math. 8 (1964) . 713. 7. K. Deleeuw and I. Glicksberg, Applications "of almost periodic compactifications, Acta Math. 105 (1961) 63-97. 8. N. Dunford and J. T. Schwartz, Linear operators, Part I, Pure and Applied Math. No. 7, Interscience, • New York, 1958. 9. W. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949) 217-240. 10. R. E l l i s , L o c a l l y compact transformation groups, Duke Math. J. 24 (1957) 119-125. 58 11. H. Frustenberg, A Poisson formula f o r semi-simple. Lie groups, Annals of Math. 77 (1963) 335~386. 12. I. Glicksberg, On convex hul l s of translates, P a c i f i c J. Math. 13 (1963) 97-113. 13« E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965) 177-197. 14. E. Granirer, Extremely amenable semigroups I I , Math. Scand. 20 (1967) 93-113-15. E, Granirer, 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 of extreme-l y amenable semigroups (to appear i n Trans. Amer. Math. Soc. 1969). 16. E. Granirer, On the invariant mean on topological semi-groups and on topological groups, P a c i f i c J. Math. 15 (1965) 107-140. 17. E. Granirer and Anthony Lau, A characterisation of l o c a l l y compact amenable groups (to appear"in I l l i n o i s J. Math.). . 18. F. P. Greenleaf, Invariant means on topological.groups and t h e i r applications (to appear i n Van Nostrand Math. Studies). 19. E. Hewitt and K..A. Ross, Abstract harmonic analysis I, Springer Verlag, Berlin-Gottingen-Heidelberg, I963. 20. A Hulanicki, Means and Fjzflner condition on l o c a l l y compact groups, Studia'Math. 27 (1966) 87-104. 59 21. J. Kelley, General topology, Van Nostrand, New York, 1955-22. J. Kelley and I. Namioka, et. a l . , Linear topological spaces, Van Nostrand, New York, 1963* 23. T. M i t c h e l l , Constant functions and l e f t invariant means on semigroups, Trans. Math. Soc. 119 (1965) 244-261. 24. T. M i t c h e l l , Fixed points and m u l t i p l i c a t i v e l e f t i n -variant means, Trans. Amer. Math. Soc. 122 (1966) 195-202 .\ 25. T. M i t c h e l l , Function algebras, means and f i x e d points, Trans. Amer. Math. Soc. 130 (1968) 117-126. 26. T. M i t c h e l l , Topological semigroups and f i x e d points (to appear). 27. I. Namioka, On certain actions of semigroups on L-spaces, Studia Math. 29 (1967) 63-77-28. J. von Neumann, Zur allgemeien theorie des ma^es, Fund. Math. 13 (1929) 73-116. 29. C. R. Rao, Invariant means on spaces of continuous or_ measurable functions, Trans. Amer. Math. Soc. 114 (1965) I87-196. 30. N. Rickert, Amenable groups and groups with the fix e d point property, Trans. Amer. Math. Soc. 127 (1967) 211-232. 6o 31. Ryll-Nardzewski, On fix e d points of semigroups of endo-morphisms of l i n e a r spaces, Proceedings of the F i f t h Berkeley Symposium on Math. S t a t i s t i c s and Pr o b a b i l i t y , I I . Berkeley, 1966. 32. R. J. Silverman, Means on semigroups and the Hahn-Banach extension property, Trans. Amer. Math. Soc. 83 (1956) 222-237-33. R. J. Silverman, Invariant means and cones with vector i n t e r i o r s , Trans. Amer. Math. Soc. 88 (1958) 75-79. 34. R. J. NSilverman and T i Yen, Addendum to "Invariant means and cones with vector i n t e r i o r s " , Trans. Amer. Math. Soc. 88 (1958) 327-330. 35. K. Witz, Applications of a compactification f o r bounded operator semigroups, I l l i n o i s J. Math. 8 (1964) 685-696. 

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