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UBC Theses and Dissertations

Invariant means on locally compact groups and transformation groups Snell, Roy Cameron 1972

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INVARIANT. MEANS ON LOCALLY COMPACT GROUPS AND TRANSFORMATION GROUPS by ROY CAMERON SNELL B.SC. (HON.), Queen's U n i v e r s i t y , 1 9 6 5 M.SC, Queen's U n i v e r s i t y , 1 9 6 7 THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f MATHEMATICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA • August, 1 9 7 2 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb i a , I a g r e e tha t the 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 ag ree 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 pu rpo se s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha 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 not 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 . Department o f M A T H Ert/vncs" The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada D a t e P\V6Vst M . I 972 T h e s i s S u p e r v i s o r : P r o f e s s o r E. G r a n i r e r ABSTRACT T h i s t h e s i s d e a l s w i t h two s e p a r a t e q u e s t i o n s i n the a r e a o f i n v a r i a n t means on l o c a l l y compact groups. G r a n i r e r has shown t h a t , f o r c e r t a i n d i s c r e t e s e m i -groups S, the range o f a l e f t i n v a r i a n t mean on the a l g e b r a m(S) i s the e n t i r e [ 0 , 1 ] i n t e r v a l and f u r t h e r , t h a t t h i s range can be o b t a i n e d on a n e s t e d f a m i l y o f l e f t a l most c o n v e r g e n t s u b s e t s o f S. We g e n e r a l i z e the f i r s t p a r t o f h i s r e s u l t t o show t h a t the range o f e v e r y l e f t i n v a r i a n t 00 mean on L (G) f o r a l o c a l l y compact group G i s [ 0 , 1 ] . I f G i s a b e l i a n we a l s o show t h a t t h i s range i s a t t a i n e d on a n e s t e d f a m i l y o f l e f t almost c o n v e r g e n t B o r e l s u b s e t s o f G. I n t h e l a s t c h a p t e r we d e a l w i t h the problem o f e x t e n d -i n g the concept o f a m e n a b i l i t y f o r a l o c a l l y compact group G t o t h e s i t u a t i o n where G a c t s on t h e space G/H o f l e f t c o s e t s o f G w i t h r e s p e c t t o a c l o s e d subgroup H ( a group a c t i n g i n t h i s way i s c a l l e d a t r a n s f o r m a t i o n g r o u p ) . We i n t r o d u c e a d e f i n i t i o n o f the amenable a c t i o n o f G on v a r i o u s c l o s e d subspaces o f L (G/H,u) (u a q u a s i - i n v a r i a n t measure on G/H) which i s e q u i v a l e n t t o the one g i v e n by G r e e n l e a f but i s o b t a i n e d by d i f f e r e n t methods. We a l s o prove analogues o f s e v e r a l well-known theorems c o n c e r n i n g the a m e n a b i l i t y o f l o c a l l y compact groups. ACKNOWLEDGEMENTS The a u t h o r wishes t o thank P r o f e s s o r E. G r a n i r e r f o r h i s a i d and a d v i c e 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 as w e l l as the Canada C o u n c i l f o r i t s f i n a n c i a l s u p p o r t . i v TABLE OF CONTENTS CHAPTER I NOTATIONS AND TERMINOLOGY P a g e 1 . 1 . N o t a t i o n 1 1 . 2 . Measure Theory 1 1 . 3 - I n v a r i a n t Means 3 1 . 4 . Group Theory 5 CHAPTER I I THE RANGE OF INVARIANT MEANS ON LOCALLY COMPACT GROUPS AND SEMIGROUPS 2 . 0 . I n t r o d u c t i o n . - 6 2 . 1 . Background 6 2 . 2 . P r e l i m i n a r i e s 9 2 . 3 . S t r u c t u r e Space o f an A l g e b r a ... 1 0 2 . , 4 . . P R O B A B I L I T Y . M E A S U R E S , . , O J O . A / J L ) , . . . . . . , 12 . 2 . 5 - Main Lemma ; 1 3 2 . 6 . O r b i t s i n ' A ( A ) 1 6 2 . 7 . Main R e s u l t s 1 9 CHAPTER I I I THE RANGE OF INVARIANT MEANS ON LOCALLY COMPACT ABELIAN GROUPS 3 . 0 . P r e l i m i n a r i e s 2 0 3 . 1 . Nested C o l l e c t i o n s i n Q u o t i e n t Groups 2 1 3 . 2 . Nested C o l l e c t i o n s i n T 3 R and Z 2 3 3 . 3 . Nested C o l l e c t i o n s i n Compact A b e l i a n Groups. 2 6 3 . 4 . Main R e s u l t s 3 1 3 . 5 . Remarks 3 3 V CHAPTER IV AMENABLE ACTIONS OP LOCALLY COMPACT GROUPS 4 . 0 . I n t r o d u c t i o n 3 4 4 . 1 . Q u a s i - I n v a r i a n t Measures 3 5 4 . 2 . Amenable A c t i o n s o f G on G/H • 4 8 BIBLIOGRAPHY 7 1 1 CHAPTER I NOTATIONS AND TERMINOLOGY 1.1 NOTATION : I n the m a t e r i a l t h a t f o l l o w s Z, R and C w i l l denote the i n t e g e r s , r e a l s and complex numbers. The p o s i -t i v e i n t e g e r s w i l l be denoted by Z and T={exp(2iTix| 0<x<_l} c o n t a i n e d i n C i s the c i r c l e group. G i v e n two s e t s A and B, A^B i s t h e i r s e t t h e o r e t i c d i f f e r e n c e and x f l t h e c h a r a c -t e r i s t i c f u n c t i o n o f the s e t A. I f we a r e d e a l i n g w i t h s u b s e t s o f a s e t Y we w i l l use 1^ ( o r s i m p l y 1 i f no co n -f u s i o n w i l l r e s u l t ) t o denote Xy ~ t h e c o n s t a n t f u n c t i o n 1 on Y. The symbol 0 w i l l denote the empty s e t . 1.2 MEASURE THEORY : A measure space i s a t r i p l e (X,J>,y) where X i s a non-empty s e t ; J) a c r - f i e l d o f s u b s e t s o f X and y a n o n - n e g a t i v e , c o u n t a b l y a d d i t i v e s e t f u n c t i o n d e f i n e d on i , w i t h y(0)= 0. I f y(X)< °° t h e n y i s c a l l e d f i n i t e or bounded and i f y(X)= 1, y i s r e f e r r e d t o as a p r o b a b i l i t y measure. G i v e n two measures y^ and on the same space we w r i t e y-^ << yg t o i n d i c a t e t h a t y^ i s a b s o l u t e l y c o n t i n u o u s w i t h r e s p e c t t o and P ^ l - ^ t o i n d i c a t e t h a t y^ and y 2 a r e e q u i v a l e n t . L^"(X,y) i s t h e space o f a b s o l u t e l y y - i n t e g r a b l e r e a l -00 v a l u e d f u n c t i o n s on X and L (X,y) the space o f y - e s s e n -t i a l l y bounded r e a l - v a l u e d f u n c t i o n s on X. L^CXjy) i s a Banach space under t h e norm II f 11-^  = / | f (x) |dy (x) and L°°(X,y) i s a Banach space w i t h || f||oo= i n f { a | y({xeX: | f ( x ) | >a})= 0} . 2 SF(J?>) w i l l denote the s i m p l e f u n c t i o n s on Jb ( i . e . a l l n r e a l - v a l u e d f u n c t i o n s f on X o f t h e form f= E a.'X. where 1=1 i cueR and {A^,...,A n} i s a c o l l e c t i o n o f p a i r w i s e d i s j o i n t members ofj?>). Note t h a t the s i m p l e f u n c t i o n s a r e norm dense i n b o t h L (X,y) and L ( X , u ) . F o r a t o p o l o g i c a l space X we denote by B(X) t h e B o r e l  s u b s e t s o f X ( t h e s m a l l e s t a - f i e l d o f s u b s e t s c o n t a i n i n g the c l o s e d s e t s ) . I f X i s a l o c a l l y compact space t h e n a measure u on X i s a r e g u l a r B o r e l measure i f t h e B o r e l s e t s a r e y-measurable and i ) u(K)< «> f o r every compact s e t K^-X i i ) f o r e v e r y measurable s e t A u(A)'= i n T {y (B)"| B-open, A ^ B }• = s u p { y ( K ) | K compact, K<=A} . I f K(X) denotes th e c o n t i n u o u s f u n c t i o n s on X which v a n i s h o u t s i d e o f a compact s e t t h e n a r e g u l a r B o r e l measure on X i s c o m p l e t e l y d e t e r m i n e d i f we s p e c i f y the v a l u e o f • / f ( x ) d y ( x ) f o r e v e r y f e K ( X ) . A l l t h e measures r e f e r r e d x t o i n C h apter IV are r e g u l a r B o r e l measures. I f G i s a l o c a l l y compact group t h e n t h e r e e x i s t s a u nique (up t o a m u l t i p l i c a t i v e c o n s t a n t ) r e g u l a r B o r e l measure A on G, c a l l e d the l e f t Haar measure on G, w i t h the p r o p e r t y t h a t f o r any measurable f u n c t i o n f on G, fnf(gx)dA(x) = /„f(x)dA(x) f o r a l l geG. There e x i s t s a c o n t i n u o u s homomorphism A_ from G i n t o t h e p o s i t i v e m u l t -i p l i c a t i v e r e a l s c a l l e d t h e modular f u n c t i o n such t h a t , 3 f o r any measurable f u n c t i o n f and any geG / f ( x g - 1 ) d X ( x ) = A G ( g ) ' / Q f ( x ) d X ( x ) . G i s c a l l e d u n l m o d u l a r i f A Q ( g ) = 1 f o r a l l geG. I f M(G) denotes t h e space o f f i n i t e r e g u l a r B o r e l measures on G t h e r e e x i s t s a c o n t i n u o u s b i l i n e a r o p e r a t o r * c a l l e d con-v o l u t i o n mapping M(G) X L 1 ( G ) - > L 1 ( G ) and M(G)xL°0(G)->L°0(G) d e f i n e d by y * f ( x ) = / _ f ( y - 1 x ) d y ( y ) f o r a l l yeM(G) and f e L 1 ( G ) o r L°°(G) . I f we d e f i n e a norm on M(G) by ||yll = y(G) t h e n l l y w f l l ^ l l y l l - l l f l ^ and H y a g l t ^ i !! y II • I! g l| r o f o r any f e L 1 ( G ) and geL ( G ) . F o r f e L (G) we can d e f i n e a f i n i t e r e g u l a r B o r e l measure yeM(G) by s e t t i n g / g g ( x ) d y ( x ) = f^g(x)f(x)dA(x) f o r each measurable f u n c t i o n g on G so we can c o n s i d e r L"*"(G) i s o m e t r i c a l l y embedded i n M(G) s i n c e ||ull= l | f | l ^ . 1 . 3 INVARIANT MEANS : A f u n c t i o n a l g e b r a A on a non-empty s e t X i s a sup norm c l o s e d , p o i n t s e p a r a t i n g a l g e b r a o f bounded r e a l - v a l u e d f u n c t i o n s on X which c o n t a i n s the c o n s t a n t f u n c t i o n s . An example i s m(X) - t h e a l g e b r a o f a l l bounded r e a l - v a l u e d f u n c t i o n s on X. I f X i s a t o p o l o g -i c a l space we may c o n s i d e r CB(X) - t h e bounded c o n t i n u o u s r e a l - v a l u e d f u n c t i o n s on X. I f X i s a t o p o l o g i c a l semigroup ( o r group) LUC(X) i s t h e a l g e b r a o f bounded c o n t i n u o u s f u n c t i o n s f on X which have the p r o p e r t y t h a t whenever x -»-x i n X we have . l i m s u p { | f ( x x ) - f ( x x) I : xeX} = 0 . a o a ^ 1 v a o 1 F o r a measure space (X,J»,y) the a l g e b r a BM(X) o f bounded y-measurable f u n c t i o n s on X i s a f u n c t i o n a l g e b r a . We w i l l CO o f t e n d e a l w i t h L (X) which i s not s t r i c t l y s p e a k i n g a 4 f u n c t i o n a l g e b r a s i n c e f u n c t i o n s a re i d e n t i f i e d i f they d i f f e r o n l y on a n u l l s e t . I f S i s a semigroup t h e n a f u n c t i o n a l g e b r a A on S 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 L feA [R f e A ] f o r a l l xeS, f e A where L f ( t ) = f ( x t ) [R f ( t ) = f ( t x ) ] f o r a l l t e S . Note t h a t each o f the spaces m(S), C B ( S ) , L U C ( S ) , BM(S) 00 and L (S) i s l e f t i n v a r i a n t . 00 A mean f on a f u n c t i o n a l g e b r a A ( o r L (X) f o r a mea-su r e space (X j v2),ii)) i s a 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 on A ( i . e . <j>eA*) withl|c}>||<_ 1 and such t h a t <$>(f)>_ 0 whenever f >_ 0 . I t i s e a s i l y checked t h a t the s e t o f means on A i s a w* compact s u b s e t o f A*.. A mean (J) on a l e f t i n v a r i a n t f u n c t i o n a l g e b r a on a semigroup S i s c a l l e d a l e f t i n v a r -i a n t mean (LIM) i f cj)(L f ) = <f>(f) f o r every xeS and feA. X I f feA i s such t h a t cj>(f)= c ( c o n s t a n t ) f o r e v e r y LIM <J> on A t h e n f i s s a i d t o be l e f t a l most c o n v e r g e n t t o c. A s e t Be S i s c a l l e d a l e f t almost c o n v e r g e n t s e t i f x B i s a l e f t almost convergent f u n c t i o n i n A. A semigroup S i s l e f t amenable i f t h e r e e x i s t s a LIM on m(S). 00 I f G i s a l o c a l l y compact group t h e n a mean u on L (G) i 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 mean (TLIM) i f u(cj>*g)= o(g) f o r e v e r y $eP(G)={f e L 1 ( G ) | f > 0 , f Qf (x) dA (x) = 1 } 00 and e v e r y geL ( G ) . I t can be shown t h a t t h e e x i s t e n c e o f oo a TLIM on L (G) i s 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 LIM on L°°(G) . A l o c a l l y compact group G i s c a l l e d amenable i f 00 t h e r e e x i s t s e i t h e r a LIM or a TLIM on L ( G ) . I t has been shown t h a t a l l compact groups and a l l l o c a l l y compact 5 a b e l i a n groups a re amenable. I n l a t e r work we w i l l use the f a c t t h a t P(G) i s w* dense i n t h e s e t o f means on L°°(G) ( s i n c e L"^(G) can be i s o m e t r i c a l l y embedded i n L°°(G)*). 1 . 4 GROUP THEORY : I f G i s a group we w i l l denote i t s i d e n -t i t y by e. I f H i s a subgroup o f G t h e n G/H i s the s e t o f l e f t c o s e t s o f G w i t h r e s p e c t t o H. The s e t {x.}. x o f 1 l e i elements o f G i s a s e t o f r e p r e s e n t a t i v e s f o r G/H i f x.HOx H= 0 f o r i / j and G=lJx.H. I n Cha p t e r IV we w i l l 1 J i e l 1 use x t o denote t h e c o s e t xH i n G/H. The c a n o n i c a l p r o -j e c t i o n II • G G/H i s g i v e n by n r r(x)= xH f o r e v e r y xeG. n n I f G i s a l o c a l l y compact group and H a c l o s e d subgroup t h e n G/H i s a l o c a l l y compact space and II i s c o n t i n u o u s and open.. A. l o c a l l y , , . c.o,mpa.c.t.„_gr.o.up,,G. i,s. .compactly, ge.nerated-.,..,, i f t h e r e e x i s t s a compact s e t K C G which g e n e r a t e s G. A l o c a l l y compact group G' i s a d i r e c t f a c t o r o f G i f G/H i s t o p o l o g i c a l l y i s o m o r p h i c w i t h G' f o r some c l o s e d , normal subgroup H o f G. CHAPTER I I THE RANGE OF INVARIANT MEANS ON LOCALLY COMPACT GROUPS AND SEMIGROUPS 0 INTRODUCTION : I n t h i s c h a p t e r we c o n s i d e r t h e range o f l e f t i n v a r i a n t means on the spaces o f bounded and e s s e n -t i a l l y bounded B o r e l measurable f u n c t i o n s on l o c a l l y compact groups and on subsemigroups of l o c a l l y compact groups. The key lemma i s Lemma 2 . 5 . 1 and the main r e s u l t s appear i n Theorems 2 . 7 . 1 and 2 . 7 - 3 . 1 BACKGROUND : L e t (X,J>,y) be a measure space. A s e t Aej> i s c a l l e d an atom i f y ( A ) > 0 and f o r eve r y Be_2> w i t h B CA we have y ( B ) = 0 or y ( A ) . The measure y i s c a l l e d non-atomic i f t h e r e are- no atoms- in-J>"; • A p a r t i c u l a r case o f L i a p o u n o f f ' s C o n v e x i t y Theorem f o r v e c t o r - v a l u e d measures (see [ 1 ] ) y i e l d s the f o l l o w i n g THEOREM 2 . 1 . 1 : Let (X,j>,y) be a f i n i t e measure space w i t h y a non-atomic measure. Then the s e t { y ( A ) | Aej>} i s c l o s e d and convex. As a consequence o f t h i s lemma we o b t a i n COROLLARY 2 . 1 . 2 : For any B ,B^eft w i t h B c B 1 we have o > i s o 1 { y ( B ' ) | B ' e | J B c B ' c B 1 } = [ y ( B Q ) , y (B±) ] . PROOF : D e f i n e a measure u on X by u(A)=y(AH(B 1^B ) ) . Then u i s c l e a r l y f i n i t e and non-atomic w i t h 0<_ u(A)<_ y ( B 1 ^ B Q ) f o r eve r y AeJ). S i n c e {u(A)| AeJ>} i s c l o s e d and convex by Theorem 2 . 1 . 1 , i t must c o n s t i t u t e the e n t i r e i n t e r v a l I = [ 0 , M ( B 1 ^ B O ) ] = [ 0 , y ( B 1 ) - u ( B o ) ] . I f a e [ u ( B Q ) , y ( B ^ ) ] t h e n a - y ( B Q ) e I so t h e r e e x i s t s Cej> w i t h u(C)=a-y (B ) . L e t B ' =B QU( Cr\(B 1^B Q )) so B c B ' c B ] L and we have y ( B ' )=y (B H y C C r t C B ^ B )) =y(B Q) +u(C) =a . From t h i s r e s u l t we see t h a t f o r non-atomic p r o b a b i l i t y measures y, the range o f y i s t h e e n t i r e [ 0 , 1 ] i n t e r v a l . The next lemma i n d i c a t e s t h a t t h e s e t s on which a measure y a t t a i n s i t s range can be assumed t o be n e s t e d . LEMMA 2 . 1 . 3 : L e t y be a non-atomic p r o b a b i l i t y measure on the measurable space (X,j>) . Then t h e r e e x i s t s a f a m i l y { A ( t ) | t e [ 0 , l ] } inj> such t h a t i ) s<_ t i m p l i e s A(s)<=A(t) and i i ) y ( A ( t ) ) = t f o r each t e [ 0 , l ] . PROOF : For n e Z + l e t Q n={k / 2 N | 0 < k< 2 n,keZ} and l e t Q= [J ,Q . Then Q i s c o u n t a b l e and Q cQ f o r each n. neZ Assume t h a t we have a f a m i l y { B ( t ) | t e Q n ) of s e t s i n J?> , n ^ l s a t i s f y i n g c o n d i t i o n s 1 ) and i i ) f o r s,teQ . I f t=k / 2 i s an element of Q n + 1 l e t Sj=k^/2n and s 2 = k 2 / 2 N be the l a r g e s t and s m a l l e s t elements o f r e s p e c t i v e l y f o r which By C o r o l l a r y 2 . 1 . 2 we have { y ( B ' ) | B ' e J ) , B ( s 1 ) c B ' c B ( s 2 ) } = [ s 1 , s 2 ] so t h e r e e x i s t s B ' ( t ) e j > w i t h B ( s , )cB » ( t )cB( s 2 ) and y ( B ' ( t ) ) = t . I n t h i s manner we o b t a i n a c o l l e c t i o n { B ' ( t ) | t £ Q n + ^ ^ o f s e t s i n J> which s a t i s f i e s c o n d i t i o n s i ) and i i ) and extends the f a m i l y { B ( t ) | t e Q n ) ( i . e . B ' ( t ) = B ( t ) f o r t e Q n + i n Q n ) . By i n d u c t i o n , t h e r e e x i s t s a f a m i l y ( A * ( t ) [ teQ} s a t i s f y i n g the r e q u i r e d c o n d i t i o n s and, f o r t e [ 0 , l ] , i f we l e t A( t ) = C\ {A ' ( s ) | seQ,t< s} t h e n A(t ) e J > s i n c e Q i s c o u n t a b l e ; { A ( t ) | t e [ 0 , l ] } i s c l e a r l y n e s t e d and f o r t<_ w i t h s ^ s ^ Q we have A* ( s 1 ) c A ( t ) c A ' ( s 2 ) c o n s e q u e n t l y s n=y(A * ( s n ) ) < y ( A ( t ) ) < y ( A ' ( s 2 ) ) = s 2 . S i n c e Q i s dense i n [ 0 , 1 ] t h i s i m p l i e s t h a t y ( A ( t ) ) = t as r e q u i r e d . A- mean- cf> on the- a l g e b r a of' bounded'-mea^U'rable-"functions-OO n ( o r L ( X ) ) o f a measurable space ( X c a n be r e g a r d e d as a f i n i t e l y a d d i t i v e measure y^ on X i f we d e f i n e y^(A)=<J>(x^) f o r AeJj.(For c o n v e n i e n c e o f n o t a t i o n we w i l l o f t e n w r i t e <j)(A) r a t h e r t h a n <(>(x A)') I n t h i s s e t t i n g i t i s p o s s i b l e t o examine t h e range of <J> i n terms of the range o f the c o r r e s p o n d i n g measure. I n p a r t i c u l a r we are i n t e r e s t e d i n the case when if i s a l e f t i n v a r i a n t mean (and hence y, i s a l e f t i n v a r i a n t f i n i t e l y a d d i t i v e measure). L i a p o u n o f f ' s Theorem does not a p p l y here even i f y^ i s non-atomic due t o the f a c t t h a t the measure i s not n e c e s s a r i l y c o u n t a b l y a d d i t i v e . G r a n i r e r showed i n [ 2 ] t h a t f o r a c e r t a i n c l a s s o f d i s c r e t e l e f t amenable semigroups the range o f each LIM 9 (J> on m(S) i s the e n t i r e i n t e r v a l [ 0 , 1 ] and f u r t h e r , t h a t t h e n e s t e d f a m i l y on which each <j) a t t a i n s i t s range can be chosen from t h e l e f t almost convergent s u b s e t s of S. THEOREM 2 . 1 . 4 : ( G r a n i r e r [ 2 ] ) I f S i s . a n i n f i n i t e d i s c r e t e r i g h t c a n c e l l a t i o n semigroup which i s not an "AB group" (an "AB group" i s an i n f i n i t e amenable d i s c r e t e group G i n w hich each element has f i n i t e o r d e r and e v e r y i n f i n i t e subgroup of G c o n t a i n s a f i n i t e l y g e n e r a t e d i n f i n i t e subgroup) t h e n t h e r e e x i s t s a n e s t e d f a m i l y { A ( t ) | t e [ 0 , l ] } o f ( B o r e l ) s u b s e t s of S such t h a t <f>(A(t))=t f o r any LIM 4> on m(S) . G r a n i r e r c o n j e c t u r e d t h a t , s i n c e i t i s u n l i k e l y t h a t "AB- group's-" exis't'y -the> re^ 'Ul-t-sh-ou»ld-'hold'--f or- a-11 i n f i n i t e -d i s c r e t e r i g h t c a n c e l l a t i o n l e f t amenable s e m i g r o u p s . Chou was a b l e t o show i n [ 3 ] t h a t f o r e v e r y i n f i n i t e amenable group G the range of each LIM <J> on m(G) i s the e n t i r e [ 0 , 1 ] i n t e r v a l . However th e s e t s he o b t a i n e d on which (J> a t t a i n s t h i s range a r e not n e s t e d and depend h e a v i l y on <j> so they a r e not l e f t almost c o n v e r g e n t . 2 . 2 PRELIMINARIES : I n t h i s c h a p t e r we g e n e r a l i z e Chou's r e s u l t s t o show t h a t the range of a LIM on BM(S) o r 00 L ( S ) , where S i s an i n f i n i t e B o r e l subsemigroup of a l o c a l l y compact group, I s the i n t e r v a l [ 0 , 1 ] . The s e t s we o b t a i n w i l l be n e s t e d , however they w i l l depend on t h e LIM <p b e i n g c o n s i d e r e d . 1 0 The method we use i s based I n p a r t on Chou's t e c h n i q u e of c o n s i d e r i n g the r e l a t i o n s h i p between l e f t i n v a r i a n t means on m(G) and r e g u l a r p r o b a b i l i t y measures on the S t o n e -Cech c o m p a c t i f i c a t i o n 3 G o f G. I n our case we c o n s i d e r t h e more g e n e r a l concept o f the s t r u c t u r e space of a Banach a l g e b r a . We w i l l r e q u i r e the f o l l o w i n g d e f i n i t i o n i n o r d e r t o s t a t e many o f our r e s u l t s i n a more c o n c i s e manner. DEFINITION 2 . 2 . 1 : L e t f be a mean on a f u n c t i o n a l g e b r a A ( o r on L°°(X) f o r a measure space (X,ji),u)) and Q a s u b s e t of [ 0 , 1 ] . The f a m i l y { A ( t ) | teQ} i s c a l l e d a n e s t e d c o l l e c t i o n on which <j> a t t a i n s range Q i f X A ( t ) e A ( ° r A(t)ej2>) f o r each t and i ) s<_ t i n Q i m p l i e s t h a t A ( s ) c A ( t ) i i ) <j)(A(t))=t f o r each teQ . 2 . 3 STRUCTURE SPACE OF AN ALGEBRA : DEFINITION 2 . 3 . 1 : L e t A be a f u n c t i o n a l g e b r a (or L°°(X) f o r a measure space (X , J>,y)). The s t r u c t u r e space A(A) o f A i s t h e s e t of a l l non-zero m u l t i p l i c a t i v e l i n e a r f u n c -t i o n a l s . on. A equipped w.it.h. t h e weak * top.o.l.o.gy as a s u b s e t of A . NOTE : As i s w e l l known, A(A) can be c o n s i d e r e d as the s e t of m u l t i p l i c a t i v e means on A f o r 4>eA(A) i m p l i e s < K l ) = l ( s i n c e 0^<j)(l) =<j>(l2 ) = (<J>( l ) ) 2 ) and l U l i i 1 (Choose feA w i t h llf||< 1 . Then f n e A w i t h l l f n | | < 1 1 1 f o r e v e r y I n t e g e r n. S i n c e | <}>(f) | n=| cf>(fn) |£ ll.<MI'llf nll< ll<f>ll<» f o r each n, we must have |cj>( f)|l. 1 so ||<J>||<_ 1 . ) . The f o l l o w i n g w e l l known r e s u l t g i v e s f u r t h e r p r o p e r t i e s o f the s t r u c t u r e space and i t s r e l a t i o n s h i p t o the a l g e b r a A. THEOREM 2 . 3 - 2 : ( [ 4 ] , Theorem C . 2 5 p . ^ 7 9 ) i ) A(A) i s compact and H a u s d o r f f i i ) I f we d e f i n e f(<j>)=<j>(f) f o r f e A and cj)eA(A) t h e n the mapping f+f i s a l i n e a r i s o m e t r y o f A onto C ( A ( A ) ) - the c o n t i n u o u s f u n c t i o n s on the s t r u c t u r e s p ace. I n t h e cases we w i l l c o n s i d e r , t h e r e i s a c o n v e n i e n t c h a - r a c t e r i z - a ^ i o n o f the--topolog-y of A-(-A)- given- by• LEMMA 2 . 3 . 3 : L e t *B be a a - f i e l d o f s u b s e t s o f a non-empty s e t X, (or*B = f o r the measure space (X,^>,y)) and l e t A be the a l g e b r a o f bounded 1B -measurable f u n c t i o n s on X ( A=L°°(X)). For B e ^ l e t UB={<J>eA(A) | <j>(B)=l} Then {Ug| Be1©} i s a base of o p e n - c l o s e d ( a l s o compact) s e t s f o r t h e t o p o l o g y on A ( A ) . PROOF :(The p r o o f i s p r e s e n t e d here f o r the sake o f c o m p l e t e n e s s ) F i r s t note t h a t X B = X B f o r a n y s e t B s o ij>(B)= 0 or 1 f o r any <j>eA(A), Be$. S i n c e x B e C ( A ( A ) ) and U B= X B 1 ( D = X ^ d / 2 ' 2 ) } w e h a v e U B o p e n - c l o s e d ( a l s o compact as a c l o s e d s u b s e t of a compact s p a c e ) . n L e t g= j _ l 2 a i ^ B e SF(1B) - the s i m p l e f u n c t i o n s on^fi -i and l e t V={ TeA(A): -[ cj> (g)-<j>0 (g) | < e} be a s u b b a s i c n e ighbourhood of <j> eA ( A ) . I f <$>Q(B^)=1 f o r some k the n <\> eUg and (J>eUg i m p l i e s cf)(g) =a k=(f) o(g) so ° k k |4>(g)- T (g)|< e and <j)eV. T h e r e f o r e <$> eUg<= V. o k I f <b ( B . ) = 0 f o r a l l i , l e t B=X^( V1B. ) eft. Then <J> ( B ) = l Y o 1 5 i = l 1 r o and f o r <J)eUg we have cj)(g) =0 = <j)o(g) so 4>QeUB^V. For a r b i t r a r y feA l e t W={cJ>eA(A): | <|>(f )-(J> ( f ) | < £> be a s u b b a s i c neighbourhood of c|> . Choose geSFC33) w i t h j| f-gj|^< e/3 ( t h i s i s p o s s i b l e s i n c e SF(*&) i s dense i n A) so we have | <J>(f )-<J>(g) | < £ / 3 f o r ev e r y <J>eA(A). I f U={(J>eA(A) : | (J) (g)-<J> (g) | < e/3} t h e n cf>oeUcW and we can f i n d B e ^ w i t h (J) euViUeW. S i n c e U n o t 3 = U n 0 U Q f o r a l l B1 ,Bne1&, t h e above i m p l i e s t h a t {Ug| Be^8} i s a base f o r the t o p o l o g y . 2.4 PROBABILITY MEASURES ON A(A) : LEMMA 2.4.1 : A mean cj> on A ( o r L (X)) g i v e s r i s e t o a r e g u l a r p r o b a b i l i t y measure y^ on A(A) (A(L°°(X))) d e f i n e d 00 by / fdy ( J )= <fr(f.) f o r a l l . feA (feL.. ( X ) ) . PROOF : T h i s f o l l o w s from the f a c t t h a t f o r any compact H a u s d o r f f space Y, C ( Y ) * i s i s o m e t r i c a l l y i s o m o r p h i c t o the space o f bounded r e g u l a r B o r e l measures on Y (see [8] p.265). S i n c e A-C(A(A)) we have a homeomorphism between A* and the bounded r e g u l a r B o r e l measures on A(A) g i v e n by ty^V^ where / fdy^= ^ ( f ) . f o r f e A . I f (J) i s a mean the n y ^ ( A ( A ) ) = / 1^ 1-1^ = d>(l) =1 and y^ i s a p r o b a b i l i t y measure. NOTE : For Be%, v n= v T T . . , . N , J > A B A U Q i n A(A) and the above f o r m u l a shows t h a t M (j )( u' B)= 4>(B). DEFINITION 2 . 4 . 2 : A bounded r e g u l a r B o r e l measure y on a l o c a l l y compact H a u s d o r f f t o p o l o g i c a l space X i s c a l l e d c o n t i n u o u s i f y ( { x } ) = 0 f o r every xeX. U s i n g t h i s d e f i n i t i o n we have LEMMA 2 . 4 . 3 : I f y i s a f i n i t e c o n t i n u o u s measure on a l o c a l l y compact H a u s d o r f f space X t h e n f o r any measurable s e t s B ,B, w i t h B c B , we have o' 1 o 1 {y(B»)| B ' e B ( X ) , B ^ B ' c B ^ = [y ( B q ) ,y (B.^ ] . PROOF : I n view of C o r o l l a r y 2 . 1 . 2 i t i s s u f f i c i e n t t o show. t h a t . y.. i.s,...n.on^ a.t,o.m.l.c,.,._I.f. .y..has...a.t.oms.. than.,,, by, reg...... u l a r i t y , t h e r e i s a compact atom K. For each xeK, s i n c e y ( { x } ) = 0 , we can f i n d an open s e t U x c o n t a i n i n g x w i t h y(U )< y ( K ) . By compactness o f K t h e r e e x i s t s a f i n i t e c o l l e c t i o n U ,...,U c o v e r i n g K. S i n c e K i s an atom, 1 n y(KflU ) = 0 f o r each i which i m p l i e s t h a t y ( K ) = 0 which X • 1 i s a c o n t r a d i c t i o n . 5 MAIN LEMMA : The f o l l o w i n g lemma i s the key t o o l used i n o b t a i n i n g the main r e s u l t s o f t h i s c h a p t e r . I n i t s p r o o f we w i l l u s e , i n p a r t , an i d e a o f Chou i n [ 3 ] and one of G r a n i r e r i n [ 2 ] . LEMMA 2 . 5 . 1 : Let X be a non-empty set^ a a - f i e l d of s u b s e t s o f X (or^=vB f o r the measure space (X,i>,y)) and A the a l g e b r a of b o u n d e d ^ measurable f u n c t i o n s on X oo (A=L ( X ) ) . Let I be a mean on A f o r which the c o r r e s p o n d i n g p r o b a b i l i t y measure y^ on A(A) i s c o n t i n u o u s . Then t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n A on w h i c h <j> a t t a i n s range [ 0 , 1 ] . PROOF : L e t A jA^elS w i t h A c A 1 and l e t Xe[<j>(A') ^ ( A , ) ] and e> 0 be g i v e n . By Lemma 2 . 4 . 3 t h e r e e x i s t s a measurable s u b s e t E o f A(A) w i t h U c E c Uft and y ( E ) = X ( s i n c e o 1 ^ * o ^ 1 o 1 f a c t t h a t i s compact and U A i s open we can use the "o 1 r e g u l a r i t y o f y^ t o f i n d a compact E^ and an open E^ w i t h U , c E , c E c E 2 c U A and ]A,(E^E^)< e. S i n c e E^ i s open and {Ug| Be^B} i s a base f o r the t o p o l o g y on A ( A ) , f o r each xeE^ t h e r e e x i s t s B^eB w i t h xeUgCrEg and by t h e compactness of E-^ t h e r e e x i s t x^, . . . , x n w i t h J V I U B ^ E 2 ' I F x. l n B= B U . . .UB z°& t h e n U = V , U n so E , c U D c E n and t h e r e -x, x B 1 = 1 B 1 B 2 I n x i | < K B ) - X | = I ^ A C U B ) - ^ * ^ ) I < E - S I N C E U A C U B C U A l m P l l e s ^ ™ o 1. A o C B e A 1 t h i s shows t h a t {<J>(B)| B'eHB, A ^ B ^ A ^ i s dense i n [ T(A o),cJ)(A ) ] . CO L e t Q n={k / 2 N | 0 < k< 2 N , k e Z + } f o r n e Z + and l e t Q=^i1%-We want t o d e f i n e a n e s t e d c o l l e c t i o n { A ' ( t ) | teQ} on which (J> a t t a i n s range Q. T h i s i s done i n d u c t i v e l y by d e f i n i n g A ( 0 ) = 0, A ( l ) = X and assuming t h a t { A ( t ) | teQ n> i s a n e s t e d c o l l e c t i o n on which <J> a t t a i n s r ange Q . 1 5 F o r t e Q n + 1 n Q n l e t A»(t)=A(t). I f t e Q n + 1 ^ Q n l e t t 1 , t 2 e Q n be the maximum and minimum elements o f f o r which t-^< t< t 2 and choose sequences a +t and $ n+t i n ( t ^ , t 2 ) . S i n c e {<j>.(B)| Be®, A(t 1)<=" B c A ( t 2 ) } i s dense i n [ t ^ t ^ , we can f i n d C ^ ^ w i t h k(t^)c c c A ( t 2 ) and 3 2 < 0 ( C 1 ) < 3-^. S i n c e ( a ^ , a 2 ) < ^ [ t-^, <J> (C^) ] we can f i n d D-^e^ w i t h A ( t 1 ) c D 1 c C 1 and o.±< <J)(D]L)< a 2 . S i n c e (3 3,6 2) CC<()(D : L) ,4)(C 1)] t h e r e e x i s t s C ^ s f f i w i t h D-^ <^2<~C1 a n d | 3 3 < $(C2^< ^ 2 * C o n t i n u i n g I n t h i s manner we o b t a i n sequences + and D.t i n ® w i t h A ( t , )<= D. c c . c A ( t n ) f o r a l l i , j and such l 1 x j 2 ' ° t h a t a.< <j>(D.)< a.,-, , 3 - , i < <J> (C . ) < 3,- f o r a l l i . I f we X X X T X X + X X X 00 l e t A ' ( t ) = U D.e'Bwe have D . C A » ( t ) c c . f o r a l l i and j i = l I I j so a.< c}>(D.)<_ <j)(A'(t))<_ ())(C.)< 3 . and s i n c e l i m a. = l i m 3 . = t t h i s i m p l i e s t h a t <f>(A'(t))= t . i->-oo j - * - 0 0 ^ I n t h i s manner we o b t a i n a n e s t e d c o l l e c t i o n { A ' ( t ) | T E < 3 n + i ^ which extends the c o l l e c t i o n { A ( t ) | teQ n> and on which <j> a t t a i n s range Q n +]_- By i n d u c t i o n t h e r e e x i s t s a c o l l e c t i o n ( A ( t ) | teQ} on which cf> a t t a i n s range Q and t h e c o l l e c t i o n i s n e s t e d f o r i f t ^ j t ^ . Q w i t h t ^ < ^ 2 t h e n t,,t„ eQ f o r some n and thus A( t-, A ( t 0 ) . 1* 2 n 1 2 As i n Lemma 2 . 1 . 3 , f o r t e [ 0 , l ] we d e f i n e B ( t ) = f U A ( s ) | seQ, t< s} . B(t ) e ? 8 s i n c e Q i s c o u n t a b l e and the d e n s i t y of Q i n [ 0 , 1 ] i m p l i e s t h a t { B ( t ) | t e [ 0 , l ] } i s a n e s t e d c o l l e c t i o n i n A on which <j> a t t a i n s range [ 0 , 1 ] . I n view o f t h i s r e s u l t the problem o f d e t e r m i n i n g the range of a mean on the a l g e b r a s mentioned i n the above lemma re d u c e s t o showing t h a t the c o r r e s p o n d i n g p r o b a b i l i t y -measure y, i s c o n t i n u o u s . 2 . 6 ORBITS IN A(A) : DEFINITION 2 . 6 . 1 :. L e t A be a l e f t i n v a r i a n t f u n c t i o n a l g e b r a on a semigroup S. I f \pek(A) t h e n the l e f t o r b i t of TJJ - 0(IJJ) = { L ^ | teS} (where L f c denotes the a d j o i n t of the l e f t t r a n s l a t i o n o p e r a t o r L f c:A ->A) . Note t h a t f o r each ^ e A ( A ) ; f,geA and t e S L ^ ( f - g ) = i K L t ( f - g ) ) = 4»(Lt.f-Ltg)= * ( L t f ) ' ^ ( L t g ) = ( L * ^ ) ( f )-(L*i(»)(g) so L ^ i s a l s o a m u l t i p l i c a t i v e mean and we have 0 ( ) <=• A(A) . The f o l l o w i n g lemma i n d i c a t e s the r e l a t i o n s h i p between l e f t o r b i t s i n A(A) and the measures y^ on A(A) c o r r e s -p o n d i n g t o l e f t i n v a r i a n t means <f> on A. LEMMA 2 . 6 . 2 : L e t S be a s e m i g r o u p , % a l e f t i n v a r i a n t a - f i e l d o f s u b s e t s of S and A t h e a l g e b r a of bounded Im-measurable f u n c t i o n s on S. I f O(^) i s i n f i n i t e f o r a l l iJ;eA(A) t h e n , f o r any LIM if on A, the. c o r r e s p o n d i n g r e g u l a r p r o b a b i l i t y measure y^ on A(A) i s c o n t i n u o u s . PROOF : Choose ifeA(A) and t e S . Note t h a t i f L,iJ;eU R where. 17 B e l S t h e n i p e U t - l B s i n c e ( X t ~ l B ) = ^ ( L t X g ) = L ^ ( B ) = 1. A l s o , due t o the f a c t t h a t y 0 ( U B ) = * ( B ) = d»(L tx B)= * ( x t - l B ) = ^ ( U t - 1 B ) , we have y ^ ( U t - l B ) = V^CUg) ( s i n c e <)> i s l e f t i n v a r i a n t ) whenever L^ijjeUg. The r e g u l a r i t y o f y^ and the f a c t t h a t {u* B| Be'£>} i s a base f o r the open s e t s i n A (A) i m p l y t h a t V < * > ) < > y { v } ) • S i n c e O(IJJ) i s i n f i n i t e , f o r any p o s i t i v e i n t e g e r n we * a can f i n d t , , . . . , t eS w i t h L.ty / L,^ f o r i ^ j . There-x n u. u . i n i * n * f o r e 1= y ^ ( A ( A ) ) > y ^ ( j y i { L t ^ } ) = ±| ^ ( { L ^ } ) > n • y ^  ( { ^ } ) and, s i n c e n i s a r b i t r a r y , t h i s i m p l i e s t h a t y (j )({^})= 0 so y, i s c o n t i n u o u s . I n the cases we a r e c o n s i d e r i n g i n t h i s c h a p t e r we are a b l e t o c a l l on the f o l l o w i n g theorem o f G r a n i r e r and Lau THEOREM 2 . 6 . 3 : ( C 5 ] Theorem 3 ) L e t S be a subsemigroup o f a l o c a l l y compact group G and assume t h a t LUC(S) ( t h e bounded l e f t u n i f o r m l y c o n t i n u o u s f u n c t i o n s on S) admits n x a LIM <j> o f the type <j> = 1 £ 1 a 1 c ! ) i where <J>1eLUC(S) a r e m u l t i p l i c a t i v e ; <J>. ^  <J>. f o r 1/ j ; a.> 0 f o r 1<_ i<_ n n and j_|]_ a-j_ = !• Then S i s a f i n i t e subgroup o f G of o r d e r n. U s i n g t h i s r e s u l t we o b t a i n THEOREM . 2 . 6 . 4 : L e t S .be an i n f i n i t e B o r e l subsemigroup 18 ( o f p o s i t i v e Haar m e a s u r e ) . i n a l o c a l l y compact group G and l e t A be the a l g e b r a of bounded B o r e l measurable f u n c t i o n s on S ( A= L°°(S) ). Then 0 (^) i s i n f i n i t e f o r a l l ipeA (A) . PROOF : F i r s t note t h a t f o r i j ^ ? ^ i n A (A) we can f i n d BeB(S) w i t h ^ ( B ) ^ ^ ( B ) . L e t C= tB where t e S . S i n c e S i s a B o r e l s e t i n G i t i s e a s i l y checked t h a t B(S)= (BeB(G)| B c S} . S i n c e f B ( f l ) c B ( G ) f o r a l l t e G , t h i s i m p l i e s t h a t CeB(S) . T h e r e f o r e X Q £ A w i t h L T X C = X B S O L ^ 1 ( C ) = ^ 1 ( L t x c ) = V - J L C B ) ji i^ 2(B)= ^ 2 ( L t x c ) = L * * 2 C 0 ) s and the mapping L^: A(A) -* A(A) i s one-to one. I f - 0(t|r) i s " f±n±tB"-f or - s'onie-^rAf *A">' we"' cair ^ fl-nd^t"-, , . . . ,1r e'S" such t h a t i ) L, ify L, \p f o r i / j and t i t j i i ) f o r any t e S we have L^TJJ = L f c i|> f o r some i . i n * L e t cj>= (.£,L. i l ; ) / n . tj) i s c l e a r l y a mean on A and f o r t e S 1 ~* JL X/ . i % n ^ j ^ : n ^ n ^ have Lt(j)= ( ^ L ^ i f O / n = ( ± | 1 L f c t _ V) /n = ( 1 g 1 L t ^ ) / n =<J» we due t o t h e p r o p e r t i e s i ) and i i ) above and the f a c t t h a t L, i s one-to-one on A(A) . T h e r e f o r e <j> i s a LIM on A and so , by r e s t r i c t i o n , a l s o a LIM on the c l o s e d s u b a l g e b r a LUC(S) o f A. However, s i n c e L f c ^ i s a m u l t i p l i c a t i v e element of LUC(S) (by r e s t r i c t i o n ) f o r each i , Theorem 2 . 6 . 3 i m p l i e s t h a t S i s f i n i t e w hich c o n t r a d i c t s our as s u m p t i o n t h a t S was i n f i n i t e . 1 9 2 . 7 MAIN RESULTS : THEOREM 2 . 7 . 1 : L e t S be an I n f i n i t e B o r e l subsemigroup of a l o c a l l y compact group G. I f A i s the a l g e b r a o f bounded B o r e l measurable f u n c t i o n s on S and c|> i s a LIM on A t h e n t h e r e e x i s t s a n e s t e d c o l l e c t i o n of B o r e l s u b s e t s of S on which <J> a t t a i n s range [ 0 , 1 ] . ' PROOF : By Lemma 2 . 6 . 4 , O(^) i s i n f i n i t e f o r each ^ e A ( A ) . T h e r e f o r e Lemma 2 . 6 . 2 i m p l i e s t h a t i s c o n t i n u o u s and the r e s u l t f o l l o w s from Lemma 2 . 5 . 1 . As a c o r o l l a r y t o t h i s theorem we o b t a i n a s t r o n g e r v e r s i o n o f Chou's r e s u l t i n [ 3 ] COROLLARY 2 . 7 . 2 : Let S be an i n f i n i t e subsemigroup of a d i s c r e t e group G. IT $ i s a LIM' on m(S') t h e r e e x i s t s a n e s t e d c o l l e c t i o n o f ( B o r e l ) s u b s e t s of S on which cf> a t t a i n s range [ 0 , 1 ] . The method o f p r o o f used i n Lemma 2 . 6 . 2 w i l l a l s o work i n the case where S i s a B o r e l subsemigroup o f p o s i t i v e oo Haar measure i n a l o c a l l y compact group G and A=L (S; so , as b e f o r e , c o m b i n i n g Lemmas 2 . 5 . 1 , 2 . 6 . 2 and 2 . 6 . 4 we o b t a i n THEOREM 2 . 7 . 3 : L e t S be a B o r e l subsemigroup o f p o s i t i v e Haar measure i n a l o c a l l y compact group G. I f (j) i s a LIM on L°°(S) t h e n t h e r e e x i s t s a n e s t e d c o l l e c t i o n o f B o r e l s u b s e t s o f S on which <J> a t t a i n s range [ 0 , 1 ] . 2 0 CHAPTER I I I THE RANGE OF INVARIANT MEANS ON LOCALLY COMPACT ABELIAN GROUPS 3 . 0 PRELIMINARIES : I n t h i s c h a p t e r we c o n t i n u e our examin-ee a t i o n o f the range o f l e f t i n v a r i a n t means on L (G) f o r a l o c a l l y compact group G. S i n c e t h e t e c h n i q u e s used a r e a l s o v a l i d f o r the a l g e b r a o f bounded B o r e l measurable f u n c t i o n s on G, the r e s u l t s which f o l l o w f o r BM(G) as 00 w e l l as f o r L ( G ) . In Chapter I I we showed t h a t the range o f any LIM <J> oo on L (G) i s the e n t i r e [ 0 , 1 J i n t e r v a l and t h a t t h i s range i s a t t a i n e d on a n e s t e d f a m i l y o f B o r e l s u b s e t s o f G, However the s e t s o b t a i n e d depend h e a v i l y on the LIM <J> b e i n g c o n s i d e r e d and are thus not l e f t almost c o n v e r g e n t as was the case of the s e t s o b t a i n e d by G r a n i r e r i n [ 2 ] . The main r e s u l t o f t h i s c h a p t e r o b t a i n e d i n Theorem 3 . 4 . 2 i s the f a c t t h a t f o r every l o c a l l y compact a b e l i a n group t h e r e e x i s t s a n e s t e d f a m i l y o f l e f t a l most c o n v e r -gent B o r e l s u b s e t s of G on which e v e r y LIM (f> a t t a i n s t h e range [ 0 , 1 ] . T h i s r e s u l t i s o b t a i n e d u s i n g s e v e r a l s t r u c -t u r e theorems f o r l o c a l l y compact, compactly g e n e r a t e d a b e l i a n groups and, i n the most d i f f i c u l t case o f compact a b e l i a n g r o u p s , by em p l o y i n g i n p a r t a t e c h n i q u e i n t r o d u c e d by G r a n i r e r i n [ 2 ] . I n o r d e r t o s t a t e many o f t h e r e s u l t s i n a more c o n c i s e form we i n t r o d u c e the f o l l o w i n g d e f i n i t i o n . 2 1 DEFINITION 3 - 0 . 1 : I f G i s a l o c a l l y compact group and Q i s a' s u b s e t of [ 0 , 1 ] t h e n a c o l l e c t i o n { A ( t ) | teQ} o f B o r e l s u b s e t s of G i s c a l l e d a n e s t e d c o l l e c t i o n w i t h range Q i f i ) s,teQ w i t h s ^ t i m p l i e s A ( s ) ^ A ( t ) and OO i i ) <j>(A(t))= t f o r any LIM I on L (G) and each t£Q. 3 . 1 NESTED COLLECTIONS IN QUOTIENT GROUPS : . In o r d e r t h a t a l o c a l l y compact amenable group G admit a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] , i t i s s u f f i c i e n t . t h a t some c o n t i n u o u s homomorphic image o f G admit a n e s t e d c o l l e c t i o n w i t h range Q where Q i s a c o u n t a b l e dense s u b s e t of [ 0 , 1 ] . T h i s f o l l o w s i m m e d i a t e l y from the f o l l o w -i n g two lemmas. LEMMA 3 - 1 . 1 : L e t G be an amenable l o c a l l y compact group and H a c l o s e d , normal subgroup of G» I f t h e r e e x i s t s a n e s t e d c o l l e c t i o n w i t h range Q i n G/H t h e n t h e r e e x i s t s a n e s t e d c o l l e c t i o n w i t h range Q i n G. PROOF : F i r s t note t h a t the a m e n a b i l i t y o f G i m p l i e s t h a t the l o c a l l y compact group G/H i s a l s o amenable (see [ 6 J p. 3 0 ) . OO oo L e t us d e f i n e a mapping T: L (G/H) L (G) as f o l l o w s -oo f o r f e L (G/H) l e t Tf= f ° n T I . S i n c e tt„ i s c o n t i n u o u s and r i ri f i s B o r e l m e a s u r a b l e , Tf i s a measurable f u n c t i o n on G. I f A. denotes l e f t Haar measure on G and u i s a l e f t Haar measure on G/H t h e n a s e t A i n G/H i s u - n u l l i f and o n l y i f n " 1 ( A ) i s X - n u l l i n G (see Chapter IV Theorem 4 . 1 . 6 ) . n For any p o s i t i v e r e a l number a, {xeG: | T f ( x ) | > a} = n~ 1{ XHeG/H : |f(xH)|> a} . S i n c e II Tf l | m= i n f { a | f\({xeG: | Tf ( x) | > a } ) = 0 > the above i m p l i e s t h a t jlTfl! O T= l l f l l ^ so Tf eL°°(G) . C l e a r l y f+Tf i s l i n e a r t h e r e f o r e T I s a l i n e a r i s o m e t r y from L°°(G/H) oo 5f i n t o L (G). C o n s e q u e n t l y the a d j o i n t T i s a l i n e a r CO S 00 . • . * i s o m e t r y from L (G) i n t o L (G/H) . I f <J> i s a mean on CO L (G) and f>_ 0 u alm o s t everywhere i n G/H t h e n Tf>_ 0 X almost everywhere i n G so 0 < <J>(Tf)=T <J>(f). A l s o T <|)( 1 Q / H )= cf>( T ( 1 Q / H ) )= <j>( 1 Q ) = 1 so T <)) i s a mean on CO L (G/H). I f , i n a d d i t i o n , <j> i s l e f t i n v a r i a n t t h e n T % ( L x R f ) = <j>(T(L x Rf))= <))(L xTf)= cf)(Tf)=. T % ( f ) so T*^ i s OO a LIM on L (G/H). L e t fl = { A ( t ) [ teQ},. be a n e s t e d c o l l e c t i o n w i t h range Q i n G/H and d e f i n e B ( t ) = n ~ 1 ( A ( t ) ) f o r each t e Q . S i n c e n R i s c o n t i n u o u s , B ( t ) i s a B o r e l s u b s e t of G f o r each t e Q and fi'={B(t)[ teQ} i s c l e a r l y n e s t e d . I f T i s a LIM on L°°(G) t h e n f o r each teQ we have t= T % ( A ( t ) ) = T ( T x A ( t ) ) = <J»(B(t)) ^ CO . s i n c e T 4> i s a LIM on L (G/H) . C o n s e q u e n t l y a ' i s a n e s t e d c o l l e c t i o n w i t h range Q i n G. LEMMA 3 . 1 . 2 : L e t G be an amenable l o c a l l y compact group and Q a c o u n t a b l e , ' dense s u b s e t of [ 0 , 1 ] such t h a t t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range Q. Then t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . PROOF : L e t ' { A ( t ) [ teQ} be n e s t e d w i t h range Q. For t e [ 0 , l ] l e t A ' ( t ) = f\{A(s) | seQ, t<_ s } . As i n the p r o o f of Lemma 2 . 1 . 3 i t i s e a s i l y checked t h a t { A ' ( t ) | t e [ 0 , l ] } i s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . .2 NESTED COLLECTIONS I N T, R AND Z : The s t r u c t u r e theorems f o r c e r t a i n l o c a l l y compact groups which we employ l a t e r i n the c h a p t e r w i l l r e q u i r e the e x i s t e n c e o f n e s t e d c o l l e c t i o n s w i t h range [ 0 , 1 ] i n the l o c a l l y compact a b e l i a n groups T, R and Z ( t h e c i r c l e g roup, r e a l s and I n t e g e r s r e s p e c t i v e l y ) , so we now prove LEMMA 3 . 2 . 1 : There e x i s t s a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] i n each o f the l o c a l l y compact a b e l i a n groups T, R and Z. PROOF : i ) ' L e t Q={R/h|" 0 < k<" n; k',neZ"+} and f o r k/neQ" d e f i n e A( k/n) = { exp ( 2iTix) | 0<_ x<k/n} . T h e r e f o r e °, = {A(k/n) | k/neQ} i s a n e s t e d f a m i l y o f B o r e l s u b s e t s o f T I f we l e t A n={exp( 2 i r i x ) | 0<_ x<l/n} we have n - 1 T= U exp (2iTim/n) • A ( d i s j o i n t u n i o n ) so f o r m=0 ^ n ° any LIM (j) on L°°(T), 1 = <J)(T)= n'<|>(A ) and c o n s e q u e n t l y <j>(A )'= 1/n . S i n c e n k - 1 A(k/n)= U exp ( 2iTim/n) • A ( d i s j o i n t u n i o n ) m=0 N t h i s i m p l i e s 4»(A(k/n))= k«<j>(A )= k/n . T h e r e f o r e !1 i s a n e s t e d c o l l e c t i o n i n T w i t h range Q and, s i n c e Q i s a c o u n t a b l e dense subset of [ 0 , 1 ] , Lemma 3 . 1 . 2 i m p l i e s the e x i s t e n c e o f a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] , i i ) Note t h a t Z i s a c l o s e d normal subgroup o f R and T= R/Z. By p a r t i ) t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n R/Z w i t h range [ 0 , 1 ] so by Lemma 3 - 1 . 1 such a c o l l e c t i o n a l s o e x i s t s i n R. i i i ) S i n c e Z i s a d i s c r e t e r i g h t c a n c e l l a t i o n , l e f t amenable group which i s a b e l i a n ( hence not an " AB group") the r e s u l t f o l l o w s from Theorem 2 . 1 . 4 (due t o G r a n i r e r ) . However we w i l l g i v e a d i f f e r e n t and c o n s t r u c -t i v e p r o o f o f t h i s r e s u l t i n o r d e r t o i n d i c a t e how the d e s i r e d f a m i l y of l e f t a l m o s t convergent s e t s can be o b t a i n e d . Our p r o o f w i l l depend on the f a c t t h a t , f o r any non-n e g a t i v e i n t e g e r n, the s e t A={m»n| meZ} i s l e f t almost c o n v e r g e n t t o 1/n s i n c e n-1 Z= (i+A) ( d i s j o i n t u n i o n ) i m p l i e s t h a t i = 0 n - 1 1 = <j>(Z)= Z <J>(i+A) = n«(J)(A) f o r any LIM <j> on i = 0 L°°(Z)= m(Z) , Thus g i v e n any number k/n where !<_ k<_ n we can o b t a i n a s e t B which i s l e f t almost c o n v e r g e n t k t o k/n by s e t t i n g B = {J ( i + A ) . i = l I n o r d e r t o o b t a i n a n e s t e d f a m i l y of l e f t a l most c o n v e r g e n t s e t s we p r o c e e d as f o l l o w s : - f o r each p o s i t i v e i n t e g e r n l e t Q = { k / 2 N | 0<_ k<_ 2 N } 00 and l e t Q= [J Q . We o b t a i n , a n e s t e d c o l l e c t i o n i n Z w i t h n=l n range 0. by i n d u c t i o n . Assume we are g i v e n a p e r m u t a t i o n {x-. , . . . ,x„n} o f the i n t e g e r s 0 , 1 , . . . , 2 n - l - a n d d e f i n e A ( k / 2 n ) = U (xi+{m-2n\ meZ}) f o r k / 2 n e _ n i = l s o , as i n d i c a t e d above, th e f a m i l y fin={A(t)| teQ n> i s a n e s t e d c o l l e c t i o n w i t h range Q . We want t o e x t e n d P t o a n e s t e d c o l l e c t i o n P . , w i t h n n+1 range Q and which i s d e f i n e d i n the same manner as P . to n+1 n D e f i n e a p e r m u t a t i o n {y-^ ...,y 2n+l} by the f o r m u l a f o r 1< i< 2 " y = x. + 2 n J 2 i 1 I t i s c l e a r t h a t y^ / y f o r i / j so we have a p e r m u t a t i o n of the i n t e g e r s 0 , 1 , . . . , 2 n + 1 - l . As b e f o r e d e f i n e k 1 = 1 P n + 1 = { B ( k / 2 n + 1 ) I k / 2 n + 1 e Q n + 1 } i s c l e a r l y a n e s t e d c o l l e c t i o n w i t h range Q n + 1- A l l t h a t must be checked i s t h a t P n + 1 extends P. . L e t t = ( k 1 / 2 n ) = ( k 2 / 2 n + 1 ) e Q n A Q n + 1 . T h i s i m p l i e s t h a t k 2=2*k 1 and we have k n , A ( t ) = U (x.+{m-2 n| meZ}). 1 = 1 1 S i n c e , f o r e v e r y i , (x,+{m-2 n| m e Z } ) = ( y 2 1 _ 1 + { m - 2 n + 1 | meZ}) U (y 2.+{m-2 n + 1| meZ}) t h i s i m p l i e s 26 A ( t ) = IjCCy^.n-K m* 2 n+1 | meZ}) U (y 9,+{m-2 n+1 | me Z} )) 1 = 1 = U 1 (yn-+{m-2 n+1 | meZ}) = B ( t ) i = l I f we l e t {0,1} be a p e r m u t a t i o n of the n o n - n e g a t i v e i n t e g e r s l e s s t h a n 2 and d e f i n e A(0)= 0 , A(l/2)={2«m| meZ} A ( l ) = {2«m| meZ}U{2-m+l| meZ} = Z, then by i n d u c t i o n we can o b t a i n a n e s t e d c o l l e c t i o n fi={A(t)| teQ} w i t h range Q. S i n c e Q i s c o u n t a b l e and dense i n [0 , 1 ] , Lemma 3.1.2 i m p l i e s the e x i s t e n c e o f a n e s t e d c o l l e c t i o n i n Z w i t h range [0,1]. NOTE : Combined w i t h Lemma 3.1.1 t h i s I m p l i e s t h a t any l o c a l l y compact amenable group G w i t h T, R or Z as a d i r e c t f a c t o r c o n t a i n s a n e s t e d c o l l e c t i o n w i t h range [0,1] f o r we must have G/H t o p o l o g i c a l l y i s o m o r p h i c w i t h T, R or Z f o r some c l o s e d normal subgroup H o f G. 3.3 NESTED COLLECTIONS IN COMPACT ABELIAN GROUPS : I n t h i s s e c t i o n we show t h a t any i n f i n i t e compact a b e l i a n group has a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] . The f o l l o w i n g s t r u c t u r e theorem w i l l be o f prime impor-t a n c e i n p r o v i n g t h i s r e s u l t . The r e a d e r s h o u l d note t h a t L (G) admits many i n v a r i a n t means i f G i s i n f i n i t e . THEOREM 3.3 1 Theorem 9-5 p. 89) L e t G be a compact a b e l i a n group and U a neighbourhood o f e i n G. Then t h e r e e x i s t s a c l o s e d ( n o r m a l ) subgroup H o f G w i t h Hczu and G/H t o p o l o g i c a l l y i s o m o r p h i c w i t h T^xF where k i s a n o n - n e g a t i v e i n t e g e r and F i s a f i n i t e a b e l i a n group. I t i s c l e a r t h a t i f G c o n t a i n s a c l o s e d (normal) subgroup H w i t h G/H t o p o l o g i c a l l y i s o m o r p h i c w i t h T^ XF as above w i t h k > 0 t h e n G/H c o n t a i n s a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] and c o n s e q u e n t l y G c o n t a i n s such a c o l l e c t i o n by Lemma 3 . 1 . 1 - I t i s t h e r e f o r e s u f f i c i e n t t o c o n s i d e r t he case where G has no c l o s e d subgroup H f o r which G/H has T as a d i r e c t f a c t o r . I n t h i s case we use i n p a r t a method i n t r o d u c e d by G r a n i r e r i n [ 2 ] t o o b t a i n t h e r e s u l t . I n o r d e r t o a p p l y t h e method we f i r s t r e q u i r e the f o l l o w i n g LEMMA 3 . 3 . 2 : L e t G be an i n f i n i t e compact a b e l i a n group such t h a t t h e r e i s no c l o s e d subgroup H o f G f o r which G/H has T as- a dire-c-t-'-f actor-.. Then»-t he-re*-e-xd-st-s-< a s t r i c t l y d e c r e a s i n g sequence {H^} of o p e n - c l o s e d subgroups of G such t h a t G/H n i s a f i n i t e group f o r each n. PROOF : L e t H =G and assume t h a t H, are d e f i n e d f o r 1 k k<_ n s a t i s f y i n g t h e r e q u i r e d p r o p e r t i e s . S i n c e H n i s open we can f i n d a neighbourhood U o f e w i t h H^ and u s i n g Theorem 3 - 3 . 1 we o b t a i n a c l o s e d subgroup H of G w i t h EC-U^E^ and such t h a t G/H i s t o p o l o g i c a l l y i s o m o r -p h i c w i t h T^xF f o r some n o n - n e g a t i v e i n t e g e r k and f i n i t e a b e l i a n group F. The c o n d i t i o n s on G i m p l y t h a t k = 0 so G/H i s f i n i t e . I f we l e t H ,.,=H the n H J_1CZ,E , G/H ' n+ 1 n + 1 r n ' n+ 1 i s f i n i t e and ^ n +^_ ^ s o p e n - c l o s e d s i n c e , f o r any s e t {a^=e,a^, . . ,.,a } o f r e p r e s e n t a t i v e s f o r G / H n + ^ w e h a v e q . q. HN + 1= G ^ yj a 1 * H n + 1 a n d >-^ ai° Hn+l i s c l o s e d because H , -, i s c l o s e d . n+ 1 The r e s u l t f o l l o w s by i n d u c t i o n . T h i s lemma a l l o w s us t o t a k e c a r e o f the r e m a i n i n g case by p r o v i n g LEMMA 3 . 3 . 3 : L e t G be an amenable l o c a l l y compact group f o r which t h e r e e x i s t s a s t r i c t l y d e c r e a s i n g sequence { H J o f c l o s e d normal subgroups such t h a t G / H N i s a f i n i t e group f o r each n. Then G c o n t a i n s a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] . PROOF : L e t p denote the o r d e r o f G / H f o r each n. *n. n Then t p n } I s a s t r i c t l y i n c r e a s i n g sequence o f i n t e g e r s ( s i n c e H ,, ^ H i m p l i e s t h a t G / H <5 G / H ) and p n+1 7* n ^ n 7- n+1 • n d i v i d e s P n + ^ f o r each n ( s i n c e G / H N i s i s o m o r p h i c w i t h a subgroup of G / H N + ^ ) • L e t Q^=^ k/P n! k = i , 2 , . . . ,p n> and note t h a t Q R + 1 f o r a l l n. We w i l l prove t h i s lemma u s i n g a more g e n e r a l f o rm o f the method employed i n Lemma 3 . 2 . 1 t o show t h e e x i s t e n c e of a n e s t e d c o l l e c t i o n i n Z w i t h range [ 0 , 1 ] . For any n, i f {a,,...,a } i s a s e t o f r e p r e s e n t a t i v e s ^n f o r G / H and we d e f i n e n k A(k/p ) = N a . ' H f o r k/p eQ *n V , i n *n n i = l t h e n Q={A(t)| teQ n> i s a n e s t e d f a m i l y o f B o r e l s u b s e t s P.n 29 o f G. S i n c e G= (J a.«H ( d i s j o i n t u n i o n ) , t h e r e f o r e 1 = <J>(G) = E n <j)(a. -H )= p • T ( H ) T 1 = 1 Y 1 n n n' oo f o r any LIM f on L (G) and c o n s e q u e n t l y <j)(H )= ( 1 / p ). So f o r any LIM cf) we must have k T ( A ( k / p ))= 0 ( U a . - H n )= k»<J>(H ) = (k/p ) i = l and fi i s a n e s t e d c o l l e c t i o n i n G w i t h range Q . I f ft i s such a c o l l e c t i o n we want t o f i n d a n e s t e d c o l l e c t i o n ft' o f the same t y p e w i t h range Qn+-]_ w h i c h extends ft. Le t ^c;]_> • • • > c m} be a s e t of r e p r e s e n t a t i v e s f o r the group H /H (which i s a subgroup, o f C/H .,_ and. t h e r e f o r e f i n i t e ) . S i n c e P n +-L = m " P n > ^ o r e v e r y 1<_ i<_ p^ we can w r i t e i u n i q u e l y as i = a(i)«m + g ( i ) where 0<_ a ( i ) < p , 1<_ 3 ( i ) < _ m. D e f i n i n g b.= a / • \,-,'c 0 , . v we o b t a i n a c o l l e c t i o n & I a ( I ) + 1 B ( I ) {b^,...,b } o f elements of G. Suppose x e b ^ * H n + i n b.*H + 1  p n + l i n J n so x= a / . v , • c„ / . >, «h, = a / . <, ,, • c„ / . N - h 0 f o r some a(i-)+-l B ( i ) - 1 a(j)-+-l B ( j ) 2 h n > ^ 2 e ^n+l' S i n c e c B ( i ) * h l 3 C $ ( j ) ' ^ 2 e H n a n d t h e a ' S a r e r e p r e s e n t a t i v e s f o r G/H n t h i s i m p l i e s t h a t a / . N . n = a i . e . a ( i ) = a ( j ) . a ( i ) + l a ( j ) + l 0 C o n s e q u e n t l y c 3 ( i ) * ^ i = c B ( j ) * h 2 a n c i s i n c e t h e c ' s a r e r e p r e s e n t a t i v e s f o r H n / H n + 1 • t h i s i m p l i e s G g ( i ) = c g ( j ) so 3 ( i ) = 3 ( j ) . T h e r e f o r e i=a(i)«m + 3(±) =a(j)-m + 3 ( j ) = j and we have shown t h a t b.*H .,Ab.*H ., = 0 f o r i ^ j , 1 n + 1 ' j n+ 1 ^ ' hence the b's a r e a s e t o f r e p r e s e n t a t i v e s f o r ^/H I n the same manner as b e f o r e we d e f i n e a n e s t e d c o l l e c t i o n ft'={B(t)| t G Q n + 1 ^ w i t h range Q n + 1 by s e t t i n g B ( k / p n + 1 ) = U b r H n + 1 . To show t h a t fi' extends Q l e t ( K 2./P n) = ( K 2 ' / p n + l ^ e ^ n + l ^ ®n' T h i s i m p l i e s t h a t k^= m'k-^ where m i s the o r d e r of H'/H ,, and note t h a t the manner i n wh i c h the b.'s were n n + 1 i d e f i n e d g i v e s us {a.'C.I l< i< k, , 1 < j< m}={b.j 1< i< k •m= k 0 } . x j 1 — — 1 — — x 1 — — 1 2 k, k m T h e r e f o r e A ( k , / p n ) = L r a, -H = U a. • ( (J c -H ) 1 = 1 1 n i = l 1 j = l J n k, m k„ = U U a i - c i ' H n + l = ^ V H n + l 1 = 1 j = l J i = l = B ( k 2 / p n + 1 ) . I t i s c l e a r t h a t by c h o o s i n g any s e t of r e p r e s e n t a t i v e s f o r G/H^ we can o b t a i n a n e s t e d c o l l e c t i o n ft^ w i t h range Q^ by the method employed above. We have shown t h a t any such c o l l e c t i o n ft w i t h range Q can be extended t o a n to n n e s t e d c o l l e c t i o n ^ n + 1 o f t h e same form w i t h range Q n + 1* By i n d u c t i o n we can o b t a i n a n e s t e d c o l l e c t i o n oo ft={A(t)| teQ} w i t h range Q where Q= I J Q n . Q i s c l e a r l y n=l c o u n t a b l e and s i n c e i s s t r i c t l y d e c r e a s i n g we have l i m p = oo so Q I s dense i n [ 0 , 1 ] . The r e s u l t now f o l l o w s n->-co from Lemma 3 . 1 . 2 . We a r e now a b l e t o prove the main r e s u l t o f t h i s s e c t i o n THEOREM 3 - 3 . 4 : L e t G be an i n f i n i t e compact a b e l i a n group. Then t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . PROOF : F i r s t note t h a t G i s amenable because i t i s a b e l i a n . I f G c o n t a i n s a c l o s e d subgroup H f o r which G1=G/H has T as a d i r e c t f a c t o r t h e n G^ i s t o p o l o g i c a l l y i s o m o r p h i c w i t h TxK f o r some c l o s e d subgroup K, c o n s e q -u e n t l y G^/K i s i s o m o r p h i c w i t h T and by Lemma 3 . 1 . 1 t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G^ w i t h range [ 0 , 1 ] . E m p l o y i n g Lemma 3 . 1 . 1 a g a i n t h i s i m p l i e s t h a t G a l s o c o n t a i n s such a c o l l e c t i o n . I f , on the o t h e r hand, t h e r e i s no c l o s e d subgroup H o f G f o r which G/H has T as a d i r e c t f a c t o r t h e n t h e r e s u l t f o l l o w s i m m e d i a t e l y from Lemmas 3 - 3 - 2 and 3 - 3 - 3 • . 4 MAIN RESULTS : I n o r d e r t o prove the main r e s u l t of t h i s c h a p t e r we w i l l r e q u i r e t h e f o l l o w i n g s t r u c t u r e theorem f o r l o c a l l y compact, comp a c t l y g e n e r a t e d a b e l i a n g r o u p s . THEOREM 3.4.1 :([4] p.90 Theorem 9-5) Eve r y l o c a l l y compact, compactly g e n e r a t e d a b e l i a n group G i s t o p o l o g -i c a l l y i s o m o r p h i c w i t h R m * z n x F f o r some n o n - n e g a t i v e i n t e g e r s m and n and some compact a b e l i a n group F. U s i n g t h i s we can now prove THEOREM 3.4.2 : L e t G be an i n f i n i t e l o c a l l y compact a b e l i a n group. Then t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . PROOF : F i r s t c o n s i d e r the case where G i s co m p a c t l y g e n e r a t e d . By Theorem 3-4.1 G i s i s o m o r p h i c w i t h R m x Z n x F . I f e i t h e r m o r n i s non-zero t h e n G has R o r Z as a d i r e c t f a c t o r and G c o n t a i n s a n e s t e d c o l l e c t i o n w i t h range [0,1] by Lemma 3-1-1- O t h e r w i s e G must be an i n f i n i t e compact a b e l i a n group which we have shown i n Theorem 3-3-4 c o n t a i n s the r e q u i r e d f a m i l y . I f G i s not compactly g e n e r a t e d , l e t U be a compact CO symmetric neighbourhood o f e. I f H= | J U n t h e n H i s a n=l subgroup o f G, H i s c l e a r l y open and t h e r e f o r e c l o s e d as w e l l (see [4] p.34). S i n c e G i s not compactly g e n e r a t e d , G/H i s i n f i n i t e and c l e a r l y a b e l i a n . Due t o the f a c t t h a t H i s open, {xH} i s open i n G/H f o r ev e r y c o s e t xH and c o n s e q u e n t l y G/H i s a d i s c r e t e group. As an a b e l i a n group, G/H cannot be an "AB group" so by Theorem 2.1.4 t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G/H w i t h range [0,1] and thus the r e q u i r e d c o l l e c t i o n e x i s t s i n G by Lemma 3-1-1-I n view of Lemma 3 - 1 - 1 we can w r i t e t h i s r e s u l t i n a s t r o n g e r form as COROLLARY 3 - 4 . 3 : L e t G be an amenable l o c a l l y compact group such t h a t t h e r e e x i s t s a c l o s e d normal subgroup H of G f o r which G/H i s i n f i n i t e and a b e l i a n . Then G c o n t a i n s a n e s t e d c o l l e c t i o n w i t h range [ 0 , 1 ] . . 5 REMARKS : We may a p p l y C o r o l l a r y 3 - 4 . 3 t o show LEMMA 3 - 5 - 1 : Let G be an i n f i n i t e amenable non-unimod-u l a r l o c a l l y compact group. Then t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . PROOF : Let H={xeG| A ( x ) = l } (where A denotes the modular f u n c t i o n on G). Then H i s a normal subgroup o f G s i n c e A ( x h x - 1 ) = A(x)«A(h)-A(x _ 1)= A ( h ) • A ( x ) • A ( x _ 1 ) = A(h)= 1 whenever xeG and heH. A l s o H i s c l o s e d because A i s c o n t i n u o u s . The f a c t o r group G/H i s a b e l i a n (as a su b s e t of the p o s i t i v e m u l t i p l i c a t i v e r e a l s ) and i n f i n i t e s i n c e t h e range o f A i s i n f i n i t e when G i s non-unimodular. C o n s e q u e n t l y , by C o r o l l a r y 3 - 4 . 3 , t h e r e e x i s t s a n e s t e d c o l l e c t i o n i n G w i t h range [ 0 , 1 ] . CHAPTER IV AMENABLE ACTIONS OP LOCALLY COMPACT GROUPS 3 4 4 . 0 INTRODUCTION : I f G i s a l o c a l l y compact group and X a l o c a l l y compact space t h e n G i s s a i d t o a c t c o n t i n u o u s l y on X i f t h e r e e x i s t s a j o i n t l y c o n t i n u o u s mapping from GxX-*X denoted by (g,x)+gx which s a t i s f i e s (g^x) = (g-|_g 2) x f ° r a n Y E>2.'&2e<~* a n < ^ xe-^- ^ o r example G a c t s c o n t i n u o u s l y on the l o c a l l y compact space G/H o f l e f t c o s e t s o f G w i t h r e s -p e c t t o a c l o s e d subgroup H i f we d e f i n e g(xH)=gxH f o r geG and xHeG/H (Note: f o r convenience o f n o t a t a t i o n , i n t h e r e -mainder o f t h i s c h a p t e r the c o s e t xH w i l l be denoted by x ) . I n s t u d y i n g i n v a r i a n t means on a l o c a l l y compact group G we are d e a l i n g w i t h the case i n which G a c t s on i t s e l f by l e f t m u l t i p l i c a t i o n . There has been i n t e r e s t r e c e n t l y i n s t u d y i n g the a c t i o n o f groups on c e r t a i n l o c a l l y compact spaces f o r the purpose of e x t e n d i n g t h e concept o f a m e n a b i l -i t y t o t h e s e more g e n e r a l s i t u a t i o n s . I n [ 1 1 ] G r e e n l e a f i n t r o d u c e d t h e concept o f the amenable a c t i o n o f a l o c a l l y compact group on a l o c a l l y compact space which s u p p o r t s a q u a s i - i n v a r i a n t measure and Eymard i n [ 9 ] s t u d i e d i n d e t a i l t he amenable a c t i o n o f a group on i t s space o f l e f t c o s e t s w i t h r e s p e c t t o a c l o s e d subgroup. I n S e c t i o n 4 . 1 we examine the concept o f a q u a s i -i n v a r i a n t measure on the space G/H o f l e f t c o s e t s o f a l o c a l l y compact group G, p a r t i c u l a r l y the r e l a t i o n s h i p between such measures and the l e f t Haar measure on G. The r e s u l t s presented, i n t h i s s e c t i o n a re w e l l known and can be f o u n d , f o r example, i n [ 7 ] and [ 1 0 ] . However the p r o o f s g i v e n i n t h e s e r e f e r e n c e s a r e somewhat s k e t c h y and the r e s u l t s t hemselves are p r e s e n t e d i n the m i d s t o f a g r e a t d e a l o f m a t e r i a l which i s o f l i t t l e i m p o r t a n c e i n the s i t u a t i o n we a r e e x a m i n i n g . We p r e s e n t p r o o f s f o r some o f t h e s e r e s u l t s s i n c e we f e e l t h a t a thorough u n d e r s t a n d i n g o f the r e l a t i o n s h i p s i n v o l v e d p r o v i d e s a much g r e a t e r f e e l -i n g f o r the m a t e r i a l t h a t i s p r e s e n t e d i n t h e r e m a i n d e r o f t h i s c h a p t e r . I n s e c t i o n 4.2 we i n t r o d u c e a d e f i n i t i o n o f t h e amenable a c t i o n o f a group on one of i t s c o s e t spaces which i s e q u i v a l e n t t o the d e f i n i t i o n used by G r e e n l e a f . However the t e c h n i c a l d e t a i l s a re v e r y d i f f e r e n t and we f e e l t h a t i t i s much e a s i e r t o u n d e r s t a n d the r e l a t i o n s h i p between our d e f i n i t i o n o f amenable a c t i o n and the concept o f an amenable group i n the u s u a l sense. We a l s o p r e s e n t p r o o f s o f v a r i o u s r e s u l t s which have been o b t a i n e d by G r e e n l e a f and Eymard. I n many cases the methods o f p r o o f used i n t h i s s e c t i o n a r e much more c l o s e l y a l l i e d w i t h those used t o prove s i m i l a r theorems f o r amenable l o c a l l y compact groups t h a n are t h o s e employed i n [ 1 1 ] . . 1 QUASI-INVARIANT MEASURES : Let G be a l o c a l l y compact group a c t i n g c o n t i n u o u s l y on a l o c a l l y compact space.' X. I f p i s a measure on X d e f i n e , f o r geG, the measure y(g)\i on X by Y ^ S ) u ( A ) = y ( g A) where g A={x eX| g x e A } . Note t h a t t h i s l m p l l e S fxf(gx)dM(x)=/xf(x)d(Y(g)y)(x) f o r a l l f e K ( X ) ( t h e c o n t i n u o u s f u n c t i o n s on X w i t h compact s u p p o r t ) . DEFINITION 4.1.1 : i ) I f y(g)y= y f o r e v e r y geG t h e n u i s s a i d t o be i n v a r i a n t under the a c t i o n o f G i i ) I f Y ( g ) H i s p r o p o r t i o n a l t o y f o r every g£G t h e n y i s s a i d t o be r e l a t i v e l y i n v a r i a n t under the a c t i o n o f G. i i i ) I f Y(g)y=y ( i . e . Y ( g ) y i s e q u i v a l e n t t o y) f o r e v e r y geG t h e n y i s s a i d t o be q u a s i - i n v a r i a n t under the a c t i o n o f G. REMARK : For a r e l a t i v e l y i n v a r i a n t measure i a , Y ( g ) y i s p r o p o r t i o n a l t o y f o r each geG so t h e r e e x i s t s a r e a l con-s t a n t a(g) w i t h y ( g ) y = a ( g ) * y . The mapping a : G->R i s c a l l e d the m u l t i p l i e r o f y. I t . i s c l e a r t h a t a ( x ) > 0 and a ( x y ) = a ( x ) a ( y ) f o r a l l x,yeG. A l s o f o r any measure y on X and f£K(X),/^f(gx)dy(x) i s a c o n t i n u o u s f u n c t i o n o f g so a i s a c o n t i n u o u s f u n c t i o n . EXAMPLES' : L e t G=X=Z i n the above c o n t e x t so we are con-s i d e r i n g the a d d i t i v e group of the i n t e g e r s a c t i n g on i t s e l f . I f we l e t y ( A ) = c a r d ( A ) f o r AcZ ( where c a r d ( A ) denotes the c a r d i n a l i t y o f A ) t h e n y i s a measure and y(m+A)=y(A) f o r any meZ c o n s e q u e n t l y y i s i n v a r i a n t . I n f a c t y i s a Haar measure on the l o c a l l y compact d i s c r e t e group Z. 37 I f we d e f i n e A ( A ) = E { e x p ( n ) | neA} t h e n A i s a r e l a t i v e l y i n v a r i a n t measure s i n c e A(m+A)=exp(m)* A ( A ) f o r any m£Z and AcZ but A i s c l e a r l y not i n v a r i a n t . I f o(A)=£{ exp(|n|)| n£A) t h e n u i s a q u a s i - i n v a r i a n t measure s i n c e the o n l y u n u l l s e t i s 0 but u i s not r e l a t i v e l y i n v a r i a n t ( i f .A={-l) and B=(l} t h e n U ( A ) = U ( B ) but u ( l + A ) = l and u(l+B)= e 2 s o Y ( - l ) u i s not p r o p o r t i o n a l t o u ) , I n the remainder o f t h i s c h a p t e r we w i l l r e s t r i c t o u r -s e l v e s t o the case o f a l o c a l l y compact group G a c t i n g on t h e l o c a l l y compact space G/H of l e f t c o s e t s o f G w i t h r e s p e c t t o a c l o s e d ( not n e c e s s a r i l y normal ) subgroup H o f G.(Note t h a t such an a c t i o n i s n e c e s s a r i l y c o n t i n u o u s ) We w i l l denote by A a f i x e d l e f t Haar measure on G and by 3 a f i x e d l e f t Haar measure on H. A t h o r o u g h t r e a t m e n t o f q u a s i - i n v a r i a n t measures on such c o s e t spaces may be found i n [ 7 ] and [ 1 0 ] . I n t h i s s e c t i o n we f i r s t p r e s e n t , w i t h o u t p r o o f s , f o u r key theorems which i n d i c a t e the r e l a t i o n s h i p between q u a s i - i n v a r i a n t measues on G/H and the measures A and- 3 mentioned, above. U s i n g t h e s e r e s u l t s we t h e n develop, f u r t h e r p r o p e r t i e s o f q u a s i - i n v a r i a n t , r e l a t i v e l y i n v a r i a n t and i n v a r i a n t measures on G/H which a r e needed i n our t r e a t -ment of the concept o f a m e n a b i l i t y f o r G/H. The key theorems we r e q u i r e a r e t h e f o l l o w i n g : THEOREM 4 . 1 . 2 : L e t u be a measure on G. Then u i s q u a s i - i n v a r i a n t i f and o n l y i f u i s e q u i v a l e n t t o X. The p r o o f o f t h i s theorem which c o m p l e t e l y c h a r a c t e r i z e s t h e q u a s i - i n v a r i a n t measures on G can be found i n [ 1 0 ] C h a p t e r V I I 1 . 9 -The next r e s u l t p r o v i d e s a r e l a t i o n s h i p between measures on G/H and t h o s e on G which a l l o w s us .to examine the i n v a r i a n c e p r o p e r t i e s of a measure on G/H by e x a m i n i n g the c o r r e s -p o n d i n g measure on G. THEOREM 4 . 1 . 3 : i ) F o r feK(G) d e f i n e Tf on G/H by the f o r m u l a Tf(x)=/„f(xh)d3(h) (Note t h a t t h e v a l u e of the n i n t e g r a l i s independent o f the c h o i c e o f r e p r e s e n t a t i v e from x s i n c e 3 i s a Haar i n t e g r a l on H). Then TfeK(G/H) and T i s a s u r j e c t i v e l i n e a r mapping f r o m K(G) onto K(G/H). i i ) L e t u be a measure on G/H. Then t h e r e e x i s t s a # unique measure u on G such t h a t / Q / H T f ( x ) d u ( x ) = / G f ( x ) d u # ( x ) f o r e v e r y feK(G) The p r o o f of t h i s r e s u l t can be found i n [ 1 0 ] Chapter V I I s e c t i o n . 2 p. 4 3 - . Now t h a t we have d e f i n e d the measure u on G i t i s p o s s i b l e t o e x t e n d the f u n c t i o n T g i v e n i n p a r t i ) above and o b t a i n THEOREM 4 , 1 . 4 : Let u be a measure on G/H i ) I f f£L 1(G,u #) t h e n the s e t o f xeG/H such t h a t h^-f(xh) i s not 3 measurable i s u n u l l ; t he f u n c t i o n "< Tf on G/H d e f i n e d a lmost everywhere by t h e f o r m u l a T f ( i ) = / ^ f ( x h ) d 3 ( h ) i s u i n t e g r a b l e and / Q / H T f ( x ) d u ( x ) = . / Q f ( x ) d u / / ( x ) 1 $ i i ) T i s a c o n t i n u o u s l i n e a r mapping o f L (G,u ) onto L 1(G/H,u) with||Tflk <||flL # f o r a l l f e L 1 ( G , u # ) . A p r o o f o f t h i s r e s u l t can be found i n [ 1 0 ] C h a p t e r V I I s e c t i o n s 3 and 4 . THEOREM 4 . 1 . 5 : Let u be a measure on G/H. A u-measurable - 1 # s e t A i n G/H i s u n u l l i f and o n l y I f rL r (A) i s u n u l l ri i n G. See [ 1 0 ] Chapter V I I S e c t i o n 2 . 3 f o r a p r o o f . S i n c e a measure i s q u a s i - i n v a r i a n t i f and o n l y i f i t s n u l l s e t s a re i n v a r i a n t under the a c t i o n o f G, Theorem 4 . 1 w i l l be e x t r e m e l y u s e f u l as we attempt t o c h a r a c t e r i z e t h e q u a s i - i n v a r i a n t measures on G/H. One such c h a r a c t e r i z a t i o n i s g i v e n by THEOREM 4 . 1 . 6 : Let u be a measure on G/H. 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 i ) u i s q u a s i - i n v a r i a n t i i ) u -\ ( i . e . u i s e q u i v a l e n t t o A ) i i i ) A i s u - n u l l i n G/H i f and o n l y i f I I ~ 1 ( A ) i s A n u l l i n G. PROOF : i)^»ii) F i r s t note t h a t i f u|< u 2 on G/H t h e n u l < < u 2 o n G ' ^^- s c a n k e seen as f o l l o w s -choose A<=G w i t h v^(A)=0. By Theorem 4 . 1 . 4 / G y H T x A ( x ) d \ j p ( x ) = u 2 ( A ) = 0 and s i n c e T x A i s non-n e g a t i v e t h i s i m p l i e s t h a t T x ^ = 0 u 2 - a l m o s t everywhere i n G/H. S i n c e u^<<u 2 i m p l i e s t h a t T x A = 0 U-^-almost e v e r y -where i n G/H we have ' l ( A ) = ' G x A ( x ) d u l ( x ) = / G T x A ( x ) d u 1 ( x ) = 0 and u*<<u*. A consequence of t h i s i s the f a c t t h a t and u 2 a r e e q u i v a l e n t i f u-^  and u 2 a r e e q u i v a l e n t . L e t u be q u a s i - i n v a r i a n t on G/H and note t h a t f o r feK(G) and geG we have fQf(x)d(Y(g)u#)(x)= / G f ( g x ) d u # ( x ) = / Q / H ( / H f ( g x h ) d 3 ( h ) ) d u ( x ) = / G / H T f ( g x ) d u ( x ) = / G / H T f ( x ) d ( Y ( g ) u ) ( x ) = / G f ( x ) d ( ( Y ( g ) u ) # ) ( x ) so y(g)v = ( Y ( S ) u ) and, s i n c e y(g)v i s e q u i v a l e n t t o u f o r ever y geG, t h i s i m p l i e s y(g)v i s e q u i v a l e n t t o u i . e . u # i s q u a s i - i n v a r i a n t on G. The f a c t t h a t u i s e q u i v a l e n t t o A f o l l o w s from Theorem 4 . 1 . 2 . i i i ) r > i > y ( g ) u ( A ) = i j ( g ~ 1 A ) = 0 i f f 0 = A ( n ~ 1 ( g ~ 1 A ) ) = x ( g " 1 n ~ 1 ( A ) ) = A ( n ~ 1 ( A ) ) ( s i n c e X i s i n v a r i a n t ) i f f u ( A ) = 0 so y(g)v=v and u i s q u a s i -i n v a r i a n t i i ) = > i i i ) Assume u =X. Then A i s u n u l l i n G/H i f f n H (A ) i s u n u l l i n G ( by Theorem 4 . 1 . 5 ) i f f n ~ 1 ( A ) i s A n u l l i n G . The next theorem p r o v i d e s an e x t r e m e l y u s e f u l c h a r a c t e r -i z a t i o n of the q u a s i - i n v a r i a n t measures on G/H which we w i l l use l a t e r t o prove t h a t G/H always admits such a measure. THEOREM 4 . 1 . 7 : I f u i s a q u a s i - i n v a r i a n t measure on G/H t h e n t h e r e e x i s t s a s t r i c t l y p o s i t i v e measurable f u n c t i o n p on G s a t i s f y i n g (*) f o r each heH, p (xh ) =p ( x ) A H ( h ) / A Q ( h ) f o r A a l m o s t a l l x eG and f o r which u =p«A. C o n v e r s e l y i f p i s a s t r i c t l y p o s i t i v e measurable f u n c t i o n on G s a t i s f y i n g (*) t h e n p•A=u f o r some q u a s i -i n v a r i a n t measure u on G/H. PROOF : Assume t h a t u i s q u a s i - i n v a r i a n t on G/H. By Theorem 4 . 1 . 6 u i s e q u i v a l e n t t o A so a p p l y i n g t h e Radon Nikodym Theorem f o r l o c a l l y compact spaces (see [ 4 ] p . l 4 4 Theorem 1 2 . 1 7 ) t h e r e e x i s t s a s t r i c t l y p o s i t i v e measurable u f u n c t i o n p on G such t h a t u =p•A . Choose f e K ( G ) , h ^ H and l e t geK(G) be d e f i n e d by g ( x ) = f ( x h " 1 ) . Then £g(x )p (x)dA ( x ) = fQf (xh~1)Q (x)dA (x) = / Q f ( x ) A Q ( h )p (xh )dX (x) ( r e p l a c i n g x by x h Q ) A l s o t g ( x ) = / H g ( x h ) d 3 ( h ) = f„f(xhh~1)d3(h) = A„-(h ) /„f(xh ) d 3(h) ( r e p l a c i n g h by hh ) n O n O - W T f U ) C o n s e q u e n t l y J r , f ( x ) A _ ( h ) p ( x h )dA(x)= / ~ g ( x ) p ( x ) d A ( x ) u u O O (J = / G g ( x ) d u # ( x ) = / Q / H T g ( x ) d u ( x ) = / G / H A H ( h . o ) T f ( x ) d u ( x ) = / Q A H ( h o ) f ( x ) d u # ( x ) = / G A H ( h 0 ) f ( x ) p ( x ) d X ( x ) . S i n c e f e K ( G ) was a r b i t r a r y t h i s i m p l i e s A G ( h o ) p ( x h o ) = A H ( h o ) p ( x ) f o r A almost a l l xeG hence p s a t i s f i e s (*). G i v e n a s t r i c t l y p o s i t i v e measurable f u n c t i o n p s a t i s f y i n g (*) we want t o d e f i n e a q u a s i - i n v a r i a n t measure u on G/H w i t h u = p«A. The o b v i o u s way t o do t h i s i s t o s e t / G / R T f ( x ) d u ( x ) = fQf(x)p(x)dA(x) f o r each f e K ( G ) . S i n c e T: K(G)-*K(G/H) i s s u r j e c t i v e and l i n e a r t h i s f o r m u l a w i l l d e f i n e a measure u on G/H p r o v i d e d t h a t the l e f t hand s i d e i s w e l l - d e f i n e d f o r each TfeK(G/H). To show t h a t u i s w e l l - d e f i n e d we choose feK(G) w i t h Tf=0 and show t h a t / _ f ( x ) p ( x ) d A ( x ) = 0 . L e t geK(G) be such t h a t Tg(x)= / R g ( x h ) d 3 ( h ) = l f o r a l l x e s u p p ( f ) ( t h i s i s p o s s i b l e due t o Theorem 4 . 1 . 3 s i n c e I L r ( s u p p ( f ) ) i s compact rl i n G/H and T maps K(G) onto K(G/H)). We t h e n have 43 / Q f ( x ) p ( x ) d A ( x ) = / G f ( x ) ( / H g ( x h ) d B ( h ) ) p ( x ) d X C x ) = /„(/,,f ( x ) g ( x h ) p ( x ) d A ( x ) )dB(h) by F u b i n i ' s Thm. = / H ( / G A G ( h _ 1 ) f ( x h _ 1 ) g ( x ) p ( x h _ 1 ) d A ( x ) ) d B ( h ) ( r e p l a c i n g x by x h - 1 ) = / H ( / G A H ( h _ 1 ) f ( x h _ 1 ) g ( x ) p ( x ) d A ( x ) ) d B ( h ) ( u s i n g «) = / Q g ( x ) p ( x ) ( / H A H ( h _ 1 ) f ( x h - 1 ) d B ( h ) ) d A ( x ) = / Q g ( x ) p ( x ) ( / R f ( x h ) d B ( h ) ) d A ( x ) ( r e p l a c i n g h by h" 1) =0 s i n c e Tf=0 i m p l i e s / „ f ( x h ) d 3 ( h ) = 0 f o r A a.a. xeG. r l T h e r e f o r e u i s w e l l - d e f i n e d on G/H and / Q f ( x ) d u # ( x ) = / G / H T f ( x ) d u ( x ) = / G f ( x ) p ( x ) d A ( x ) ft f o r a l l feK(G) hence u =p«A. S i n c e p i s s t r i c t l y p o s i t i v e we have p*A e q u i v a l e n t t o A c o n s e q u e n t l y u sA and u i s q u a s i - i n v a r i a n t by Theorem 4.1.6. We can now use t h i s r e s u l t t o o b t a i n a s i m i l a r c h a r a c -t e r i z a t i o n of the r e l a t i v e l y i n v a r i a n t and i n v a r i a n t measures on G/H. The f o l l o w i n g appears i n p a r t as Theorems 15-22, 15-24 of [ 4 ] . THEOREM 4 ..1....8 : I f u Is. a r e l a t i v e l y i n v a r i a n t measure on G/H the n t h e r e e x i s t s a s t r i c t l y p o s i t i v e c o n t i n u o u s f u n c t i o n p on G s a t i s f y i n g ( * * ) p ( x y ) = p ( x ) p ( y ) / a f o r a l l x,yeG p(h)= a •A„(h)/A„(h) f o r a l l heH jf. where a>0 i s some c o n s t a n t , and f o r which u =p*A. C o n v e r s e l y i f p i s a s t r i c t l y p o s i t i v e c o n t i n u o u s •n f u n c t i o n on G s a t i s f y i n g (**) t h e n p-A=u f o r some r e l a t i v e l y i n v a r i a n t measure u on G/H. PROOF : Assume u i s a r e l a t i v e l y i n v a r i a n t measure on G/H w i t h m u l t i p l i e r a i . e . f o r e v e r y yeG / G / R T f ( y x ) d u ( x ) = a ( y ) / G / R T f ( x ) d u ( x ) f o r a l l TfeK(G/H) Fo r any feK(G) we have / Q f ( y x ) d u # ( x ) = / G / R T f ( y x ) d u ( x ) = a ( y ) / G / H T f ( x ) d u ( x ) = a ( y ) / G f ( x ) d o # ( x ) # so u i s r e l a t i v e l y i n v a r i a n t on G w i t h m u l t i p l i e r a. ft L e t y=a*u . For any geG and f e K ( G ) / Q f ( g x ) d y ( x ) = / Q f ( g x ) a ( x ) d u # ( x ) ft = (l/a(g.) ) / G f ( g x ) a ( g x ) d u (x) s i n c e a ( x ) = a ( g x ) / a ( g ) = / G f ( x ) a ( x ) d o (x) s i n c e u i s r e l a t i v e l y i n v a r i a n t = / G f ( x ) d v i ( x ) T h e r e f o r e y i s an i n v a r i a n t measure on G so t h e r e e x i s t s a c o n s t a n t a>0 f o r which y=a*A (due t o the uniqueness o f Haar measure). By Theorem 4.1.7 t h e r e e x i s t s a s t r i c t l y p o s i t i v e measurable f u n c t i o n p s a t i s f y i n g (*) f o r which u =p«A. ft Thus we have a*p*A= au =y=a*A which i m p l i e s t h a t a*p=a A almost everywhere i n G i.e.p=a/a almost everywhere. T h e r e f o r e we can assume t h a t p i s c o n t i n u o u s (due t o the c o n t i n u i t y o f a) and e q u a l t o a/a which I m p l i e s t h a t 45 p(xy)= a/a(xy)= a / ( a ( x ) a ( y ) ) = p ( x ) p ( y ) / a f o r a l l x,yeG and s i n c e p s a t i s f i e s ( *), f o r any heH^xeG we have p ( x ) A H ( h ) / A G ( h ) = p ( x h ) = p ( x ) p ( h ) / a . so p(x)>0 i m p l i e s p(h)= a»A H(h)/Ag(h) . C o n v e r s e l y assume t h a t we are g i v e n a s t r i c t l y p o s i t i v e c o n t i n u o u s f u n c t i o n p on G which s a t i s f i e s ( * * ) . Then p(xh) = p ( x ) p ( h ) / a = p ( x ) (1/a) • a ' A H ( h ) / A ( J C h ) = p ( x ) A H ( h ) / A G ( h ) f o r a l l x.eGjheH. S i n c e p a l s o s a t i s f i e s (*) of Theorem 4.1.7 t h e r e e x i s t s a measure u on G/H w i t h o =p*X. Fo r any feK( G ) and geG we have / G / H T f ( g x ) d u ( x ) = / Q f ( g x ) d u # ( x ) = / Q f ( g x ) p ( x ) d A ( x ) = ( l / p ( g ) ) / Q f ( g x ) p ( g x ) d A ( x ) s i n c e p(x)=p.(gx)/p(g.). = ( l / p ( g ) ) / g f ( x ) p ( x ) d A ( x ) s i n c e X i s i n v a r i a n t = ( 1 / P ( g ) ) / G f ( x ) d u # ( x ) = ( l / p ( g ) ) / Q / H T f ( x ) d u ( x ) c o n s e q u e n t l y u i s r e l a t i v e l y i n v a r i a n t w i t h m u l t i p l i e r 1/p. COROLLARY 4.1.9 : There e x i s t s an i n v a r i a n t measure u on G/H i f and o n l y i f A n ( h ) = A u ( h ) f o r a l l heH. F u r t h e r m o r e u n i f u i s i n v a r i a n t t h e n every i n v a r i a n t measure on G/H i s o f the form b«u f o r some c o n s t a n t b>0. PROOF : Assume t h a t u i s an i n v a r i a n t measure on G/H. Then u i s r e l a t i v e l y i n v a r i a n t w i t h m u l t i p l i e r a ( y ) = l f o r e v e r y yeG. S i n c e the s t r i c t l y p o s i t i v e c o n t i n u o u s p f o r w hich u =p*A e q u a l s a/a t h i s i m p l i e s t h a t p(x)=a f o r a l l xeG hence a=p(h)=a»A H(h)/A G(h) f o r a l l heH and t h e r e -f o r e A„(h)=A„(h). u ri I f A Q ( h ) = A H ( h ) f o r a l l heH, l e t a>0 and d e f i n e p(x)=a f o r a l l xeG. Then p s a t i s f i e s (**) and t h e r e e x i s t s a r e l a t i v e l y I n v a r i a n t measure u on G/H w i t h u =a*A. For TfeK(G/H) and geG we have / G / H T f ( g x ) d u ( x ) = / Q f ( g x ) d u # ( x ) = / G f ( g x ) - a dX(x) = / Q f ( x ) . a dA(x)= / Q f ( x ) d u # ( x ) = / G / H T f ( x ) d u ( x ) and u i s i n v a r i a n t . ft I f U-^  and u 2 are i n v a r i a n t measures on G/H t h e n u^=a^«A ff and u 2=a 2«A. f o r some p o s i t i v e c o n s t a n t s a^ and a 2 . T h i s ft ft i m p l i e s t h a t Uj^Ca^/a^O- • and., i t . i s e a s i l y . .che.c,ke.d... that., t h i s g i v e s \)^= ( a 1/a- 2) • u 2=b • L>2 where b=a 1/a 2>0. We now a p p l y Theorem 4.1.7 t o prove THEOREM 4.1.10 : There e x i s t s a q u a s i - i n v a r i a n t measure on G/H. PROOF : Choose any f^O i n K +(G) and l e t A={xeG| f ( x ) > 0 } so A' i s compact and A i s open. By means' of a p u r e l y t o p o -l o g i c a l argument (see [ 7 ] p . l ' 6 l f o r d e t a i l s ) t h e r e e x i s t s a s u b s e t Y of G such t h a t the f a m i l y {AyE}^^ c o v e r s G and i s l o c a l l y f i n i t e ( i . e . e v e r y p o i n t of G has a n e i g h b o u r -hood which meets a t most f i n i t e l y many of the s e t s AyH). For each yeG d e f i n e the f u n c t i o n f^eK(G) by f ( x) =f (xy -" 1") f o r every xeG and l e t F(x)=£ „f ( x ) . S i n c e x i s an element of a t most yeY y' f i n i t e l y many of the s e t s {Ay} y, ^ ^ s w e l l - d e f i n e d as a f i n i t e sum and c o n t i n u o u s s i n c e {Ay} y i s l o c a l l y f i n i t e D e f i n e p(x)= ( / R P ( x h ) / A ( h ) d 3 ( h ) ) / A Q ( x ) . S i n c e { A y H } y £ y c o v e r s G, f o r each xeG t h e r e I s some heH and yeY w i t h xhy~"^eA hence F(xh)>0. S i n c e F i s c o n t i n u o u s t h i s i m p l i e s t h a t p(x)>0 and p i s c l e a r l y c o n t i n u o u s as a q u o t i e n t of c o n t i n u o u s f u n c t i o n s . I f h eH t h e n o p ( x h o ) = ( / H ( F ( x h o h ) / A H ( h ) ) d 3 ( h ) ) / A Q ( x h o ) = ( / H ( F ( x h ) / A H ( h ~ 1 h ) ) d 3 ( h ) ) / A G ( x h o ) r e p l a c i n g h by h " 1 ] ! = ( A H ( h o ) / A G ( h o ) ( / H ( F ( x h ) / A H ( h ) ) d 3 ( h ) ) / A G ( x ) = ( A H ( h Q ) / A G ( h Q ) ) - p ( x ) S i n c e p s a t i s f i e s (*) of Theorem 4 . 1 . 7 t h e r e e x i s t s a q u a s i - i n v a r i a n t measure u on G/H w i t h u =p«A. REMARK : For t h e f u n c t i o n F d e f i n e d above, i f we s e t P 1 ( x ) = / R F ( x h ) d 3 ( h ) t h e n F^ i s c o n t i n u o u s and s t r i c t l y p o s i t i v e and i f we d e f i n e k Q ( x ) = F ( x ) / F ^ ( x ) f o r each xeG we have a non-n e g a t i v e c o n t i n u o u s f u n c t i o n k Q on G such t h a t f o r any compact s e t B i n G, k Q c o i n c i d e s on BH w i t h a f u n c t i o n i n K + ( G ) and T k ( j . ) = f . ( x h ) d 3 ( h ) = x f o r a l l x e G t o H o I f u i s a q u a s i - i n v a r i a n t measure on G/H and geL^CG/HjU) i t can be v e r i f i e d t h a t the f u n c t i o n g' =k o • ( g c l l ^ ) i s i n L (G,u w) and c l e a r l y Tg»=g. T h i s i s the method which i s used i n t h e p r o o f of Thm. 4 . 1 . 4 t o show t h a t T i s s u r j e c t i v e . 48 4.2 AMENABLE ACTIONS OF G ON G/H : I n d e f i n i n g the c o n c e p t s 0 0 o f i n v a r i a n t and t o p o l o g i c a l i n v a r i a n t means on L (G) f o r a l o c a l l y compact group G, s t r o n g use i s made o f the con-v o l u t i o n o p e r a t i o n s which p r o v i d e c o n t i n u o u s b i l i n e a r mappings from M(G) x L 1 ( G ) ^ L 1 ( G ) and M(G) XL°°(G)-^h"(G) (where M(G) denotes the space of f i n i t e Radon measures on G). I n o r d e r t o e x t e n d t h e d e f i n i t i o n s t o i n c l u d e the case where G a c t s on t h e space G/H of l e f t c o s e t s o f G w i t h r e s p e c t t o a c l o s e d subgroup H, we must d e f i n e s i m i l a r 1 1 0 0 OO o p e r a t i o n s from M(G)xL (G/H)-»-L (G/H) and M(G)xL (G/H)+L (G/H) which c o i n c i d e w i t h c o n v o l u t i o n i n t h e case t h a t H={e} and G/H=G. The o p e r a t i o n s we d e f i n e are e q u i v a l e n t t o those i n t r o d u c e d by G r e e n l e a f i n [11] however our approach i s d i f f e r e n t due t o the f a c t t h a t we r e l y h e a v i l y on the m a t e r i a l d e v e l o p e d i n S e c t i o n 4.1 which a l l o w s f o r a more e x p l i c i t d e f i n i t i o n of t h e r e q u i r e d o p e r a t i o n s . I n view of Theorem 4.1.10 we know t h a t t h e r e e x i s t q u a s i - i n v a r i a n t measures on G/H and any two such measures a r e e q u i v a l e n t , s i n c e by Theorem 4.1.6 the n u l l s e t s of a q u a s i - i n v a r i a n t measure on G/H a r e p r e c i s e l y t h e s u b s e t s A of G/H f o r which IT""'"(A) i s A n u l l i n G. F o r the r e m a i n d e r n of t h i s c h a p t e r u w i l l denote a f i x e d q u a s i - i n v a r i a n t  measure on G/H and, s i n c e we w i l l be e x a m i n i n g the space 00 L (G/H,u) of u e s s e n t i a l l y bounded B o r e l measurable f u n c t i o n s on G/H, our r e s u l t s w i l l h o l d f o r any o t h e r q u a s i -00 i n v a r i a n t measure because L (G/H) does not depend on u. F o r t h e q u a s i - i n v a r i a n t measure u we have chosen, we know, by Theorem 4.1.7,' t h a t t h e r e e x i s t s a s t r i c t l y p o s i t i v e measurable f u n c t i o n p on G w i t h u =p-A . I n view o f Theorem 4.1.10 e s t a b l i s h i n g the e x i s t e n c e o f a q u a s i - i n v a r i a n t measure on G/H, we w i l l assume t h a t u has been chosen so t h a t p i s a l s o c o n t i n u o u s w i t h p(xh)= ( p ( x ) - A T I ( h ) )/A_(h) f o r a l l xeG, heH . n u F o r f e L 1 ( G , A ) l e t T f = T ( f / p ) ( w i t h T d e f i n e d as i n . Theorem 4.1.4). S i n c e / n ( f ( x ) / p ( x ) ) d u # ( x ) = / p f ( x ) d A ( x ) t h e f u n c t i o n ( f / p ) i s i n L 1 ( G , u # ) c o n s e q u e n t l y T feL 1(G/H,u) w i t h T f ( x ) = /„(f(xh ) / p(xh))d3(h) almost everywhere n i n G/H. S i n c e T i s c o n t i n u o u s , l i n e a r and s u r j e c t i v e , T i s a l s o a c o n t i n u o u s l i n e a r s u r j e c t i o n from L^(G,A) onto I / W H . O ) w i t h l l T f | / l j U = | | T ( f / p ) J | l i U < \\f/p]]ltu# = H f H l 3 A ' T h e r e f o r e L 1(G/H,u) (which w i l l be r e f e r r e d t o h e n c e f o r t h as L 1 ( G / H ) ) i s i s o m e t r i c a l l y i s o m o r p h i c w i t h L 1 ( G ) / J ( T ) -where J ( T ) = { f e L 1 ( G ) | Tf=0} denotes the k e r n e l o f T. I t i s shown i n [ 7 ] C h a p t e r 8 S e c t i o n 2.5 t h a t J(T") can be c h a r a c t e r i z e d as the c l o s e d l i n e a r span of { A h f - f | heH, f e K ( G ) } where A h f ( x ) = f ( x h ) - A Q ( h ) f o r a l l xeG. oo I n w o r k i n g w i t h t o p o l o g i c a l i n v a r i a n t means on L (G) f o r a l o c a l l y compact group G r e f e r e n c e I s o f t e n made t o t h e s e t P ( G ) = { f e L 1 ( G ) | f> 0, / _ f ( x ) d X ( x ) = 1}. I n the m a t e r i a l t h a t f o l l o w s we w i l l r e f e r t o P(G/H)={feL 1(G/H)| f > 0, / Q / H f ( x ) d u ( x ) = 1}. I f f e P ( G ) t h e n Tf> 0 and / Q / H T f ( x ) d u ( x ) = / Q f ( x ) d X ( x ) = 1 so TfeP(G/H). L e t geP(G/H) and l e t kQ be t h e n o n - n e g a t i v e c o n t i n u o u s f u n c t i o n on G d e f i n e d i n the remark f o l l o w i n g Theorem 4.1.10. I f g'= k n ' ( f » n „ ) t h e n g'e^CG.u*) and U r l Tg'=g. I f g i s d e f i n e d on G by g= g'*p t h e n g>_ 0 s i n c e k g , f , p and II^ are n o n - n e g a t i v e . A l s o / Q g ( x ) d X ( x ) = / G g ' ( x ) p ( x ) d X ( x ) = / G g ' ( x ) d u # ( x ) = / G / H T g ' ( x ) d u ( x ) = / Q / H g ( x ) d u ( x ) = 1 ( s i n c e geP(G/H) ), c o n s e q u e n t l y geP(G) w i t h Tg= T(g/p).= T(.g.!)= g and. T map.s P(G.). onto P(.G/H)... The o p e r a t o r T p r o v i d e s a c o n v e n i e n t method f o r w o r k i n g w i t h f u n c t i o n s i n L 1(G/H) by d e a l i n g w i t h t h e i r pre-images i n L^"(G) . The f o l l o w i n g lemma p r o v i d e s a CO s i m i l a r r e l a t i o n s h i p between f u n c t i o n s i n L (G/H) and L ( G ) . LEMMA 4.2.1 : L e t feL°°(G/H) and d e f i n e f on G by y\ /v z\ f = f ° n H - Then feL°°(G) w i t h l l f l l ^ II f 1 1 ^ . CO F u r t h e r m o r e i f geL (G) i s such t h a t f o r e v e r y heH g(xh)= g(x) f o r almost a l l xeG — 00 — t h e r e e x i s t s a f u n c t i o n geL (G/H) such t h a t g = g. PROOF : S i n c e II^ i s c o n t i n u o u s and f i s m e a s u r a b l e , we have f measurable on G. By Theorem 4.1.6 a s e t A i n G/H i s u - n u l l i f and o n l y i f n^"'"(A) i s X - n u l l i n G. F o r a g i v e n r e a l number a l e t A = {xeG/H: | f ( x ) | >a } so we have a n H1 ( A r v ) = { x e G : | f ( x) { > a} S i n c e u(A )= 0 i f and o n l y i f x ( n T T 1 ( A ))= 0 t h i s i m p l i e s a i i a /\ /\ t h a t llfll = U f)! c o n s e q u e n t l y the mapping f + f i s an i s o m e t r y from L°° {G/R) i n t o L°°(G). 0 0 L e t geL (G) be such t h a t , f o r e v e r y heH, g(xh)= g(x') f o r almost a l l xeG. F i r s t note t h a t t h i s p r o p e r t y i s independent o f t h e c h o i c e o f r e p r e s e n t a t i v e from the e q u i v a l e n c e c l a s s denoted by g. T h i s can be seen as f o l -lows : l e t g^jg^BMCG) w i t h g = g 2 almost everywhere and such t h a t , f o r eve r y heH, g^(xh)= g -^ (x) f o r a l m o s t a l l xe,G„ Let.. he.H, .b.e f i x e d and, le.t... N={xeG| g x ( x ) ^ g 2 ( x ) }, A={xeG| g±(xh)^ g 1 ( x ) } so X(N) = X(A) = X ( N h - 1 ) = 0. I f B= A U N U N h - 1 t h e n X(B)= 0 and f o r xjzfB we have g 2 ( x h ) = g 1 ( x h ) = g 1 ( x ) = g 2 ( x ) s i n c e xhg'N, xg'A and xg'N c o n s e q u e n t l y g 2 ( x h ) = g 2 ( x ) .almost everywhere and t h e p r o p e r t y i s independent on the c h o i c e o f r e p r e s e n t a t i v e . C h o o s i n g h^eH and feK(G) we have f Qf ( x h Q ) A G ( h 0 ) g ( x ) d X (x) = / Q f ( x ) g ( x h ~ 1 ) d X ( x ) r e p l a c i n g x by x h " 1 = / Q f ( x ) g ( x ) d X ( x ) s i n c e g(x)= gCxh" 1) a.e. T h e r e f o r e / „ (A. f - f ) (x)g(x)dX (x) = G hQ = / Q ( f ( x h 0 ) A G ( h 0 ) - f ( x ) ) g ( x ) d X (x)= 0 f o r a l l h QeH, f e K ( G ) . S i n c e J ( T ) i s the c l o s e d l i n e a r span o f {A f - f | f e K ( G ) , heH} and f -»•/ f ( x ) g ( x ) d X ( x ) i s c o n t i n u o u s , we have fnf(x)g(x)dX(x)= 0 f o r a l l f e J ( T ) . I f we d e f i n e <£>(f)= fnf(x)g(x)dX(x) f o r f e L 1 ( G ) t h e n u $ i s a 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 on L"^(G) which v a n i s h e s on J ( T ) . S i n c e L 1 ( G / H ) = L 1 ( G ) / J ( T ) we can d e f i n e t h e 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 $' on L^"(G/H) by s e t t i n g $ '(Tf)= $ ( f ) f o r T f e L 1 ( G / H ) . S i n c e L 1(G/H)*= L°°(G/H) (see [ 4] p . l 4 8 Theorem 12.18) _ CO t h e r e e x i s t s a f u n c t i o n geL (G/H) such t h a t * ' ( T f ) = / G / R T f ( x ) i ( x ) d u ( x ) . S i n c e T ( f - g ) ( x ) = / R ( f ( x h ) g ( x h ) / p ( x h ) ) d g ( h ) = / R ( f ( x h ) g ( x ) / p ( x h ) ) d B ( h ) = g ( x ) T f ( x ) t h i s i m p l i e s t h a t / Q f ( x ) g ( x ) d X ( x ) = / G / H g ( x ) T f ( x ) d u ( x ) = $ t ( T f ) = $ ( f ) = fQf(x)g(x)dX(x) f o r a l l f e L 1 ( G ) so g= g. With the o p e r a t o r s T: L (G)->L (G/H) and : L (G/H)-*L (G) d e f i n e d we now i n t r o d u c e the c o n v o l u t i o n type o p e r a t o r s w hich w i l l be used i n the m a t e r i a l t h a t f o l l o w s . DEFINITION 4.2.2 : a) I f yeM(G) and f eL°°(G/H) l e t y®f CO denote the unique f u n c t i o n i n L (G/H) f o r which y®f = y * f on G. b) I f yeM(G) and f e L 1 ( G / H ) t h e n u«feL 1(G/H) i s d e f i n e d by y®f= T(y«T _ 1(f)) . CO / CO t REMARKS : 1. S i n c e f e L (G) f o r a l l f e L (G/H) we have oo y * f e L (G) w i t h , f o r eve r y heH, y * f ( x h ) = fG?(y~1xh)dy(y)= fQfoIL^y^xlOdy(y) = / G f o n H ( y _ 1 x ) d y ( y ) = > G f ( y _ 1 x ) d y ( y ) = y # f ( x ) f o r a lmost a l l xeG so by Lemma 4.2.1 t h e r e e x i s t s a oo s^***** ^ f u n c t i o n y®feL (G/H) w i t h y®f = y * f and y®f i s w e l l - d e f i n e d . 2. I f h QeH and feK(G) t h e n , f o r yeM(G), T ( y * A . f ) ( x ) = /„(y*A.f ( x h ) / p ( x h ) dg(h) h Q H h Q = / R / G ( f ( y _ 1 x h h 0 ) A G ( h 0 ) / p ( x h ) ) d y ( y ) d B ( h ) = / Q / H ( f ( y _ 1 x h h 0 ) A G ( h 0 ) / p ( x h ) ) d 3 ( h ) d y ( y ) = / G / H ( f ( y " 1 x h ) A G ( h 0 ) A H ( h ~ 1 ) / p ( x h h G 1 ) ) d 3 ( h ) d y ( y ) ( r e p l a c i n g h by h h ^ ) = / G / H ( f ( y _ 1 x h ) / p ( x h ) ) d 3 ( h ) d y ( y ) ( s i n c e p ; ( x h h Q 1 ) = p ( x h ) A H ( h ~ 1 ) / A G ( h Q 1 ) ) = / H / G ( f ( y ~ 1 x h ) / p ( x h ) ) d y ( y ) d 3 ( h ) = / H ( ( y * f ) ( x h ) / p ( x h ) ) d 3 ( h ) = T ( y * f ) ( x ) so T(y«(A hf-f))= 0 f o r a l l feK(G) and h QeH. S i n c e the mapping g-*-T(y*g) i s c o n t i n u o u s t h i s i m p l i e s t h a t T(y*g)= 0 whenever geJ(T) c o n s e q u e n t l y t h e f u n c t i o n H©feL]-(G/H) i s w e l l - d e f i n e d f o r a l l yeM(G) and f e L 1 ( G / H ) . 3 . I t s h o u l d be noted t h a t the f u n c t i o n u@f d e f i n e d above i s independent o f the q u a s i - i n v a r i a n t measure we have chosen on G/H. I f and are d i f f e r e n t q u a s i - i n v a r i a n t measures on G/H t h e n they a re e q u i v a l e n t by Theorem 4.1.6 so t h e r e e x i s t s a s t r i c t l y p o s i t i v e measurable f u n c t i o n k on G/H such t h a t , f o r eve r y f e L 1 ( G / H , u 1 ) we have f'= f • k e L 1 ( G / H , u 2 ) w i t h / Q / H f ( x ) d u 1 ( x ) = / Q / H f ' ( x ) d u 2 ( x ) . Thus A: f + f ' i s 1 1 an i s o m e t r i c i s omorphism o f L (G/H,u^) onto L (G/H,u 2). I t i s e a s i l y checked t h a t Af,,i» f\ — ,,sa f hp \ f^-v, n i l , , ^ m / / - i . \ f ~T 1 I n /tr , i "\ (where ®^ , ® 2 denote t h e c o n v o l u t i o n type o p e r a t i o n s c o r r e s p o n d i n g t o u, and u„ r e s p e c t i v e l y ) . Some p r o p e r t i e s o f the o p e r a t i o n s ® and ® which we w i l l need i n l a t e r work a r e the f o l l o w i n g . LEMMA 4.2.3 : Let u ^ e M f G ) ; f eL°°(G/H) and geL 1(G/H) Then i ) ®: M(G)xL (G/H)-*L (G/H) i s a j o i n t l y c o n t i n u o u s b i l i n e a r o p e r a t o r w i t h l!y®fllco <_ )1 y 11 • I I f . i i ) ®: M(G)xL 1(G/H)->L 1(G/H) i s a j o i n t l y c o n t i n u o u s b i l i n e a r o p e r a t o r w i t h ||y@g||-^  < |!y|l • l l g l l ^ . i i i ) ( y * y , )®f = y®(y_, ®f) PROOF : i ) S i n c e y®f = y # f , we have by Lemma 4.2.1, II y®.f II =IJy®fll = l l y * f l l <»/yl!-llf|l = || nil • l l f )l which i m p l i e s O O C O OO ^ 0 0 ' i i " ' CO ^ t h a t ® i s j o i n t l y c o n t i n u o u s . I t i s e a s i l y checked t h a t the mapping f->f i s l i n e a r and s i n c e (y , f)->-y*f ,where yeM(G), feL°°(G),is b i l i n e a r , t h e f a c t t h a t ® i s b i l i n e a r f o l l o w s i m m e d i a t e l y . i i ) I n Remark 2 above we showed t h a t T(y*k)= 0 whenever k e J ( T ) . L e t g e L 1 ( G ) w i t h Tg= g and l e t k e J ( T ) . Then 11 ia©g|/:L = 11 T (P « g ) - j ^ = < | l y * ( g+k)|/ 1 s i n c e II T i l < 1 < || y 11-11 g + k l ^ S i n c e L 1 ( G / H ) = L 1 ( G ) / J ( T ) we have 1| g)L = i n f I) g+k!L and 1 keJ.(T) 1 t h e f a c t t h a t k e J ( T ) was a r b i t r a r y i n the c a l c u l a t i o n above i m p l i e s t h a t || y®gf)^ <_ II y II • l l g / / ^ and ® i s j o i n t l y c o n t i n u o u s . S i n c e T i s l i n e a r and (y,g)-»-y*g , where yeM(G) and geL^CG), i s b i l i n e a r , i t i s c l e a r t h a t © i s b i l i n e a r . i l l ) Note t h a t ("y^ y^ T®?" = ( y * y 1 ) * f = y«(y 1*f) = y%(y 1®f) = y®(y 1®f) CO CO and the f a c t t h a t i s an i s o m e t r y from L (G/H) i n t o L (G) i m p l i e s t h a t (y«y 1)®f = y®(y 1®f) . We a r e now a b l e t o d e f i n e the co n c e p t s o f the amenable a c t i o n o f a group on i t s c o s e t spaces as f o l l o w s DEFINITION 4.2. j4 : L e t G be a l o c a l l y compact group and H a c l o s e d subgroup o f G. I f X i s any o f the spaces L°°(G/H) , CB(G/H), LUC(G/H) and y i s a mean on X t h e n y i s s a i d t o be i ) a l e f t i n v a r i a n t mean (LIM) i f u(6 ®f) = y ( f ) S f o r e v e r y geG, feX (where 6 eM(G) denotes p o i n t mass a t g ) . i i ) 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 mean (TLIM) i f y(cj)®f) - y ( f ) f o r every <j>eP(G) = { g e L 1 ( G ) | f >0, f Qf (x) dX (x) =1} (note t h a t <j)®f i s d e f i n e d s i n c e L 1 ( G ) can be embedded i n M(G)). We say t h a t X i s G ( t o p o l o g i c a l l y ) amenable I f such a (TLIM) LIM e x i s t s . REMARKS : 1. Note t h a t ' 5-©r(x-)=- f Cg" 1-x)'=- L^-l f ' ( x " ) " f o r feL°°(G/H) . 2. I n the above LUC(G/H) denotes the space o f bounded c o n t i n u o u s f u n c t i o n s f on G/H such t h a t whenever g g Q G we have II6 ®f- 6 ®fll -> 0 . 3. I f feL°°(G/H) and (J>eL1(G) t h e n cf>®feLUC(G/H) so P(G)®Xcx f o r any o f the spaces mentioned above and y(<j>®f) i s d e f i n e d when y i s a mean on X . ( T h i s f o l l o w s from t h e f a c t t h a t the mapping t-»-L^(J> i s l e f t u n i f o r m l y c o n t i n u o u s so whenever x ->x i n G we can f i n d a such t h a t a> a a o o — o i m p l i e s || L <j> - L x <J> 1^ < (e/l| f If ^ ) . Hence f o r any xeG/H a o | c))®f (xax)-cj)®f ( X Q X ) | = | / Q ( j ) ( y ) f ( y " 1 x a x ) d X ( y ) - / G ( } ) ( y ) f ( y _ 1 x o x ) d X ( y ) | = |/ < i > ( x a y ) f ( y _ 1 x ) d X ( y ) - U ( x y ) f ( y _ 1 x ) d A ( y ) | 57 = | / r ( L <J>- L <j>)(y)f(y 1 x ) d X ( y ) | T a o < H f l l ^ . / |L 4>- L cf>| ( y ) d X ( y ) a o < H f l l ^ |i L x L x 4 . 1^ < e a o w h e n a> a Q s o <j»©f e L U C ( G / H ) . ) T h i s d e f i n i t i o n o f l e f t i n v a r i a n t a n d 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 s c o i n c i d e s w i t h t h e u s u a l o n e . i n t h e c a s e w h e r e H i s n o r m a l i n G a n d G/H i s a l o c a l l y c o m p a c t g r o u p . I n t h i s s i t u a t i o n l e t d e n o t e a l e f t H a a r m e a s u r e o n G / H . B y T h e o r e m 4 . 1 . 2 u i s e q u i v a l e n t t o ( s i n c e u i s q u a s i -i n v a r i a n t ) s o , a s i n R e m a r k 3 f o l l o w i n g D e f i n i t i o n 4 . 2 . 2 , t h e r e e x i s t s a s t r i c t l y p o s i t i v e m e a s u r a b l e f u n c t i o n k o n G/H s u c h t h a t f o r e v e r y f e L 1 ( G / H , u ) we h a v e f ' = f • k e L 1 ( G / H , u 1 ) w i t h / Q / H f ( x ) d u ( x ) = / G / H f » ( x ) d u 1 ( x ) . I f (f>eP(G) we h a v e s h o w n t h a t T(|>eP(G/H,u) c o n s e q u e n t l y Tc|> • k e P ( G / H , u ] L ) a n d c o n v e r s e l y e v e r y e l e m e n t i n P ( G / H , U - ^ ) i s o f t h e f o r m T t f i ' k f o r s o m e <f>eP(G) . F o r <J>eP(G) a n d f e L ° ° ( G / H ) i f we l e t q ( y ) = < j ) ( y ) f ( y _ 1 x ) t h e n T q ( y ) = / H ( < J ) ( y h ) f ( h ~ 1 y " 1 x ) / p ( y h ) ) d 3 ( h ) = / H ( c J ) ( y h ) f ( ( y _ 1 ) x ) / p ( y h ) ) d 6 ( h ) s i n c e H i s n o r m a l = f ( ( y - 1 ) x ) - T c j > ( y ) t h e r e f o r e 4 > ® f ( x ) = / Q ( J ) ( y ) f ( y ~ 1 x ) d X ( y ) = / G / H T < J > ( y ) f ( ( y _ 1 ) x ) d u ( y ) = / G / H T ( J ) ( y ) k ( y ) f ( ( y ~ 1 ) x ) d u 1 ( y ) = (T<J>-k) * f ( x ) . 58 I n view o f the remarks above t h i s i m p l i e s t h a t U ® f | <f>eP(G)} = { (Tc|>'k)*f | (T<j)-k)eP(G/H,u 1) } = {<f>»*f| <$> 'eP(G/H,u )} so y i s a TLIM on G/H i n the u s u a l sense i f f y(<|>'#f) = y ( f ) f o r a l l (|) ,eP(G/H,o 1) i f f y(cf)®f) = y ( f ) f o r a l l <j>eP(G) i f f y i s a TLIM on G/H i n the sense o f D e f i n i t i o n 4.2 . 4 As mentioned i n Remark 1 above, 6 ®f = L - 1 ( f ) f o r g g geG and f e L (G/H) so i t i s c l e a r t h a t y i s a LIM on G/H i n t he u s u a l sense i f and o n l y i f y i s a LIM on G/H i n the sense o f D e f i n i t i o n 4 . 2 . 4 . As i n the case o f the u s u a l d e f i n i t i o n o f the a m e n a b i l i t y o f a group a c t i n g on i t s e l f , the f o l l o w i n g theorem shows t h a t t h e n o t i o n s o f X b e i n g G amenable o r G t o p o l o g i c a l l y CO amenable a r e e q u i v a l e n t when X i s any o f the spaces L (G/H), CB(G/H) o r LUC(G/H). THEOREM 4.2.5 : L e t G be a l o c a l l y compact group and H a c l o s e d subgroup o f G. C o n s i d e r the f o l l o w i n g s t a t e m e n t s 0 - G i s amenable ( i n the u s u a l sense) 1 [1A] - t h e r e e x i s t s a [ t o p o l o g i c a l ] LIM on L°°(G/H) 2 [2A] - t h e r e e x i s t s a [ t o p o l o g i c a l ] LIM on CB(G/H) 3 [3A] - t h e r e e x i s t s a [ t o p o l o g i c a l ] LIM on LUC(G/H). . Then statement (0) i m p l i e s each o f t h e o t h e r s and the r e s t are e q u i v a l e n t . PROOF : We w i l l show t h a t 0->l+2->-3+3A-*2A->lA->-l . 0+1 ) L e t y be a LIM on L (G) and f o r f e L (G/H) d e f i n e y ( f ) = y ( f ) . I f f> 0 on G/H then f> 0 on G so y ( f ) = y ( f ) > 0. A l s o I Q / J J ~ 1 Q hence ^ ( 1 Q / ^ ) = 1 A N < 3 y i s a mean on L (G/H) I f geG t h e n (lf®f) = ( 6 *f ) = L - 1 ( f ) so y(S ®f) = p ( L - 1 ( f ) ) = y ( f ) = y ( f ) and y i s a LIM 0 0 on L (G/H). l->2->3 ) S i n c e CB(G/H) i s a c l o s e d i n v a r i a n t subspace o f L°°(G/H) and LUC(G/H) i s a c l o s e d i n v a r i a n t subspace o f CB(G/H) the r e q u i r e d l e f t i n v a r i a n t means can be o b t a i n e d by r e s t r i c t i o n . 3+3A ) L e t {e f l} ^ be an approximate i d e n t i t y i n L"^(G) . T h i s means t h a t e a e P ( G ) f o r each a i n the d i r e c t e d s e t & and, f o r any neighbourhood V o f the i d e n t i t y i n G, t h e r e e x i s t s a z(h such t h a t a> a i m p l i e s e v a n i s h e s o u t s i d e o — o . a o f V. A l s o f o r any g e L 1 ( G ) we have l^m||g*e -g|)^= 0 and l^m||ea#g-g||^= 0. (The e x i s t e n c e o f such an approx-imate i d e n t i t y f o r L^(G) i s w e l l known. A p r o o f can be fo u n d i n [12] p-.124-.) I f feLUC(G/H) we can show t h a t lim||e ®f-f|J = 0 as CX CX oo f o l l o w s : g i v e n e> 0 we use the f a c t t h a t f i s l e f t u n i f o r m l y c o n t i n u o u s t o choose a neighbourhood V o f the i d e n t i t y i n G w i t h || 6 ®f-flloo< e f o r a l l yeV. I f a ed i s chosen so t h a t e v a n i s h e s o u t s i d e o f V f o r a> a th e n a — o we have 60 |e a®f(x)-f(x) h | / G e a ( y ) f ( y _ i x ) c l X ( y ) - / & e a ( y ) f ( x ) d X ( y ) | ( s i n c e / Q e a ( y ) d A ( y ) = 1) = |/ Ge a(y)(6 y®f(x)-f(x))dX(y) | < ]j6 y®f-f|| c o-/ Ge a(y)dA(y) f o r a> a Q ( s i n c e e a v a n i s h e s o u t s i d e o f V) < e c o n s e q u e n t l y H e a®f-fll 0 0< e whenever a>_ a Q so limllea®f-f||00= 0. With t h i s f a c t e s t a b l i s h e d we can employ a method o f Namioka t o o b t a i n the d e s i r e d r e s u l t . Let u be a LIM on LUC(G/H), feLUC(G/H) and d e f i n e a f u n c t i o n a l Q f on L ! ( G ) by the f o r m u l a Q f($)= u(<f>©f). Q f i s c l e a r l y a bounded l i n e a r f u n c t i o n a l ' on.L^CG')' and s i n c e , f o r g£*G' (L * ) o f ( x ) = / G < j > ( g y ) f ( y _ 1 x ) d A ( y ) = / G < j > ( y ) f ( y - 1 g x ) d A ( y ) ( r e p l a c i n g y by g - 1 y ) = (<J>®f)(gx) = L (<j>©f) (x) o f o r a l l xeG/H we have Qf.(L0.c|))= y((Ld>)®f) = y ( L (c)>®f)) = y(<j)®f) = Q-CcjO t h e r e f o r e Q f i s l e f t i n v a r i a n t . By the- uniqueness- o f Haar-measure on G, t h e r e must e x i s t a c o n s t a n t k ( f ) w i t h y(cj)®f)= k ( f ) - / G ( j ) ( y ) d A ( y ) f o r a l l <|)eL1(G) . T h e r e f o r e f o r any cj)eP(G), Qf(<j>) = y(<J>®f) = k ( f ) and | y ( f )-y((J>®f) | < | y ( f ) - y ( e ®f) | + | i i ( e ®f)-y((<()#e )®f) | + | y( (<J»*e )®f )-y(c|)®f) | 6 1 < ijyfj • l ! f - e a ® f ll^+l Q f ( e a ) - Q f (cb«e a) | + | y ( ( c f r K e ^ )©f) | <_ || f - e a ® f ! l o o + | k ( f ) - k ( f ) |+l!yi|'ll,cp«e a~c{)|J 1-llf 1)^  ( s i n c e <f>#e eP(G)) and s i n c e !| f-e_,®fII and ||c|>*e -c|>|j, t e n d t o 0 we have a oo a 1 y ( f ) = y(cb®f) so y i s a l s o a TLIM on LUC(G/H) . 3A+2A+1A ) F i r s t note t h a t 1A+2A+3A as i n the p r o o f o f 1+2+3 so i t s u f f i c e s t o show t h a t 3A+1A. L e t y be a TLIM on LUC(G/H) and l e t E be a compact neighbourhood o f the i d e n t i t y i n G w i t h cbg i t s n o r m a l i z e d c h a r a c t e r i s t i c f u n c t i o n ( i . e . (j>E= (1/A ( E ) ) *Xg) . As mentioned i n Remark 3 f o l l o w i n g D e f i n i t i o n 4.2 . 4 <f> ®feLUC(G/H) f o r any feL°°(G/H) ( s i n c e cbgeP(G)) so we can d e f i n e a mean y on L°°(G/H) by y ( f ) = y(<j>E®f) . Ag a i n l e t ^ e a ^ a e C i b e a n approximate i d e n t i t y i n L^(G) and choose cJ)eP(G), feL°°(G/H). We have y(<J>©f) = y(cJ>E®(rb®f)) = y ((cj>E*<}))©f) by Lemma 4 . 2 . 3 (c) = l i m y ((cbpKcb) ® (e ®f)) s i n c e y i s c o n t i n u o u s a * a and II f-e ®fII + 0 a 0 0 = l i m y ( e ®f) s i n c e (|>„*<}>eP(G) , e ®feLUC(G/H) CX 06 CX and y i s a TLIM = l i m y(AT_,®(e ®f)) s i n c e cb„eP(G) and y i s a E a E a TLIM = l i m y((<j> #e )®f) by Lemma 4 . 2 . 3 (c) a E cx = y(4> E®f) s i n c e y i s c o n t i n u o u s and II <|>E*e -4> )| 0 = y ( f ) Thus y i s a TLIM on L (G/H) and we have shown t h a t 3A+1A. CO 1A->1 ) Let y be a TLIM on L (G/H) and choose cpeP(G) xeG and f e L (G/H). Then y(<5 ®f) = y(4>®(5 ®f)) = y((<f>*6v)®f) = y(f) A A .A. s i n c e <j>*<$ = A(x _ 1)«R -l((J))e P(G) f o r a l l xeG, <J>eP(G) X X so y i s a LIM on L (G/H). I n view o f t h i s Theorem we may adopt the t e r m i n o l o g y t h a t G a c t s amenably on G/H t o i n d i c a t e the e x i s t e n c e o f 0 0 a LIM or a TLIM on any o f the spaces L (G/H), CB(G/H) and LUC(G/H). NOTE , : The methods used i n t h e p r o o f o f t h i s theorem a r e v e r y c l o s e t o the ones employed i n [6] Theorem 2.2.1 t o e s t a b l i s h the same type o f r e s u l t c o n c e r n i n g t h e e x i s t e n c e o f l e f t i n v a r i a n t and t o p o l o g i c a l l e f t i n v a r i a n t means on the spaces L°°(G), CB(G), LUC(G) and UCB(G) f o r a l o c a l l y compact group G. Our Theorem 4.2.5 i s e x a c t l y Theorem 3-3 o f G r e e n l e a f ' s paper [11] however, due t o the approach he uses- i n d e f i n i n g the c o n v o l u t i o n type o p e r a t i o n s , h i s . p r o o f s o f many of t h e i m p l i c a t i o n s are q u i t e d i f f e r e n t . A v a l u a b l e t o o l f o r s t u d y i n g amenable groups has been Day's concept o f n e t s o f f i n i t e means ( i . e . elements o f P(G)) c o n v e r g i n g t o l e f t i n v a r i a n c e . We are a b l e t o i n t r o d u c e a s i m i l a r concept which i s u s e f u l i n e x a m i n i n g the amenable a c t i o n o f G on G/H. The d e f i n i t i o n which f o l l o w s i s e q u i v a l e n t t o the one used by G r e e n l e a f i n [ 1 1 ] . DEFINITION 4.2,6 : A net {<f> } c P(G/H) i s s a i d t o converge i ) weakly [ s t r o n g l y ] t o l e f t I n v a r i a n c e i f (6 ©0 -<J> )-> 0 i n the weak* t o p o l o g y on L 1(G/H) f o r a l l geG § ot ot ( i . e . (6 ©tj) -cj) ,f)= fn/„(6 ®4> -<J> ) ( x ) f ( x ) d u ( x ) -> 0 f o r g Y a a' G/H g Y a T a ' v CO e v e r y geG and f e L (G/H)) [ i f II 6 ®<j> -(}> ||_ 0 f o r a l l geG ] . g Y a r a '1 & i i ) weakly [ s t r o n g l y ] t o t o p o l o g i c a l l e f t i n v a r i a n c e i f (4>®<|>a-<|> ) -> 0 i n the weak* t o p o l o g y on L 1(G/H) f o r a l l <{»eP(G). [ i f lU®4' a- (f > all 1 * 0 f o r a 1 1 *eP(G).] The s i g n i f i c a n c e o f t h e s e d e f i n i t i o n s and t h e i r u s e f u l -n e s s , i n examining, the- concept- a£ • the-.-ame-na-b-le,. a c t i o n o f G on G/H i s i n d i c a t e d by the f o l l o w i n g theorem. I n t h e p r o o f we use e s s e n t i a l l y the same methods as a r e employed t o prove a s i m i l a r r e s u l t f o r amenable groups ( [ 6 ] Theorems 2 . 4 . 2 , 2 . 4 . 3 ] THEOREM 4.2 . 7 : I f G i s a l o c a l l y compact group and H i s a c l o s e d subgroup o f G t h e n the f o l l o w i n g a re e q u i v a l e n t 1 ) G a c t s amenably on G/H 2 [2A] ) t h e r e e x i s t s a net {<J> } £ P(G/H) c o n v e r g i n g weakly [ s t r o n g l y ] t o l e f t I n v a r i a n c e 3 [3A] ) t h e r e e x i s t s a net {<(> } c P(G/H) c o n v e r g i n g weakly [ s t r o n g l y ] t o t o p o l o g i c a l l e f t i n v a r i a n c e . 6 4 PROOF : F i r s t note t h a t f o r <j>eP(G), cf^ePCG/H) , feL°°(G/H) and geG we have the f o l l o w i n g e q u a l i t i e s -( t ) / G / H(6 g®<j> 1)(x)f(x)du(x) = / G / H < | ) 1 ( x ) ( ( 6 -l)®f)(x)du(x) ( t t ) / G / R ( (|)®<|)1)(x)f(x)du(x) = / G / R(J) 1(x)(?®f)(x)du(x) where ? ( y ) = ( A G ( y ~ 1 ) - ^ ( y - 1 ) f o r yeG (hence ^ e P ( G ) ) . To e s t a b l i s h t h e s e e q u a l i t i e s l e t ^ eP(G) be such t h a t T^= <|> . Then / G / H(6 g®<J) 1)(x)f(x)du(x) = / G / H T ( 6 * i p ) ( x ) f ( x ) d u ( x ) = /G(<5 *<//)(x)f(x)dA(x) ( s i n c e T((6 »^).f)= T(6 *<H'f ) O O = / G ^ ( g - 1 x ) f ( x ) d A ( x ) = / G i | ) ( x ) f ( g x ) d A ( x ) r e p l a c i n g x by gx = /Qi(»(x)((6 - l ) * f ) ( x ) d A ( x ) = / G / H T i ( ; ( x ) ( ( 6 -l)®f)(x)du(x) ( s i n c e T(i|<«((6 -1) * f ) ) =TiJ> • ((<5 -l)®f) ) = / G / H < ( ) 1 ( x ) ( ( 6 -l)®f)(x)du(x) I n a s i m i l a r f a s h i o n /G/H(«J>»(j>1)(x)f(x)du(x) = / G ( < J ) * ^ ) ( x ) f ( x ) d A ( x ) = / G / G ( f ) ( y ) ^ ( y ~ 1 x ) f ( x ) d A ( y ) d A ( x ) = • r G - r G ( A G ( y " 1 ) < } ' ( y " 1 ) ) ^ ( y x ) f ( x ) d A ( y ) d A ( x ) ( r e p l a c i n g y by y" 1) = / G / G 7 ( y ) ^ ( y x ) . f ( x ) d X ( x ) d X ( y ) 65 = / G / G 4 > ( y ) ^ ( x ) f ( y 1x)dX(.x)dA(y) ( r e p l a c i n g x by y 1 x ) = / G ^ ( x ) ( / G | ( y ) f ( y - 1 x ) d A ( y ) ) d A ( x ) = / Q^(x)(?«f)(x)dA(x) = / G / H T i p ( x ) ("<J>®f) ( x ) d u ( x ) = / G / Hc() 1(x)(7®f ) ( x ) d u ( A ) 2+1 ) L e t {<J> } c p ( G / H ) converge weakly t o l e f t i n v a r i a n c e . S i n c e P(G/H) i s c o n t a i n e d i n the weak* compact s e t s o f means on L°°(G/H), t h e r e e x i s t s a subnet {cf> } CO • 1 c o n v e r g i n g i n the weak* t o p o l o g y t o a mean y. F o r any 00 f e L (G/H) and geG we have u ( 6 ®f) = l i m 6 (6 ®f) = l i m /_ ...d> ( x ) ( 6 ©fVx)du(x) g a ± ' a ± g G/ri • g l i m / G / H ( (6 g-l)«<j> a <)(x)f(x)du(x) ( u s i n g t ) x =' l i m ( ( 6 - I ) ® * ) ( f ) ° i = l i m $ ( f ) = y ( f ) a i i 00 so y i s a- l e f t i n v a r i a n t mean on L (G/H) and G a c t s amenably on G/H. 3+1 ) I f {(J)^} c P(G/H) converges weakly t o t o p o l o g i c a l l e f t i n v a r i a n c e t h e n , as above, t h e r e i s a subnet {d> } a. l c o n v e r g i n g i n the weak* t o p o l o g y t o a mean y. I f feL°°(G/H) and cbeP(G) t h e n y(4)®f) = l i m / G / R * a .(*) (<J>®f) ( x ) d u ( x ) l l 66 l i m / Q / H ( W a > ) ( * ) f ( x ) d u ( x ) ( u s i n g t t ) = l i m ("<f>»<|> ) ( f ) i = l i m <f> ( f ) ( s i n c e f e P ( G ) ) a i . a i = y ( f ) 00 so y i 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 mean on L (G/H) and c o n s e q u e n t l y G a c t s amenably on G/H. CO 1+2 ) L e t y be a LIM on L (G/H). S i n c e P(G/H) i s dense i n the s e t o f means w i t h the weak* t o p o l o g y , t h e r e e x i s t s a net' {<j>a}c P(G/H) w i t h w * - l i m cba= y. Con s e q u e n t l y f o r 00 , , . f e L (G/H), geG we have l i m (6g«4»a)(f) = l i m cj>a( ( 6 - 1 ) o f ) ( u s i n g t ) = y ( ( 6 -l)®f) = y ( f ) = l i m <j> ( f ) a a so {$ a} converges weakly t o l e f t i n v a r i a n c e . 1+3 ) I f y i s a TLIM on L°°(G/H) and ' {<J> } c p(G/H) i s such t h a t w * - l i m cj>a = y t h e n f o r <j>eP(G) and feL°°(G/H) , l i m (4>®<j>a)(f) = l i m ) ( u s i n g t t ) = y(?®f) = y ( f ) ( s i n c e f e P ( G ) ) = l i m 4 ( f ) a a so {^a} converges weakly t o t o p o l o g i c a l l e f t i n v a r i a n c e . We have e s t a b l i s h e d the e q u i v a l e n c e o f 1,2 and 3- S i n c e t h e norm t o p o l o g y on L^"(G/H) i s s t r o n g e r t h a n the weak* t o p o l o g y , i t i s c l e a r t h a t 2A+2 and 3A+3. The r e m a i n i n g i m p l i c a t i o n s can be e s t a b l i s h e d u s i n g a method due t o Namioka. 3+3A ) For each 4>eP(G) ta k e a copy o f L (G/H) and form the l o c a l l y convex p r o d u c t space E = n { L 1 ( G / H ) | <j,eP(G)} w i t h the p r o d u c t o f the norm t o p o l o g i e s . D e f i n e the l i n e a r mapping S: L 1(G/H) -> E by Sf(4>) = <j>®f-f f o r cbeP(G), f e L 1 (G/H) . Let ^ * ~ " P(G/H) converge weakly t o t o p o l o g i c a l l e f t i n v a r i a n c e so we have S<f> (cb ) = cb@cb - cb ->0 i n the weak* ot ot ct t o p o l o g y on L 1(G/H) f o r every cbeP(G). S i n c e the weak t o p o l o g y on E c o i n c i d e s w i t h the p r o d u c t o f the weak* t o p o l o g i e s (see [ 1 3 ] p . l 6 0 ) t h i s i m p l i e s t h a t w * - l i m S<1> = 0 and t h e r e f o r e 0 l i e s i n the weak c l o s u r e a a o f S(P(G/H)) E. S i n c e E i s l o c a l l y convex and S(P(G/H)) i s a convex s e t , the weak and s t r o n g c l o s u r e s c o i n c i d e (see [ 8 ] Chapter V Theorem 3 . 1 3 p.422) so t h e r e i s some net { ^ C J } 0 P(G/H) such t h a t s ^ a ) ^ 0 i n the norm t o p o l o g y on E i . e . f o r ev e r y <J>eP(G) l l ^ a A l l l * 0 C o n s e q u e n t l y {i^^} <^  P (G/H) converges s t r o n g l y t o t o p o l o g i c a l l e f t i n v a r i a n c e . 2->2A ) I n t h e same manner as above, f o r ev e r y ge G t a k e a copy of L 1(G/H) and form the l o c a l l y convex p r o d u c t space E' = II{ L 1 ( G / H ) | ge G} w i t h the p r o d u c t o f the norm t o p o l o g i e s . D e f i n e the l i n e a r mapping S': L 1(G/H) •> E' by S ' f ( g ) = 6 ©f-f f o r geG, f e L 1 ( G / H ) . I f {$a)<^ ?(G/H) converges weakly t o l e f t i n v a r i a n c e we have w * - l i m S'(cj>a)=0 i n E' and, as b e f o r e , the weak and s t r o n g c l o s u r e s o f S'(P(G/H)) c o i n c i d e so t h e r e e x i s t s a net' {ip } P(G/H) w i t h II 6 ®\b -\\> |L -> 0 f o r a l l geG. T h e r e f o r e {i) } converges s t r o n g l y t o t o p o l o g i c a l l e f t i n v a r i a n c e . REMARK : As we have mentioned b e f o r e , g i v e n any two q u a s i - i n v a r i a n t measures and on G/H the f a c t t h a t U 1 E U 2 i m P l i e s t h a t L^(G/H,u^) i s i s o m e t r i c a l l y i s o m o r p h i c w i t h L 1(G/H,u 2) under an isomorphism A which s a t i s f i e s A C y ^ f ) = y» ( A f ) f o r a l l yeM(G), feL 1.(G/H,u 1) . I f {(J>a}c: P(G/H,u^) converges weakly ( s t r o n g l y ) t o l e f t i n v a r i a n c e t h e n {Acf> } i s a net i n P(G/H,u 2) w i t h <5g®2(A(*)a'> " A * a = A ( < Sg®l*a~ < i >a ) s o { A * a } c o n v e r e e s weakly ( s t r o n g l y ) t o l e f t i n v a r i a n c e as w e l l . A s i m i l a r argument shows t h a t the n o t i o n o f n e t s o f f i n i t e means c o n v e r g i n g t o t o p o l o g i c a l i n v a r i a n c e i s a l s o independent o f the q u a s i - i n v a r i a n t measure b e i n g c o n s i d e r e d . One i m p o r t a n t a p p l i c a t i o n ^ t h i s theorem o c c u r s i n t h e p r o o f o f the f o l l o w i n g r e s u l t . THEOREM 4 . 2 . 8 : G a c t s amenably on G/H I f and o n l y i f g i v e n e >0 and a compact s u b s e t K o f G t h e r e e x i s t s <J>eP(G/H) such t h a t \\S ©cj) - <J»|| < e f o r a l l xeK. X J_ PROOF : +- ) L e t r = { ( e , K ) | e> 0 , K compact s u b s e t o f G} d i r e c t e d by the p a r t i a l o r d e r i n g (e^,K^)< ( e 2 , K 2 ) i f e 2 < e l a n d K . j C K 2 . F o r each a=(e,K)er t h e r e e x i s t s <J> eP(G/H) w i t h 116 ®<b -d> Ik < e f o r a l l xeK. 1 x ra a 1 1 ! 69 G i v e n e> 0 and xeG choose a =(e.K ) e f where xeK . I f a> a o 5 o o — o then 11 6 ©cb -<(> H,< e so (cb } converges s t r o n g l y t o l e f t X Ot CL J_ i n v a r i a n c e and, by Theorem 4 . 2 . 7 5 G a c t s amenably on G/H. -> ) L e t {<)> } cr P(G/H) converge s t r o n g l y t o t o p o l o g i c a l l e f t i n v a r i a n c e ( t h e e x i s t e n c e o f such a net i s due t o Theorem 4 . 2 . 7 and t h e f a c t t h a t we are assuming t h a t G a c t s amenably on G/H). F i x 3eP(G) and choose a s m a l l compact neighbourhood E o f t h e i d e n t i t y i n G such t h a t ||(|>E*3-3II 1< e and II 6 * 3 - 3 l l 1 < e f o r a l l xeE (where cb = ( 1/A(E)) 'X-n denotes the n o r m a l i z e d c h a r a c t e r i s t i c E E f u n c t i o n o f the s e t E ) . Choose x n=e ,x n,...,x eG f o r which 1 5 2 ' ' n n K c r l J x . - E and l e t 4>.= cb „ = 6 «cb„ f o r i = l , 2 , . . . , n . • -l l x. X . . TE• • ' ' »• i = l l I S i n c e (<^a^ converges s t r o n g l y t o t o p o l o g i c a l l e f t i n v a r -i a n c e t h e r e e x i s t s <J> (which we s h a l l denote by 4> ) o w i t h (J 3»<I>0-4)0||1< e and || ^ ..©c^-cbjl^ e f o r i = l , 2 , . . . , n . L e t <b = 3@<j>0 e P(G/H) and note t h a t , f o r every xeE, < |)(bE®(6@(J)o)-3®(J,o|S1 + 11 3©<f0-5x®(3®(b0)||1 <1|((|>E*3-3)®<|)0II1 + 11 (3-6 x*3)@cb o|l 1 <. | l * E * 3 - 3 l l 1 - l l < l > 0 l l 1 + ) | 3 - 6 x » 3 l l 1 - l | c | > 0 l l 1 ( u s i n g Lemma 4 . 2 . 3 i i ) ) < 2*e ( due t o the manner i n which E was d e f i n e d and the f a c t t h a t II <J> 11-^1) Thus i f xeE we have i l ( 6 *<j)E)®((>-6x X©*II1=11'5X ®C<(>E®*)-6x 8 ( 6 ^ ) 1 ^ i i i i < || 6 x II -||<(>E»c|)-6x®(|)|| < 2-e. . i T h e r e f o r e II ^x.x®*-*^ £ 1|<SX>X®*-(5X>**E)®*II1+II(«X_**E)®*-*II — X . ill O U J. 1 < 2-e + ||(6 *4> ) ® C 3® 4> > — C «S *(J)E)®(J)o||1+ i i 11 (6 * * ) • * - < ! . || + l l v e * * o l l i i < 2-e + H (6 *<|>E)®(S«<j> H> 0) )| ± ' i +IUi®(j)o-(i)oil1 + ||<j>0-3«<yj 1 <_ 5 ' £ f o r a l l xeE. S i n c e , f o r any t e K , we have t= x^x f o r some i and xeE t h e r e f o r e || 6^ ®$-<j>|l^  < 6-e f o r a l l teK and the r e s u l t f o l l o w s . T h i s theorem i s an analogue o f Theorem 3.2.1 o f [6] f o r l o c a l l y compact groups. I t appears w i t h o u t p r o o f as Theorem 3-5 i n [ 1 1 ] . 71 BIBLIOGRAPHY [ 1 ] . J . L i n d e n s t r a u s s , A s h o r t p r o o f o f L i a p o u n o f f ' s c o n v e x i t y theorem, J . of Math, and Mech. 1 5 ( 1 9 6 6 ) , 9 7 1 - 9 7 2 [ 2 ] . E. G r a n i r e r , On the range o f an i n v a r i a n t mean, T r a n s . Amer. Math. Soc. 1 2 5 ( 1 9 6 6 ) , 3 8 4 - 3 9 4 [ 3 ] . C. Chou, On a c o n j e c t u r e o f E. G r a n i r e r c o n c e r n i n g the range o f an i n v a r i a n t mean, P r o c . Amer. Math. Soc. 2 6 ( 1 9 7 0 ) , 1 0 5 - 1 0 7 [ 4 ] , E. H e w i t t and K.A. Ross, A b s t r a c t Harmonic A n a l y s i s I , S p r i n g e r - V e r l a g ( 1 9 6 3 ) [5.1 • E.. G r a n i r e r and.A. Lau, I n v a r i a n t means on l o c a l l y compact gr o u p s , 1 1 1 . J . o f Math. 1 5 ( 1 9 7 1 ) , 2 4 9 - 2 5 7 [ 6 ] . P.P. G r e e n l e a f , I n v a r i a n t means on t o p o l o g i c a l g r o u p s , Van N o s t r a n d Math. S t u d i e s #16 ( 1 9 6 9 ) [ 7 ] . H. R e i t e r , C l a s s i c a l harmonic a n a l y s i s and l o c a l l y compact g r o u p s , O x f o r d Math. Monographs ( 1 9 6 8 ) [ 8 ] , N. Dunford and J.T. S c h w a r t z , 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 ( 1 9 5 8 ) [ 9 ] . P. Eymard, Sur l e s moyennes i n v a r i a n t e s et l e s r e p r e s e n t a t i o n s u n i t a i r e s , C.R. Acad. Sc. P a r i s 2 7 2 ( 1 9 7 1 ) , 1 6 4 9 - 1 6 5 2 72 [ 1 0 ] . N. B o u r b a k i , Elements de Mathematique L i v r e VI I n t e g r a t i o n Chap. 7 et 8 , Hermann ( 1 9 6 3 ) [ 1 1 ] . P.P. G r e e n l e a f , Amenable a c t i o n s o f l o c a l l y compact gro u p s , J . o f F u n c t . A n a l . 4 ( 1 9 6 9 ) , 2 9 5 - 3 1 5 [ 1 2 ] . L.H. Loomis, An i n t r o d u c t i o n t o a b s t r a c t harmonic a n a l y s i s , Van N o s t r a n d ( 1 9 5 3 ) [ 1 3 ] - J . K e l l e y , I . Namioka et a l . , L i n e a r T o p o l o g i c a l Spaces, Van N o s t r a n d (1963) 

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