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UBC Theses and Dissertations

Summability and invariant means on semigroups Mah, Peter Fritz 1970

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. SUMMABILITY AND INVARIANT MEANS ON SEMIGROUPS by PETER FRITZ MAH B. A., U n i v e r s i t y of Saskatchewan, 1963 M. A., U n i v e r s i t y of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF : DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the req u i r e d standard . THE UNIVERSITY OF BRITISH COLUMBIA March, 1970 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a 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 reference and study. I f u r t h e r agree t h a t p e r m i s s i o n - f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g ain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of MfiTHeMATlCS The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date APHIL- ? , 11 76 Supervisor: P r o f e s s o r E. G r a n i r e r i i ABSTRACT This t h e s i s c o n s i s t s of two p a r t s . In the f i r s t p a r t , we study summability i n l e f t amenable semigroups. More e x p l i c i t l y , v a r i o u s summability methods defined by matrices are considered. Necessary and (or) s u f f i c i e n t c o n d i t i o n s are given f o r matrices to be r e g u l a r , almost r e g u l a r , Schur, almost Schur, s t r o n g l y r e g u l a r and almost s t r o n g l y r e g u l a r , g e n e r a l i z i n g those of 0. T o e p l i t z , J . P. K i n g , J . Schur, G. G. Lorentz and P. Schaefer f o r the semigroup of a d d i t i v e p o s i t i v e i n t e g e r s . The theorems are of i n t e r e s t even f o r the semigroup of m u l t i p l i c a t i v e p o s i t i v e i n t e g e r s . Let S be a t o p o l o g i c a l semigroup which i s amenable as a d i s c r e t e semigroup. Denote by LUC(S) the set of bounded r e a l - v a l u e d l e f t u n i f o r m l y continuous f u n c t i o n s on S . I t i s shown by E. G r a n i r e r that i f S i s a separable t o p o l o g i c a l group which is.amenable as a d i s c r e t e group and has a c e r t a i n property (B) then LUC(S) has "many" l e f t i n v a r i a n t means. In the second part of t h i s t h e s i s , we extend t h i s r e s u l t to c e r t a i n t o p o l o g i c a l subsemigroups of a t o p o l o g i c a l group. In p a r t i c u l a r , we show that i f S i s a separable close d non-compact subsemigroup of a l o c a l l y compact group which i s amenable as a d i s c r e t e semigroup then LUC(S) has "many" l e f t i n v a r i a n t means. F i n a l l y , an example i s given to show that t h i s r e s u l t cannot be extended to every t o p o l o g i c a l semigroup. i i i TABLE OF CONTENTS Page INTRODUCTION . . . . . . . . . . 1 CHAPTER I : SUMMABILITY IN AMENABLE SEMIGROUP 1. D e f i n i t i o n s and no t a t i o n s . 3 2. Convergence and l e f t almost convergence . . . . . . 5 3. Almost r e g u l a r matrices . . . . . 6 4. Regular matrices 12 , 5. Strongly r e g u l a r matrices 13 6. Almost Schur matrices 23 7. Schur matrices . . . . . . . . . . . . . 26 8. Almost s t r o n g l y r e g u l a r matrices 28 CHAPTER I I : THE INVARIANT MEAN ON A TOPOLOGICAL SEMIGROUP 9. D e f i n i t i o n s and no t a t i o n s 33 10. T e c h n i c a l lemmas 34 11. The Main theorem . . . 39 12. Examples . 48 BIBLIOGRAPHY 53 i v ACKNOWLEDGEMENTS I wish to thank Profes s o r E. G r a n i r e r f o r h i s guidance, encouragements and v a l u a b l e suggestions during the p r e p a r a t i o n of t h i s t h e s i s . I would a l s o l i k e to thank Profes s o r J . Fournier f o r h i s c a r e f u l reading of the d r a f t form of t h i s t h e s i s , and to Mrs. Mo n l i s a Wang f o r typing t h i s t h e s i s w i t h patience and care. F i n a l l y the f i n a n c i a l support given me by the U n i v e r s i t y of B r i t i s h Columbia and the N a t i o n a l Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. / 1 INTRODUCTION In [11] G. G. Lorentz introduced a new method of summation which assigns a general l i m i t to c e r t a i n bounded sequences x = (x(n)} . He c a l l e d the sequences which are summable by t h i s method almost convergent. In v a r i o u s summability methods defined by m a t r i c e s , an i n f i n i t e m a t r i x A i s s t r o n g l y r e g u l a r i f the sequence Ax i s convergent to k whenever x i s almost convergent to k , where Ax i s the sequence {^A(m,n)x(n)} whenever the sum i s convergent f o r each m . I t i s almost n r e g u l a r i f Ax i s almost convergent to k whenever x i s convergent to k , and i t i s almost s t r o n g l y r e g u l a r i f Ax i s almost convergent to k whenever x i s almost convergent to k . Necessary and (or) s u f f i c i e n t c o n d i t i o n s f o r matrices to be s t r o n g l y r e g u l a r , almost r e g u l a r and almost s t r o n g l y r e g u l a r have been obtained by G. G. L o r e n t z , J . P. King and P. Schaefer r e s p e c t i v e l y i n [11], [10] and [14]. Now the method of G. G. Lorentz i s connected w i t h the theory of amenable semigroups. One of the purposes of t h i s t h e s i s i s to extend the v a r i o u s summability methods described above to a c l a s s of amenable semigroups. This w i l l be our concern i n Chapter I . Our s e t t i n g then i s as f o l l o w s : S i s a l e f t amenable semigroup without any f i n i t e l e f t i d e a l s . I t i s shown that only i n such semigroups i s almost convergence a g e n e r a l i z a t i o n of convergence i n the sense defined i n s e c t i o n 1 (theorem 2.1). I f A i s an i n f i n i t e m a t r i x on S , we have the f o l l o w i n g cases: (1) Af (see s e c t i o n 1 f o r the d e f i n i t i o n of Af) i s convergent for. every bounded r e a l - v a l u e d f u n c t i o n f on S . (Schur matrices) 2 (2) Af i s l e f t almost convergent f o r every bounded r e a l - v a l u e d f u n c t i o n f on S . (Almost Schur matrices) (3) Af i s convergent to k whenever f i s convergent to k . (Regular matrices) (4) Af i s l e f t almost convergent to k whenever f i s convergent to k . (Almost r e g u l a r matrices) (5) Af i s convergent to k whenever f i s l e f t almost convergent to k . (Strongly r e g u l a r matrices) (6) Af i s l e f t almost convergent to k whenever f i s l e f t almost convergent to k . (Almost s t r o n g l y r e g u l a r matrices) We give necessary and (or) s u f f i c i e n t c o n d i t i o n s f o r matrices to s a t i s f y (1) to (6). I t . s h o u l d be pointed out here that the r e s u l t s f o r (1) and (3) do not i n any way depend on the a l g e b r a i c s t r u c t u r e of S , so that the r e s u l t s of 0. T o e p l i t z [17] and J . Schur [15] can e a s i l y be extended. The only a d d i t i o n a l argument needed i s given i n lemma 3.1. Examples are given to i l l u s t r a t e our r e s u l t s . The second purpose of t h i s t h e s i s i s the study of the set of l e f t i n v a r i a n t means of a t o p o l o g i c a l semigroup. This w i l l be our concern i n Chapter I I . Our goal i s to extend to t o p o l o g i c a l semigroup a theorem of E. G r a n i r e r which s t a t e s that the set of l e f t i n v a r i a n t means on the l e f t u n i f o r m l y continuous f u n c t i o n s of an amenable separable t o p o l o g i c a l group i s "huge" i f i t has an unbounded l e f t u niformly continuous f u n c t i o n . However, we are only able to extend the above r e s u l t to c e r t a i n t o p o l o g i c a l subsemigroups of a t o p o l o g i c a l group. Examples are then given to show that t h i s extension i s n o n - t r i v i a l . A l s o an example i s given to show that the extension to any t o p o l o g i c a l semigroup i s not p o s s i b l e . CHAPTER I 3 SUMMABILITY IN AMENABLE SEMIGROUPS 1. DEFINITION AND NOTATIONS. Let S be a s e t . A f u n c t i o n f on S w i t h values i n a l i n e a r t o p o l o g i c a l space L i s c a l l e d u n c o n d i t i o n a l l y summable to g i n L i f l i m £ f ( s ) = g , where £ i s the f a m i l y of a l l f i n i t e subsets of aeE sea S d i r e c t e d by i n c l u s i o n . We s h a l l denote t h i s by g = £ f ( s ) and say seS the sum ][ f ( s ) converges to g [ 2 ] , In p a r t i c u l a r , we may take L seS to be the r e a l s . Then £ f ( s ) = g i f f o r every e > 0 there i s a seS f i n i t e subset a^ such that i f a "D a^ then | \ f ( s ) - g| < e . I t i s sea w e l l known that the above d e f i n i t i o n i m p l i e s only countably many f ( s ) are d i f f e r e n t from 0 [8, p. 19, theorem 1 ] . Let S be a set and S U {°°} be the one-point c o m p a c t i f i c a t i o n of S when S has the d i s c r e t e topology. Let m(S) be the l i n e a r space of a l l bounded r e a l - v a l u e d f u n c t i o n on S w i t h the sup norm, and l e t C^ be the closed l i n e a r subspace of a l l those f i n m(S) such that l i m f ( s ) e x i s t s . From now on, we s h a l l w r i t e l i m f ( s ) f o r s->°° •'' s l i m f ( s ) , so that l i m f ( s ) = k means that f o r every e > 0 there i s a s-x» s f i n i t e subset a C S such that | f ( s ) - k| < e i f s e S ^ a . I f , i n a d d i t i o n , S i s a'semigroup, then, f o r f e m(S) , a e S , p ( f ) = f ( a ) , cl and ^ [r ] i s the l e f t [ r i g h t ] t r a n s l a t i o n operator on m(S) defined by 3. 3. ^ f ( s ) = f ( a s ) [r f ( s ) = f ( s a ) ] . The conjugate mapping of I w i l l be Si '• SL 3. denoted by L . I f CoA denotes the convex h u l l of A then elements i n J a Co{p : a e S} are c a l l e d f i n i t e means.. A l i n e a r f u n c t i o n a l (J) on m(S) 4 i s a l e f t i n v a r i a n t mean (LIM) i f <j>(f) >_ 0 f o r f >_ 0 , <j>(l) = 1 and <(>(£ f ) = <j)(f) f o r a l l f e m(S) and a l l a e S , where 1 i s the constant one f u n c t i o n on S , and f >_ 0 means f (s) >_ 0 f o r a l l s e S . We denote the set of a l l l e f t i n v a r i a n t means by M£(S) . I f MJl(S) ^ 0 , , where 0 denotes the empty s e t , then the semigroup S i s s a i d to be l e f t amenable (LA) . I f , i n a d d i t i o n , <j> . i s m u l t i p l i c a t i v e , i . e . , <Ji(fg) = <j)(f)<j>(g) f o r a l l f,g e m(S) then S i s s a i d to be extremely l e f t amenable (ELA) . Examples of l e f t amenable semigroups are: commutative semigroups, s o l v a b l e groups and l o c a l l y f i n i t e groups. For d e t a i l s and an e x c e l l e n t reference see [ 1 ] . Extremely l e f t amenable semi-groups are p r e c i s e l y those semigroups i n which every two elements have a common r i g h t zero. For d e t a i l s and other i n t e r e s t i n g r e s u l t s see [ 3 ] , [ 4 ] , [5] and.[13]. I f S i s LA , 1 then a f u n c t i o n f e m(S) i s s a i d to be l e f t almost convergent to k i f <j>(f) = i|>(f) = k f o r every ( J > e M £ ( S ) •;. We s h a l l denote the set of a l l almost convergent f u n c t i o n s by F , and w r i t e f i s &ac to k to mean f i s l e f t almost convergent to k . I f A = ( A ( s , t ) ) i s an i n f i n i t e m a t r i x on S and f e m(S) , l e t Af be the f u n c t i o n defined on S by A f ( s ) = £ A ( s , t ) f ( t ) , t whenever the sum on the r i g h t hand s i d e converges f o r each s e S . We say f i s F -summable to k i f f l i m £ A ( s , t ) f ( t b ) = k uniform l y i n b, A s t where b e S . This g e n e r a l i z e s the d e f i n i t i o n by G. G. Lorentz [11, p.171]. 5 2. CONVERGENCE AND LEFT ALMOST CONVERGENCE. We show i n t h i s s e c t i o n that l e f t almost convergence i s a g e n e r a l i z a t i o n of convergence i n a LA semigroup without f i n i t e l e f t i d e a l s . ' s 2.1. THEOREM. Let S be a LA semigroup. Then f i s £ac to k whenever f i s convergent to k i f f S does not c o n t a i n any f i n i t e l e f t i d e a l s . PROOF. Suppose S does not c o n t a i n any f i n i t e l e f t i d e a l s . We f i r s t show <j)(l ) = 0 f o r any LIM <J> and any a e S , where 1. , here a A and elsewhere, denotes the c h a r a c t e r i s t i c f u n c t i o n of A . We s h a l l always w r i t e <)>(A) f o r <Kl A) • I f <f>(a) > 0 then s i n c e <j> i s l e f t i n v a r i a n t , <j)(sa) >_ <J>(a) > 0 f o r a l l s e S . Since cfi(S) = 1 , Sa has to be a f i n i t e l e f t i d e a l , which cannot be. Suppose now f e C b and l i m f ( s ) = 0 . For e > 0 s l e t H be the f i n i t e subset of S f o r which | f ( s ) | < e whenever s e S ^  H . Let M = max If (s) I . Then |f (s) | < Y Ml + el,, ' . „ 1 1 1 1 — L„ a S'vH seH aeH / Hence i f ij> i s any LIM then | < j)(f)| < e • And s i n c e e i s a r b i t r a r y , we see that <j)(f) = 0 . I f now l i m f ( s ) = k then by c o n s i d e r i n g f-k , we see th a t f i s &ac to k . Conversely, suppose S has a f i n i t e l e f t i d e a l A . Let G be a minimal l e f t i d e a l of S contained i n A . We now show that G i s a group. Let G = {g^, g^} . Then by the m i n i m a l i t y of G , Gg = G f o r each g e G . Hence f o r 1 <_ i , j <_ n , there i s some 1 < k < n 6 such that g vg. = g. • Then f o r any LIM o) , we have <i>(g.) = <j> ( g i g . ) > <J>(g-) • And s i n c e t h i s argument i s symmetric we conclude 3 K. x 1 that < K S j ) = M g . ^ ) f ° r each g ^ , g e G . I f now g e G i s a r b i t r a r y then <{>(gG) >_<j>(G) = n ^ C g g ^ >_mtj)(gg^) , where m i s the c a r d i n a l i t y of gG . Since gG C G we see that n = m and so gG = G . This shows that G i s a group. Let now <J> be defined on m(S) by <J>(f) = ~ I f ( a ) , n aeG f c m(S) . Then i t i s easy to see that <j> i s indeed a LIM . C l e a r l y the f u n c t i o n 1 i s convergent to 0 , w h i l e 1 i s not lac to 0 s i n c e Cjr G §(ln) = 1 . This completes the proof. G In view of the above theorem, whenever we consider LA semigroups we w i l l always assume the semigroup to be without any f i n i t e  l e f t i d e a l s , even though we might not e x p l i c i t l y mention so. 3. ALMOST REGULAR MATRICES. We f i r s t prove the f o l l o w i n g u s e f u l lemma. We p o i n t out here that unless S i s a countable set the u s u a l proof does not work. 3.1. LEMMA. Let A be an i n f i n i t e m a t r i x on S . A necessary and s u f f i c i e n t c o n d i t i o n f o r Af E m(S) whenever f e C i s that there e x i s t s CO an M > 0 such that sup £|A(s,t)| <M . S t PROOF. Suppose sup £|A(s,t)| <_ M f o r some M > 0 . Then \ S t c l e a r l y f o r each s e S the sum ^ A ( s , t ) f ( t ) converges and || Af || <_ MJ| f || t 7 f o r each f e C . 00 Conversely, suppose Af e m(S) f o r every f e C . For each f i x e d s e S l e t a be the countable subset such that A ( s , t ) = 0 f o r a l l t £ a. Let be an i n c r e a s i n g sequence of f i n i t e subsets such that U c = a . For each f i n i t e subset a d e f i n e the continuous l i n e a r n n n f u n c t i o n a l A on C by A ( f ) = I A ( s , t ) f ( t ) . Then A ( f ) tea n converges to A ( f ) f o r each f e C^ , where A ( f ) = ][ A ( s , t ) f ( t ) • tea Hence f o r a l l n , llAnl| — M(s) f o r some constant M(s) . This i m p l i e s | A II < M(s) . Let now A be the l i n e a r f u n c t i o n a l on C defined by . 11 a" — . • s °° A (f) = A f ( s ) . Then A i s continuous s i n c e ||A II = ||A || < M(s) . s s 11 s 1 1 " a" — Now the set {A g : s e S} i s a pointwise bounded set of continuous l i n e a r f u n c t i o n a l on C s i n c e f o r each f e C sup |A (f) I = sup IAf (s) I < 0 0 by hypothesis. By the p r i n c i p l e of uniform- «• s s s boundedness there i s an M > 0 such that |A | < M f o r a l l s e S'. s Once again l e t s e S be f i x e d , and f o r each f i n i t e subset a d e f i n e the f u n c t i o n f e C by n n co J r sgn A ( s , t ) i f t e a f n ( t ) = > 0 otherwise C l e a r l y f e C^ and | | f j | <_ 1 . Therefore I | A ( s , t ) | = H A ( s , t ) f n ( t ) | = | A s ( f )| < ||As|j ||f || < M . tea t n Consequently, £|A(s,t)| = ^ | A ( s , t ) | <_M . And s i n c e t h i s i s true f o r t tea each s e S , i t f o l l o w s that sup £|A(s,t)| <_M . s t , • ' 8 3.2. THEOREM. Let S be a LA semigroup. Then a matrix A i s almost r e g u l a r i f f the f o l l o w i n g c o n d i t i o n s are s a t i s f i e d : (3.2.1) sup I A ( s , t ) <_ M f o r some M > 0 . S t (3.2.2) A ( s , t ) ,as a f u n c t i o n of s , i s £ac to 0 f o r each t e S . (3.2.3) ^A(s,t) , as a f u n c t i o n of s , i s Jlac to 1 . t PROOF. Suppose A i s almost r e g u l a r . Then (3.2.1) f o l l o w s from lemma 3.1. Conditions (3.2.2) and (3.2.3) f o l l o w s i f we note that A l t ( s ) . = A ( s , t ) and A l ( s ) = ^ A ( s , t ) . Conversely, suppose (3.2.1) and (3.2.2) and (3.2.3) h o l d . Then (3.2.1) together w i t h 3.1 i m p l i e s Af e m(S) f o r every f e C . Using (3.2.2) and (3.2.3) Af i s Hac to l i m f ( s ) whenever f i s i n the s set B = {1, 1 : t e S} . The proof i s then completed by n o t i n g that B i s fundamental i n C^ , i . e . , the uniform c l o s u r e of the l i n e a r span of B i s C CO . , 3.3. REMARK. (a) J . P. King was the f i r s t to consider almost r e g u l a r m a t r i c e s , and f o r the semigroup of a d d i t i v e p o s i t i v e i n t e g e r s , 3.2 y i e l d s King's theorem 3.2 i n [10]. (b) Let (N,+) and (N,-) denote the semigroup of a d d i t i v e p o s i t i v e i n t e g e r s and-!the semigroup of m u l t i p l i c a t i v e p o s i t i v e i n t e g e r s r e s p e c t i v e l y . Define the matrix A by 9 f<5(l,n) i f m i s odd A(m,n) = \ <^5(m,n) i f m i s even , where <S(m,n) = 1 i f f m = n and 0 otherwise. Then w i t h (N,+) , A i s not almost r e g u l a r because the sequence (1,0,1,0,...) which appears i n the f i r s t column of the matrix i s iac to 1/2 . However, w i t h (N,')» the sequence (1,0,1,0,...) r e s t r i c t e d to the i d e a l 2N i s the i d e n t i c a l l y 0 sequence and hence by p r o p o s i t i o n 3.5 below, i s lac to 0 . I t can e a s i l y be checked that A s a t i s f i e s the c o n d i t i o n s of 3.2, so that A i s almost r e g u l a r when the semigroup i s (N,-) . We f e e l that t h i s example together w i t h those that w i l l f o l l o w j u s t i f y the study of summability i n LA semigroups. In the f o l l o w i n g l e t S be a LA semigroup, C be the constant f u n c t i o n s on S , and H = { I f - ( g . " * 8 , ) : a e S , f ,g. e m(S) , n = 1,2,...} K = j I ( f . - I f.) : a. £ S , f. e m(S) , n = 1,2, ...} l i = 1 a ± 1 1 We denote the uniform c l o s u r e of any set A i n m(S) by CI(A) . For <)> e m(S) d e f i n e : m(S) ->• m(S) by ( T , f ) ( s ) = <j>(r f ) . In p a r t i c u l a r , 9 9 s n i f cj) i s a f i n i t e mean, i . e . cj> = £ <K t.)P t > then i = l i n ( T . f ) ( s ) = 7 <j)(t.)f(t.s) . Therefore, i f L 0 ( f ) = {SL f : s £ S> then T^f £ CoLO(f) . 9 We now b r i n g i n a r e s u l t of E. G r a n i r e r [5, p. 71, theorem 7] which w i l l be used throughout t h i s chapter. For completeness, 10 we w i l l give i t s proof. 3.4. THEOREM. Let S be a LA semigroup. Then F = C © C1(K) , and f i s Jlac to k i f f f e k l + C1(K) . Furthermore, ( i ) • I f f i s Jlac . to k and * s a n e t °f means such that l i m ||L <J> - <J) |j = 0 f o r each s e S then l i m ||T, f - k l |j = 0 . a s a a a Y a ( i i ) I f k l e ciCoLO(f) then f i s lac to k . PROOF. I f f i s Jlac to 0 then f e C1(K) . Otherwise, there would be some <f> e m(S) X such that <|>(f) ^  .0 and <j>(Cl(K)) = 0 by the Hahn-Banach theorem. But then <j> i s l e f t i n v a r i a n t and hence <j> = a<f>^  - 3<f>2 » where <j> , ^ are LIM on m(S) and a,g >_0 [6, p. 55]. Since <j)^(f) = ^ ( f ) = 0 i t f o l l o w s that <j>(f) = 0 . w h i c h cannot be. Thus f E C1(K) . And s i n c e <j>(Cl(K)) = 0 f o r any LIM <j> we get C1(K) co i n c i d e s w i t h the set of f u n c t i o n s Jiac to 0 . I f f i s Jlac to k then f - k l i s Jlac to 0 , so that f e k l + C1(K) . I f k l e C1(K) then k = <j>(kl) = 0 . Hence F = C ® C1(R) . I f <j> i s a LIM then we note <|>(f) = <Kg) f o r every g e CoLO(f) . Hence i f k l £ ClCoLO(f) then i t f o l l o w s from the f a c t that k l can be uniforml y approximated by a sequence g^ £ CoLO(f) we have' <j)(f) = k . This proves ( i i ) . Suppose now f = g - I g f o r some g e m(S) , a e S . I f now {d) } i s any net of means such t h a t l i m | L <f> - d) || = 0 f o r each Ct Q s ot ot s £ S , then for.„ t e S , 11 (T f ) ( t ) | = U a ( r g - r * g)| a = I 9 (r g - I r g) | y a t 6 a t & Thus l i m || T f || = 0 . Since each a , || T |{ <_ || 9 |f = 1 , and T " i s u a T a a l i n e a r i t i s easy to see that l i m f II = 0 whenever f E C1(K) . I f a 9 A f i s lac to k then f = k l + g f o r some g.e C1(K) . Thus l i m ||T f - k l | | = l i m ||T (f - k l ) || = l i m ||T g|| = 0 . This proves ( i ) . a a a a a . a 3.5. PROPOSITION. Let f e m(S) and A be any r i g h t i d e a l of S . I f Trf e m(A) i s the r e s t r i c t i o n of f to A then iTf i s lac to k i f f f i s lac to k . PROOF. The map IT : m(S) -»• m(A) i s defined by i r f ( s ) = f (s) f o r s c A , f £ m(S) . Let 9^ be a net of f i n i t e means converging to l e f t i n v a r i a n c e i n norm, i . e . , l i m ||L 9 - 9 || = 0 f o r each s £ S [1, Q s ot 01 p. 524, theorem 1 ] . We may assume that each 9^ have t h e i r support i n A, s i n c e otherwise, we r e p l a c e 6 by L 9 f o r some f i x e d a £ A . Let r rct J a a n 9 ( f ) = 7 9 ( t . ) f ( t . ) . I f f i s lac to k then by theorem 3 . 4 ( i ) , i t x=l f o l l o w s f o r a l l s E S , n (T TTf)( S) - k| = I I 9 a ( t . ) T T f ( t . s ) - k| a i = l = 1 1 9 a ( t . ) f ( t . s ) - k| i = l < || T f - k l | | 0 a 12 Hence f i s Jlac to k by-theorem 3 . 4 ( i i ) . S i m i l a r l y by t a k i n g 'd) to be a net of f i n i t e means on m(A) such that l i m L d> —. <±> 11= 0 f o r each s e S , we can show f " s a a" i s Jlac to k whenever Trf i s lac to k . . • • 4. REGULAR MATRICES. The f o l l o w i n g i s the extension of the w e l l known T o e p l i t z theorem f o r r e g u l a r matrices. However, t h i s extension does not depend on the a l g e b r a i c s t r u c t u r e of S . The only a d d i t i o n a l argument needed i n the usual proof i s given i n lemma 3.1. 4.1. THEOREM. Let S be a LA semigroup. The f o l l o w i n g c o n d i t i o n s are both necessary and s u f f i c i e n t f o r an i n f i n i t e m a t r i x A to be r e g u l a r : (4.1.1) sup £|A(s,t)| <_ M f o r some M > 0 . ' 1 s t (4.1.2) l i m A ( s , t ) = 0 f o r each t e S . s I 9 (4.1.3) l i m £A(s,t) = 1 . s / PROOF. Suppose A i s r e g u l a r . Then (4.1.1) f o l l o w s from lemma 3.1. Conditions (4.1.2) and (4.1.3) f o l l o w i f we note that A l (s) = A ( s , t ) and A l ( s ) = \ A ( s , t ) . teS Conversely, suppose (4.1.1), (4.1.2) and (4.1.3) h o l d . Then (4.1.1) and lemma 3.1 i m p l i e s Af e m(S) f o r every f e C m . Using (4.1.2) and (4.1.3) Af i s convergent to l i m f (s) whenever f i s i n s 13 the set B = {1, 1 : t e S} . We then complete the proof by n o t i n g that B i s fundamental i n C 00 4.2. REMARK. I t i s i n t e r e s t i n g to compare theorem 3.2 and 4.1. More s p e c i f i c a l l y , •. we draw to the reader's a t t e n t i o n the s i m i l a r i t i e s i n the con d i t i o n s of the theorems as w e l l as the pro o f s . 5. STRONGLY REGULAR MATRICES. We w i l l f r e q u e n t l y need the f o l l o w i n g : I f S i s a l e f t c a n c e l l a t i v e semigroup, b e S , sup £|A(s,t)| £. M then s t ( i ) I A ( s , t ) = I A(s,bt) tebS teS ( i i ) I. | A ( s , t ) | = I | A ( s , t ) | - I | A ( s , t ) | teS^bS teS tebS = I (|A(s,t)| - |A(s,bt)|) teS (5.0.1) <_ I |A(s,t) - A(s,bt) | teS The f o l l o w i n g theorem 5.1 contains one of the main r e s u l t s of t h i s chapter. When S i s the semigroup of a d d i t i v e p o s i t i v e i n t e g e r s , theorem 5.1 y i e l d s G. G. Lorentz's theorem 8 i n [11, p. 181]. 5.1. THEOREM. Let S be a l e f t c a n c e l l a t i v e LA semigroup generated by B CZ S . The f o l l o w i n g c o n d i t i o n s are necessary and s u f f i c i e n t f o r an .» ... i n f i n i t e m a t r i x A to be s t r o n g l y r e g u l a r : (5.1.1) sup £|'A(s,t)| <_ M f o r some M > 0 . 14" (5.1.2) li m l k ( s , t ) = 1 . s t (5.1.3) lim £|A(s,t) - A(s,at)| = 0 for each a e B . s t PROOF. We f i r s t show (5.1.3) implies that li m y|A(s,t) - A(s,bt)| = 0 for every b E S . Let b = a a , ... a', , o I n n—1 1 t where a. E B for i = 1, 2, .... n . Let b. = a.a. .. ... a, , i J J J-1 . 1 j = 2, 3, n-1 . Then the desired r e s u l t follows from the following: X|A(s,t) - A ( s , b t ) | <.I|A(s,t) - A ( s , a 1 t ) | (5.1.4) +.. I |A(s,t) - A(s,a t) | tea S + I |A(s,t) - A(s,a,t)| + ... teb 2S • + I |A(s,t) - A(s,a_t)| . tsb .S n-1 Now suppose f i s Jlac to k . Then Af e m(S) by 3.1. Let 9^ be a net of f i n i t e means converging i n norm to l e f t invar iance, i . e . , l i m ||L 9 - 9 || .= 0 [1, p. 524, theorem 1]. Let s ot ot n 9 (f) = £ 9 (t.)f('t'.) . Let e > 0 . Then there i s an a 0 such that a 1=1 a 1 1 n i f a >_ ao then | £9 ( t . ) f ( t . t ) - k| < e for a l l t e S by theorem i = l 3 . 4 ( i ) . Then f o r a l l t,s E S , and a l l a >_ arj. , n (5 . 1.5) HA(s,t) I <j> ( t . ) f ( t t) - l A ( s , t ) k | < ME t i = l t From ( 5 . 1 . 2 ) there i s a f i n i t e subset such that for s £ then 15 |^A(s,t) - l | < e . Hence f o r s £ H , t 1 .. (5.1.6) |£A(s,t)k - k| < e|k| t F i x an a >_ ag . From (5.1.4) there i s a f i n i t e subset B.^ such that I f s (j: H 2 , then £ | A(s ,'t.^t) • - A ( s , t ) | < e f o r i = 1, 2, . . . , n . Hence f o r s £ H 2 , n |Af(s) - ^A(s.t) I * (t ) f ( t . t ) | t i = l a 1 1 = | I <j> a(t.)Af(s) - I ( l ) a ( t i ) ^ A ( s , t ) f ( t i t ) | i = l i = l t = | 1$ ( t )l£A(s,t)f(t) - j A C s , t ) f C t . t ) ] | . i = l • t t n (5.1.7) = \l 4 <t ) [ I A ( s , t ) f ( t ) + I- A ( s , t ) f ( t ) - £A(s,t)f(t t ) ] i = l t s t . S teS'vt.S t l l l f l l I 4»a(t.)I|A(s,t.t) - A ( s , t ) | + | | f | | I <f> ( t ) I | A ( s , t ) | i = l a 1 t 1 i = l a 1 teSMi.S n < 2|f|| I * (t,)I|A(s,,.t t) - A ( s , t ) | (see (5.0.1)) 1=1 t < 2||£||e . Then f o r the f i x e d a >_ ag and s £ H^ = H^VJ H , i t f o l l o w s from (5.1.5), (5.1.6) and (5.1.7) that •- • n |Af(s) - k| £ |Af(s) - £A(s,t) I d> ( t . ) f ( t t ) | t i = l a 1 1 n + H A ( s , t ) I * ( t ) f ( t t ) - l A ( s , t ) k [ t i = l t 16 + |)>(s,t)k ~ k | t < (21If II + M + |k| )e Conversely, suppose A i s s t r o n g l y r e g u l a r . Then A i s re g u l a r and hence, (5.1.1) and (5.1.2) f o l l o w s . I f (5.1.3) does not hol d f o r some a , then there i s an e > 0 such that £|A(s,t) - A ( s , a t ) | > 5E t f o r i n f i n i t e l y many s e S . Using t h i s and the f a c t that l i m [A(s,t) - A ( s , a t ) ] = 0 f o r each t e S we now choose an i n c r e a s i n g sequence s cr(k) of f i n i t e subsets of S and an i n f i n i t e subset {s, } of S as k f o l l o w s : For convenience, denote A ( s , t ) - A(s,at) by B ( s , t ) . In general, f o r k = 1, 2, i f a(k) 3 a ( k - l ) (where a(0) = 0) l e t s, e S be such that k (5.1.8) I |B(s , t ) | > 5e and teS • (5.1.9) I |B(s, , t ) | < e . tecr(k) And s i n c e the sum £ |B(s , t ) | i s convergent there i s a f i n i t e subset teS a(k+l) D a(k) such that (5.1.10) I. |B(s , t ) | < e teS ^ a(k+l) Then from (5.1.8), (5.1.9) and (5.1.10) we have -' . I | B ( s k , t ) | tea(k+l)^a(k) (5.1.1D = ( I - I ~ . I )|B(s , t ) | teS tea(k) teS^a(k+l) > 5 e - e - e = 3 e " 17 We now d e f i n e f e m(S) by sgn B(s, ,t) i f t e a(ff(k+l)^a(k)) f ( t ) = ; k 0 otherwise. Since S i s l e f t c a n c e l l a t i v e , f i s w e l l - d e f i n e d . Moreover, | f || <_ 1 and £ f - f i s iac to 0 . But f o r k = 1, 2, i t f o l l o w s from cL (5.1.9), (5.1.10) and (5.1.11) that |A(£ f - f ) ( s )| = | I A ( s , t ) f ( a t ) - I A ( s , t ) f ( t ) teS teS I A(s t ) f ( a t ) - I A(s t ) f ( t ) - I. A(s , t ) f ( t ) | teS teaS R teS^aS •I [A(s t) - A(s, , a t ) ] f ( a t ) teS k . I B(s t ) f ( a t ) | teS fc = K I + ' I . + • • I ) B ( s k , t ) f ( a t ) teo(k) tea(k+l)^a(k) teS^a(k+l) L < I ~ I . I )|B(s ,t) tea(k+l)^a(k) tea(k) teS^a(k+l) > 3 e - e - . e = e But t h i s cannot be s i n c e A(£ f - f ) converges to 0 . Thus (5.1.3) holds, 3. 5.2. REMARK. I f A i s a s t r o n g l y r e g u l a r m a t r i x we cannot hope that (5.1.3) hold i n general as the f o l l o w i n g example shows: Let S be the set of p o s i t i v e i n t e g e r s w i t h m u l t i p l i c a t i o n * defined by i * j = k , where 18 k i s the s m a l l e s t odd i n t e g e r greater than or equal to i V j - m a x ( i , j ) . Then * i s a s s o c i a t i v e s i n c e f o r each i , j , k e S , both ( i * j ) * k and i * ( j * k ) are equal to the s m a l l e s t odd i n t e g e r greater than or equal to max(i,j,k) . Moreover, f o r every i , j e S then e i t h e r i v j or ( i V j ) + 1 i s a r i g h t zero f o r i and j . Hence S i s an ELA semigroup. Let now A be a m a t r i x defined on S by ( i ) A(m,n) = 0 whenever n i s even, or n < 2m-l . ( i i ) A(m,2n-1) >_ A(m,2n+1) > 0 whenever 2 n - l >_ 2m-l . ( i i i ) ^A(m,n) = 1 f o r each m . n Then A does not s a t i s f y (5.1.3) s i n c e f o r example, l i m £|A(m,n) - A(m,3*n)| = 1 . However, A i s s t r o n g l y r e g u l a r as theorem m n 5.4 below shows. We leave the d e t a i l s f o r the reader to check. 5.3. REMARK. I f S i s a c a n c e l l a t i v e LA semigroup without any f i n i t e l e f t i d e a l s then i s a proper subset of F , s i n c e otherwise, the i d e n t i t y m a t r i x would have to s a t i s f y (5.1.3). Then there e x i s t f i n i t e subsets a^, a^, a, b c S , a ^ b , such that at = t f o r t e S V and" "' bt = t f o r t -e S ^ a 2 . Hence i f t e S ^ (o^ a^) then a t = b t = t . Since S i s r i g h t c a n c e l l a t i v e , a = b , which cannot be. 5.4. THEOREM. I f S i s ELA semigroup then the f o l l o w i n g c o n d i t i o n s are both necessary and s u f f i c i e n t f o r an i n f i n i t e m a t r i x on S to be s t r o n g l y r e g u l a r : (5.4.1) sup £|A(s,t)| <_M f o r some M > 0 . s t 19 (5.4.2) l i m l k ( s , t ) = 1 . s t (5.4.3) . l i m £ | A ( s , t ) | = 0 f o r every a E S such that a e Sa. s teS'vaS PROOF. Let f = g - £ bg , g E m(S) , g ^ 0 , and b e S . Let a e S be such that ba = a . For E > 0 , l e t be a f i n i t e subset such that i f s ij: H then £ | A ( s , t ) | < e/2||g|| . Then f o r s ^ H teS^aS U we have A f ( s ) | = | I A ( s , t ) g ( t ) - I A ( s , t ) g ( b t ) | teS teS <\l A ( s , t ) g ( t ) - I A ( s , t ) g ( b t ) ' | +. 2||g|| I. | A ( s , t ) teaS tsaS tsS^aS < e Thus Af i s convergent to 0 . And s i n c e A i s l i n e a r , Af i s convergent to 0 whenever f e K . Suppose now f e C1(K) , and l e t g e K be such that l i m II g — f 11 = 0 Then °n n n l i m || Ag - Af || < l i m M] jg - f || = 0 . Since l i m i s a continuous l i n e a r n n n . n f u n c t i o n a l on C t h i s i m p l i e s l i m A f ( s ) = 0 . I f now f i s lac to k . s then f-k i s lac to 0 and hence l i m A(f - k ) ( s ) = 0 , i . e . , s l i m A f ( s ) = k . s Conversely, i f A Is s t r o n g l y r e g u l a r then (5.4.1) and (5.4.2) h o l d . I f (5.4.3) does not h o l d there i s an e > 0 and an a E S such thatN • £ | A ( s , t ) | > 5E f o r an i n f i n i t e number of s £ S . tsS^aS Using t h i s together w i t h the f a c t that l i m A ( s , t ) = 0 f o r each t E S , 20 we can choose, as i n the proof of 5.1, an i n c r e a s i n g sequence c?(k) of f i n i t e subsets of S ^ aS and an i n f i n i t e subset {s, } of S so that the k f o l l o w i n g c o n d i t i o n s h o l d s : (5.4.4). I | A ( s , , t ) | < e . tea(k) K (5.4.5) I. |A(s t ) | < c t£S^aS^o-(k+l) (5.4.6) I; |A(s t ) | > 3£ . t£CT(k+l)'VCT(k) We can choose the sets a(k) to be subsets of S ^ aS s i n c e S ^ aS i s i n f i n i t e (otherwise £ | A ( s , t ) | would be a f i n i t e sum of convergent tES'vaS f u n c t i o n s ) and the sum £ | A ( s , t ) | i s f i n i t e . Define teS^aS t sgn A ( s , , t ) i f t £ a(k+l)^a(k) f ( t ) = j k ^ 0 otherwise. Now observe that || f || <_ 1 , £ f = 0 , the support of f i s contained i n S aS , and that f - £ f i s £ac to 0 . Using (5.4.4), (5.4.5) and 3. (5.4.6) we have f o r a l l k , |A(f - i f ) ( s ) | = | I A(s. , t ) f (t) - I A(s, , t ) f ( a t ) | = | \ A ( s , , t ) f ( t ) a K teS teS tcS-vaS k B K I + I • + • I )A(s , t ) f ( t ) | tea(k) tea(k+l) ,va(k) teS^aS^aCk+l) L ( I " I ~ • I >|A(s , t ) | tea(k+l)^a(k) tea(k) t.eS^aS^a(k+l) >3e - £ - £ = £ . 21 But t h i s cannot be since A(f - I f) i s convergent to 0 . Thus (5.4.3) hold. • 5.5. REMARK. (a) In the proof of the necessity i n 5.4, we did not use the f a c t that a i s a r i g h t zero of some element i n S , so that i n any LA semigroup, (5.4.1), (5.4.2) and (5.4.3) are necessary conditions whenever A i s strongly regular. (b) We note that i n the proof of 5.1, i f f i s replaced by r^'f for any b e S , the same a 0 i s obtained since ||T ( r b f ) II l l ^ A ( f) II • X t a a follows from t h i s that the same f i n i t e subset there i s obtained i f f i s replaced by r ^ f for any b e S . Also i n the proof of 5.4, i f f i s replaced by r ^ f for any b e S , the same f i n i t e subset HQ i s obtained. In his proof, f o r the semigroup of ad d i t i v e p o s i t i v e integers, G. G. Lorentz made the same observation ( i t should be pointed out that our proof d i f f e r s - i n many ways from his) and used i t i n the proof of the following theorem 5.6 for the a d d i t i v e p o s i t i v e integers [11, p. 181]. We now use t h i s observation i n the following theorem. 5.6. THEOREM. Let S be a l e f t c a n c e l l a t i v e LA [ELA, not n e c e s s a r i l y l e f t c a n c e l l a t i v e ] semigroup and A be an i n f i n i t e matrix on S s a t i s f y -ing the conditions of theorem 5.1 [theorem 5.4], Then f i s F -summable to k i f f f i s lac to k . 22 PROOF. I f f i s lac to k then, as known, r f i s lac to k f o r every t e S . This can e a s i l y be seen from the f a c t that the l e f t t r a n s l a t i o n operator commutes w i t h the r i g h t t r a n s l a t i o n operator. By remark'5.5(b) l i m A(r f ) (s) = l i m T A ( s , t ' ) f ( t ' t ) = k uniforml y i n t , s t s t t i . e . , f i s F -summable. A The converse f o l l o w s e a s i l y from c o r o l l a r y 5.8 to the f o l l o w i n g theorem, proved f i r s t f o r the semigroup of a d d i t i v e p o s i t i v e i n t e g e r s by P. Schaefer [14, p. 51]. 5.7. THEOREM. Let S be a LA semigroup. I f A i s almost r e g u l a r and f i s F -summable to k then f i s Jiac to k . A PROOF. We b a s i c a l l y adapt Schaefer's proof to the general semigroup case. Suppose f i s F -summable to k . Then l i m ^ A ( s , t ) f ( t b ) = k uniformly i n b . Let g be a f u n c t i o n of s and s t ' ^ b defined by g(s,b) = ^ A ( s , t ) f ( t b ) . Then g(s,b) = k + h(s,b) , where t h i s a f u n c t i o n of s and t such that h , as a f u n c t i o n of s , i s convergent to 0 uniforml y i n b e S . Now f o r each f i n i t e subset a of S , d e f i n e g as a f u n c t i o n of s and b by g (s,b) = £ A ( s , t ) f ( t b ) tea .. .. Then g^ converges unifor m l y to g f o r each f i x e d s e S s i n c e || g - g|| = sup |g (b) - g(b)| <_ I. | A(s, t) | ||f|| and t h i s can be made as CT b ° teS^a small as we please. I f now c() i s any LIM then f o r each f i x e d s e S , 23 9(g) = <f>(lim g ) = lim I A ( s , t ) 9 U f) = £ A(s,t)<Kf) = * (k+h) = k + 9(h). 0 0 a tea C teS. Thus <j>(f)£A(s,t) = k + 9(h) . Since h , as a function of s converges t to 0 uniformly i n b e S , for every e > 0 there i s a f i n i t e subset H such that i f s £ H then |9(h)| < e , i . e . , 9(h) , as a function of S; , i s convergent to 0 . If now i}> i s any LIM then i i i (5.7.1) *[<Kf)EA(s,t)] = 9 ( f M l A ( s , t ) ] ="*(k) + i K < K h ) ) . ! t t ; f Since A i s almost regular, ^ [ ^ A ( s , t ) ] = 1 and iK<t>(h)) = 0 . Therefore, t we see from (5.7.1) that 9(f) = if>(k) = k , i . e . , f i s £ac to k . This completes the proof. The following c o r o l l a r y , which i s due to G. G. Lorentz for the ad d i t i v e p o s i t i v e integers [11, p. 171], i s an immediate consequence of, 5.7 since every regular matrix i s almost regular. 5.8. COROLLARY. Let S be a LA semigroup. If A i s regular and f i s F -summable to k then f i s Jlac to k . A / 6. ALMOST SCHUR.MATRICES. The following theorem gives s u f f i c i e n t conditions f o r a matrix to be almost Schur. 6.1. THEOREM. Let S be a LA semigroup. Let A be an i n f i n i t e matrix on S s a t i s f y i n g the following conditions: 24 (6.1.1) sup £|A(s,t)| £ M f o r some M > 0 . s t (6.1.2) The sum £|A(s,t)| converges uniformly i n s . t (6.1.3) A ( s , t ) , as a f u n c t i o n of s , i s Jlac to a t f o r each t e S Then Af i s Jlac to ^ a £ f ( t ) f o r each f e m(S) . PROOF. Let I be the f a m i l y of a l l f i n i t e subsets of S d i r e c t e d by i n c l u s i o n . Let f e m(S) . For each a e £ d e f i n e g^ by g (s) = 1 A ( s , t ) f ( t ) . Then c l e a r l y g i s Jlac to £ « tg(t) by 0 tea tea (6.1.3). Now (6.1.1) i m p l i e s Af e m(S) . And using (6.1.2), one can r e a d i l y show that Af i s the uniform l i m i t of g^ . Hence Af i s Jlac and i f <f i s any LIM then <j>(Af) = d)(lim g ) = l i m <KgJ = l i m I a f ( t ) = ^ a . f ( t ) . ° a o CT tea • t C 6.2. COROLLARY. I f A i s an almost r e g u l a r matrix then A cannot be an almost Schur matrix. PROOF. I f A i s an almost r e g u l a r m a t r i x A ( s , t ) i s Jlac to 0 and ^A(s,t) i s Jlac to 1 . I f A i s a l s o an almost SchUr m a t r i x t " then Af i s ilac to 0 by the theorem. In p a r t i c u l a r , ^ A(s,t) i s Z • \ Jlac to 0 , which cannot be. ' 6.3. REMARK. I t i s easy to see that i f A i s an almost Schur m a t r i x then (6.1.1) and (6.1.3) are necessary. However, (6.1.2) i s not necessary 25 as the f o l l o w i n g example shows: Let S be the semigroup of o r d i n a l s l e s s i I i than the f i r s t uncountable o r d i n a l ft w i t h the usual a d d i t i o n of order j; j types. Then S i s a non-commutative, l e f t c a n c e l l a t i v e , ELA semigroup [5, p. 73]. Define A on S by i <5(s,t) i f l < s < o o , t e S A ( s , t ) = 0 otherwise, where OJ i s the f i r s t countable i n f i n i t e o r d i n a l . Then f o r any f e m(S) , i t i s easy to see that A f ( s ) = 0 f o r s e a + S f o r any a > u) . By 3.5 Af i s Hac to 0 . But c l e a r l y (6.1.2) i s not s a t i s f i e d . > 6.4. EXAMPLE. Let S = {(m,n) : m = l , 2, . . . , n = l , 2,...} Define the operation * on S by (a) (m^n )*(m 2,n ) = (ra^n^, 1^+^) i f ^ ± 1 and m 2 ^ 1 . (b) (m1,n1)5'«(l,n2) = ( l , ^ ) * ^ , ^ ) = ( l , n 2 ) i f ± 1 . (c) ( l , n ) * ( l , n 2 ) = ( l j ^ v n 2) , where n^y xi^ = max ( n ^ n ^ . That S i s an ELA semigroup a c t u a l l y f o l l o w s from the f o l l o w i n g general c o n s t r u c t i o n : Let 5 = 5 ^ ^ 8 2 , where i s any semigroup and S 2 i s any ELA semigroup. For a,b e S , d e f i n e the product a*b to be the product of a and b i n , i = 1, 2, i f both a,b e . I f a e , b e S 2 then a&b = b*a = b . Now f o r each k and each £ f i x e d , d e f i n e r ( l / 2 ) k + A ' i f m = 1 f(m,n) = j ^ 0 otherwise . 26 Then g(m,n) = J l ( 1 1 )f(m,n) = f[(1,1)*(m,n)] = (1/2) f o r a l l m and ^ n . Then g(m,n) - f(m,n) i s Jlac - to 0 . Define the ma t r i x A on S by A(m,n;k,Jl) = g(m,n) - f(m,n) . Then A(m,n;k,Jl) , as a f u n c t i o n of (m,n) , i s Jlac to 0 f o r each k and Jl ; and £ | A(m,n;k, Jl) | = £ ( l / 2 ) k + ^ converges uniformly i n (m,n) to 1 . By 6.1 A i s an almost Schur matrix. 7. SCHUR MATRICES. The f o l l o w i n g i s the extension of Schur's theorem. However, t h i s extension does not depend i n any way on the a l g e b r a i c s t r u c t u r e of S . 7.1. THEOREM. The f o l l o w i n g c o n d i t i o n s are both necessary and s u f f i c i e n t f o r an i n f i n i t e m a t r i x A to be a Schur m a t r i x : (7.1.1) l i m A ( s , t ) e x i s t s f o r every t e S . s (7.1.2) The sum £|A(s,t)| converges uniformly i n S . t Moreover, i f l i m A ( s , t ) = a then Af converges to £a f ( t ) f o r every s t t f e m(S) . PROOF. Suppose (7.1.1) and (7.1.2) are s a t i s f i e d . I f £ denote the f a m i l y of a l l f i n i t e subsets of S d i r e c t e d by i n c l u s i o n , f o r each o e 2 a n d f e m ( S ) de f i n e g (s) = I A ( s , t ) f ( t ) . Then (7.1.1) and tea „ 27 (7.1.2) imply that each g^ e and Af i s the uniform l i m i t of g^ ; and thus Af e C . Since l i m i s a continuous l i n e a r f u n c t i o n a l on C CO 00 l i m A f ( s ) = £a f ( t ) . s t C Conversely, suppose A i s a Schur m a t r i x . Then sup £|A(s,t)| <_M f o r some M > 0 and l i m A ( s , t ) = a e x i s t s f o r each s t s fc t e S . Define g^ as above. Then g^ i s a norm-bounded net converging pointwise to Af i n C^ . By [12, p. 249, lemma 3] g^ converges w* to Af i . e . , f o r every 9 e (S) , l i m cf>(g ) = <f>(Af) . In p a r t i c u l a r , 1 0 0 . l i m A f ( s ) = l i m l i m g (s) = )a f ( t ) . Hence the matri x B defined on S s • : a s a t t by B ( s , t ) = A ( s , t ) - ot i s a Schur m a t r i x such that Bf converges to 0 f o r every f s m(S) . I f now (7.1.2) i s not s a t i s f i e d then there i s an e > 0 such that f o r a l l f i n i t e subsets a there i s an i n f i n i t e subset S(a) such that £. | B ( s , t ) | > 5e f o r a l l s e S(a) . Using t h i s , teS^a together w i t h l i m B( s , t ) = 0 f o r a l l t e S , choose an i n c r e a s i n g s • sequence a(k) of f i n i t e subsets of S and an i n f i n i t e subset ^ s j c ^ °f S as f o l l o w s : In general, f o r k = 1, 2, i f a(k) 3 o ( k - l ) (where a(0) = 0) , l e t s^ e S be such that / . (7.1.3) I |B(s , t ) I > 5c and tester (k) (7.1.4) I |B(s , t ) | < E . tea On) And s i n c e the sum £ |B(s, , t ) | i s convergent, there i s a f i n i t e subset t£S K a(k+l) 3 o(k) such that (7.1.5) I |B(s , t ) | < £ .' teS^crCk+l) Then from (7.1.3), (7.1.4) and (7.1.5) i t f o l l o w s that I • | B ( s t).| = ( I - I - I )|B(s t ) tea(k+l)^cr(k) * teS tea(k) teS^a(k+l) (7.1.6) > 5e - e - e = 3e Define now f e m(S) by i sgn B(s ,t) i f t e a(k+l)^a(k) f ( t ) = ; k 0 otherwise Using (7.1.4), (7.1.5), (7.1.6) and ||f||<.-l , we have, f o r k = l , 2 , : f' | B f ( s k ) | = |. I B ( s k , t ) f ( t ) | j t £ S j > ( I • " I ' l )|B(s t ) | tea(k+l)^a(k) tea(k) teS'va(k+l) >3e - £ - e = e . But t h i s cannot be s i n c e Bf i s converging to 0 . 8. ALMOST STRONGLY REGULAR MATRICES. The f o l l o w i n g theorem gives s u f f i c i e n t c o n d i t i o n s f o r a ma t r i x to be almost s t r o n g l y r e g u l a r . We strengthen the theorem when the semigroup S i s ELA . 8.1. THEOREM. Let S be a l e f t c a n c e l l a t i v e LA semigroup generated by B C S . Let A be an i n f i n i t e m a t r i x on S such that the f o l l o w i n g 29 c o n d i t i o n s h o l d : (8.1.1) sup £|A(s,t)| <_ M f o r some M> 0 . s t . (8.1.2) ^A(s,t) , as a f u n c t i o n of s , i s lac to 1 . t (8.1.3) £|A(s,t) - A ( s , a t ) | , as a f u n c t i o n of s , i s Jlac to 0 f o r t every a e B . Then Af i s lac to k whenever f. i s lac to k . ' PROOF. Con d i t i o n (8.1.3) together w i t h (5.1.4) i m p l i e s £|A(s,t) - A ( s , a t ) | , as a f u n c t i o n of s , i s lac to 0 f o r a l l a £ S. t I f f e F then (8.1.1) i m p l i e s Af e m(S) , w h i l e (8.1.2) i m p l i e s Af i s lac to f ( s ) whenever f i s a constant f u n c t i o n . Suppose f e K and f = g - l^g f o r some g e m(S) , a E S . Let <f>a be a net of f i n i t e means converging i n norm to l e f t i n v a r i a n c e , i . e . , l i m J L 9 - 9 || = 0 f o r each s E S [1, p. 524, theorem 1] . Let \ a s a a y n d> ( f ) = Z d> ( t . ) f ( t . ) . Then (8.1.3) together wxth theorem 3.4 shows that \ a . , a I l . .. i = l f o r every E > 0 , there i s an arj /such that i f a >_ ag then f o r a l l n s E S , £ <j> ( t . ) ^ | A ( t . s , t ) - A ( t . s , a t ) | < E . Then f o r a l l s E S , i = l a 1 t . 1 1 a >_ otQ , n I * ( t )[Ag - A(£ g ) ] ( t s ) | i = l n n = j I 9 a ( t . ) A g ( t . s ) - I * a ( t . ) A ( A a g ( t . s ) ) ' i = l i = l 30 n n I 6 (t ) I A ( t s , t ) g ( t ) - I A ( t ) I A ( t s , t ) g ( a t ) | i = l a 1 teS. 1 i = l teS 1 i l l * I [ A ( t s,at) - A ( t s , t ) ] g ( a t ) | i = l a 1 teS + I I 4> ( O I A ( t s , t ) g ( t ) | , i = l teS'vaS lllsll I <0 (t ) I |A(t s,at) - A ( t s , t ) | + ||g|| I * (t ) I |A(t s y t i j l .1=1 a 1 teS i = l • teS^aS 1 ' \ ! i i i n i l 2 l | g | | I *„(tJ I |A(t s,t) - A ( t s , a t ) | (see (5.0.1)) j i = l a teS <2||g|| £ • I t f o l l o w s from theorem 3.4 that Af i s ilac to 0 . Since A i s l i n e a r , Af i s &ac to 0 whenever f s K . Suppose now f s C1(K) , and l e t g e K be such that l i m II g - f II = 0 . Then n n n l i m || Ag - Af || <_ l i m M ||g - f || = 0 . Hence i f § i s any LIM then n n n n l i m | cb (Ag ) -.(j>(Af)| = 0 . Thus Af, i s lac to 0 s i n c e cj)(Ag ) = 0 n n n for each n . This completes the proof. 8.2. REMARK. (a) I f S i s the a d d i t i v e p o s i t i v e i n t e g e r s theorem 8.1 y i e l d s P. Schaefer's theorem 2 [14, p. 52]. Our proof i s e n t i r e l y d i f f e r e n t from h i s . His proof does not seem to c a r r y over to the general case. 31 (b) I t i s c l e a r that i f A i s an almost s t r o n g l y r e g u l a r m a t r i x then (8.1.1) and (8.1.2) are both necessary c o n d i t i o n s . However, (8.1.3) does not always h o l d , s i n c e the i d e n t i t y m a t r i x A i s almost s t r o n g l y r e g u l a r but f o r the a d d i t i v e p o s i t i v e i n t e g e r s , l i m Y|A(m,n) - A(m,n+1)I = 2 . When S i s ELA (not n e c e s s a r i l y m n l e f t c a n c e l l a t i v e ) we have the f o l l o w i n g stronger r e s u l t . 8.3. THEOREM. Let S be ELA , and A be an i n f i n i t e m a t r i x on S s a t i s f y i n g the f o l l o w i n g c o n d i t i o n s : (8.3.1) sup £'|A(s,t)| <_ M f o r some M > 0 . s t (8.3.2) £A(s,t) , as a f u n c t i o n of s , i s lac to 1 . t (8.3.3) £ | A ( s , t ) | , as a f u n c t i o n of s , i s lac to 0 f o r teS'vaS every a s S such that a e Sa . Then Af i s lac to k whenever f i s lac to k . PROOF. Let f £ac to k . By [5, p. 72, theorem 8] f o r every e > 0 there i s a b e S such that i f t e bS then | f ( t ) - k| < e. j Let a e S be such that ba = a . Then i f t e aS C bS , | f ( t ) - k| < e. By (8.3.2) and (8.3.3) l e t c,d £ S be such that i f s £ cS then i |^A(s,t) - l| < e , and i f s £ dS then £ | A ( s , t ) | < e . Now ! t t£S^aS ! Af £ m(S) by (8.3.1) and i f s £ cSOdS + 0 ( s i n c e S i s ELA), then 1 32 |Af(s) - k| <_ |£A(s,t)(f(t) - k ) | .+ |^A(s,t)k - k| t t j 1 ( 1 + 1 ) | A ( s , t ) | | f ( t ) - k| + |k| H A ( s , t ) - 1 teS^aS teaS t < (||f || + |k|')e + Me + |k|e | = (2|k|+ ||f || + M)e . ' v . By [5, p. 72, theorem 8] , Af i s Jlac to k . 8.4. REMARK. I f A i s non-negative, i . e . , A ( s , t ) >_ 0 f o r a l l s,t e S , then (8.3.1), (8.3.2) and (8.3.3) are necessary a l s o . For, I | A ( s , t ) | = I A ( s , t ) = A l _(s) . Since i s ^ac to 0 teS^aS teS^aS . i t f o l l o w s that (8.3.3) holds. 8.5. EXAMPLE. Let S be the semigroup described i n 5.2. Let A be defined f o r each m,n by ( i ) A(2m-l,2n-l) ^  A(2m-l,2n+l) > 0 whenever 2 n - l _> 2m-l and 0 otherwise. ( i i ) A(2m,n) = 1 only i f n = 1 and 0 otherwise. ( i i i ) A(m,n) = 0 whenever n i s even. ( i v ) ^A(m,n) = 1 f o r each m . n By 8.3 A i s almost s t r o n g l y r e g u l a r . However, i f we r e p l a c e the op e r a t i o n * by the or d i n a r y a d d i t i o n , then A i s not almost s t r o n g l y r e g u l a r s i n c e the sequence f = (1,0,1,0,...) i s Jlac to 1/2 w h i l e Af i s the sequence (1,1,1,...) , which i s Jlac to 1 . 33 CHAPTER I I THE INVARIANT MEAN ON A TOPOLOGICAL SEMIGROUP 9. DEFINITIONS AND NOTATIONS. For general t o p o l o g i c a l terms we w i l l f o l l o w K e l l e y [ 9 ] . We r e c a l l i f (S,/7) i s a uniform space, then f o r each V^, V^, V e U , x e S , A C S , ( i ) V[x] = {y e S : (x,y) e V} . ( i i ) V[A] = U V[x] . xeA ( i i i ) Vi° V2 = ^ x > y ) e S x S : f o r some z e S , (x,z) e and (z,y) e V ( i v ) V _ 1 = {(x,y) e SxS : (y,x) e V} . (v) V n = V°V° .. °V (n terms) ( v i ) (V 1»V 2)[A] = V 1 [ V 2 [ A ] ] . For each V e U we say V t o t a l l y covers S i f f o r k \ some f i n i t e subset, {a. , a. } C S > 'S (2 U V f a , ] . We say S has J- K. n 1 i = i property (B) i f there i s some V e U such that f o r each n and each k f i n i t e subset {a , a, ) C S , S ^  U V [ a . ] ^ 0 . Examples of uniform 1=1 spaces having property (B) are non-compact l o c a l l y compact group and any uniform space which has an unbounded r e a l - v a l u e d u n i f o r m l y continuous f u n c t i o n on S [7, p. 118]. To see the l a t e r , l e t V = ((x,y) e SxS : |f(x) - f ( y ) | < 1} . Then (x,y) e V 2 i m p l i e s 2 y £ V [x] = V[V[x]] and hence y e V[s] f o r some s £ V[x] .; I t f o l l o w s that |f(x) - f ( y ) | <_ |f(x) - f ( s ) | + | f ( s ) - f ( y ) | < 2 . By i n d u c t i o n , i f (x,y) £ V n then |f(x) - f ( y ) | < n . Therefore, i f S does not have 34 n property (B) then S = L J V [a.] f o r some n and some f i n i t e subset i = l 1 { a . , a } C S . Hence f o r s e S , I f ( s ) I < n + max I f ( a . ) I , which l n — , . 1 l<i<n. cannot be. Throughout t h i s chapter we are mainly i n t e r e s t e d i n a t o p o l o g i c a l subsemigroup S of a Hausdorff t o p o l o g i c a l group G . For such a semigroup S we s h a l l adopt the f o l l o w i n g n o t a t i o n s throughout: I f U i s the l e f t u n i f o r m i t y f o r a t o p o l o g i c a l group G then f o r each V z U , • the set V D S x S w i l l be denoted by V* , so that U* = {V* : V* = V r\ S x S , V £ U) i s the r e l a t i v e l e f t u n i f o r m i t y f o r the t o p o l o g i c a l subsemigroup S of G . The space of bounded r e a l - v a l u e d l e f t u n i f o r m i l y continuous fu n c t i o n s w i t h respect to U* w i l l be denoted by LUC(S) . I t i s w e l l known that LUC (S) i s a closed l e f t t r a n s l a t i o n j i n v a r i a n t ( i . e . , £^f £ LUC(S) f o r each f £ LUC (S) , a £ S ) subspace of \^  m(S) c o n t a i n i n g the constants. I f L C m(S) i s a l e f t i n v a r i a n t subspace v l e t i ' : L -> L be the r e s t r i c t i o n of i to L . We w i l l use the symbol s s 3 £ g i n s t e a d of £ g' so that L g w i l l be the conjugate mapping £ g'* a s w e l l as £ g* . I t w i l l be c l e a r from the context which mapping we have i n mind. Let J£(S) = (d) e m(S)* : L g d) = d) f o r each s e S> and J U £ ( S ) =' {<(> £ LUC(S)* : L g d) = § f o r each s e S} . Then J£(S) or J A(S) i s i n f i n i t e dimensional i f considered as v e c t o r spaces, i t i s dimensional. not f i n i t e 10. . TECHNICAL LEMMAS. We w i l l need the f o l l o w i n g lemmas. The c r u c i a l lemma i s 35 10.4, and the proof fo'llows more or l e s s the proof of lemma I I I - l i n [7, p. 119]. r 10.1. LEMMA. I f W i s a neighbourhood of the i d e n t i t y e i n G and V = {(x,y) e G * G : y e Wx} then f o r every a e S , V*[a] = W a f l S . PROOF. V*[a] = {y £ S : (a,y) £ V} = V[a] O S = Wa/^S . i 10.2. LEMMA. I f W i s a symmetric neighbourhood of the i d e n t i t y e i n G and V = {(x,y) £ G x G : y £ Wx} then V* n C V n* f o r every n . ' i •' 2 ;| PROOF. . Suppose (p,q) e V* . Then f o r some r e S , (p,r) £ V O S x s and (r ,q ) e V ^ S x s . Hence (p,q) £ V2f~\ s x s = !V2 The lemma now f o l l o w s by i n d u c t i o n . 10.3. LEMMA. I f W i s a symmetric neighbourhood of the i d e n t i t y e 2 i n G , V = {(x,y) £ G x G : y £ Wx} , and £ S ^  V ^[a^] then V*{a ] n V * [ a 2 ] = 0 . PROOF. Suppose q £ V*[a 1] C\ V*[a 2] . Then q £ Wa OWa^S'. \ - 1 - 1 2 Hence q = w-|_a^ = w 2 a 2 , which i m p l i e s a 2 = w 2 w - j _ a ] _ e w ^ a ^ = W a^ . 2 2 Thus a„ e W a^C\ S = V - v[a 1] , which cannot be. 36 10.4. LEMMA. Let S be a topological'subsemigroup of a t o p o l o g i c a l . 0 0 i group G , and {p } .. be a countable dense subset of S i n the r e l a t i v e n n=l i topology. Let . V* E U* be symmetric and such that V* n does not t o t a l l y cover S f o r any p o s i t i v e i n t e g e r n . Then there i s an unbounded non-negative l e f t u n i f o r m l y continuous f u n c t i o n F on S such that k+2 F _ 1 [ o , k ] C U v * 2 ( k + 2 ) [ P . ] . 1=1 PROOF. We choose a sequence of i n c r e a s i n g subsets of S as f o l l o w s : In ge n e r a l , i f A^, has been chosen such that f o r I £ j £ n-1 , (1) v * [ P l l \J ... UV* [p ] C A-j » (2) V*.[A J_ 1]CA J , (3) A c v * 2 j [ p a ] u • •. U v * 2 j lp. ] , . we l e t A = V*[A , t j V * [ p ]] , where A = 0 , n = l , 2, ... . -(Here A n n - 1 w n 0 denotes the c l o s u r e of A i n S .) Then we have (4) v * [ P l ] U . . . U v * [ p n ] C A N • (5) V A [ A n _ 1 ] C A n • (6) A n C V*[A Q] C V * 2 [ A N _ 1 ( J V * [ p n ] ] C v * 2 n [ P l ] u • . . U v * 2 n - 1 [ p n ] C v * 2 n [ P l ] u.-.UV* 2 n[p n] . Continuing this~-.way we o b t a i n the sequence ^AN} w i t h the f o l l o w i n g p r o p e r t i e s : 37 We (7) U A = S . n=l (8) V * [ A j C A n + 1 • i = l Conditions (8) and (9) are c l e a r from (5) and ( 6 ) , w h i l e (7) f o l l o w s 00 CO 00 because {p } , i s dense i n S and so S = V J V A [ p . 1 C l _ J A C S , n n=l ; L t i J v— v—; n v— x=l n=l may even assume ^ A n_i ^ $ > sin c e otherwise, we could choose a | subsequence w i t h t h i s property. That t h i s i s p o s s i b l e f o l l o w s from (9) and the f a c t that V* n does not t o t a l l y cover S . I t i s proved by A. Weil i n [18, p. 13] that i f (E,£/) i s a uniform space, p E E , V £ <7 , i s a sequence of symmetric members i n U such that V ,,°V ,.. C V C V then there i s a uniforml y continuous n+1 n+1 n J f u n c t i o n f : E + [0,1] such that f ( p ) = 0 , f ( E ^  V[p]) = 1 and |f(q) - f ( r ) | < l / 2 n 1 whenever (q,r) e . Moreover, as noted by E. G r a n i r e r [6, p. 121] i f P C S then the f u n c t i o n f can even be chosen i n such a way that f ( P ) = 0 and. f ( g ^  v[P]) = 1 . Applying t h i s to our s i t u a t i o n , i f V* , n = 0 , 1, 2, ... i s any sequence of symmetric member of U* such that V* =. V* and V*,.. °V*,.. C V* , then there i s , f o r each . 0 n+1 n+1^— n ' k = 1, 2, ... , a l e f t u niformly continuous f u n c t i o n f ^ : S [0,1] such that (10) f ( A ^ = 0 . (11) f k ( S ^ V * ^ ] ) = 1 and hence f (S * ^ ) = 1 . (12) | f k ( p ) " f k ( q ) | < l / 2 n _ 1 i f (p,q) e V* . 38 Define now the sequence of f u n c t i o n s ^ n^^ ky h. (s) = f (s) + k-1 f o r k = 1, 2, ... . Then 1£ tc k-1 i f s e h k ( s ) = ' 1 k i f s e S > A k + 1 , and i f (p,q) £ V* then f o r every k , |h k(p) - h k ( q ) | =. | f k ( p ) - f k ( q ) | < l / 2 n 1 Then Define the r e q u i r e d f u n c t i o n F on S by r h (s) i f s £ A-F(s) = 1 h (s). i f s £ A k + 1 f o r k > 2 . F i s unbounded s i n c e ^ 4 0 • Suppose now s £ * Then s £ A , ^ A f o r n > k+1 , and so F(s) = h (s) > n-1 > k+1-1 = k . n+1 n k + 2 n Thus F" 1[0,l]C-A v + 9CU v* 2 ( k + 2 )[P.] • k + Z i = l 1 We now prove F i s l e f t u n i f o r m l y continuous. Let £ > 0 and choose n so l a r g e so that l / 2 n 2 < £ . Let (p,q) £. V* . We have three cases to consider: ( i ) I f both p and q are i n ^ +^_ ^ ^ f o r some k >_ 2 , then |F(p) - F ( q ) | = |h k(p) - h k ( q ) | < l / 2 n _ 1 < £ . ( i i ) I f both p and q are i n A^ then |F(p) - F(q) | = |h 1(p) - h ^ q ) | < £ . ( i i i ) Assume that i i s the f i r s t index f o r which p e A_^  and j i s the f i r s t index f o r which q £ A. . Assume a l s o i < j . Since i 3 39 q £ V*[p]C.V*[p]C V*[A ±]C. V * [ A ± + 1 ] C A ± + 1 , we have j = i + ,1 . A l s o , we may assume i > 2. Then s i n c e p e A. ^ A.,., and q e A.,n ^ A. ' J — r 1. 1+1 1+1 | 1 i we have |F(q> " F(p) I = l\(q) - h.^Cp)] = |h.(q) - ( i - 1 ) + ( i - 1 ) - h.^Cp) = | h . ( q ) - h . ( p ) + h . ^ C q ) - h . ^ C p ) ! < | h . ( q ) - h . ( p ) | + | h i _ 1 ( q ) - h . ^ C p ) < l / 2 n _ 1 + l / 2 n _ 1 = l / 2 n _ 2 < £ This completes the proof of the lemma. 11. THE MAIN THEOREM. The f o l l o w i n g theorem i s due to E. G r a n i r e r [7, p. 124, theorem 1 ] : 11.1. THEOREM. Let G be a separable Hausdorff t o p o l o g i c a l group which i s amenable as a d i s c r e t e group and s a t i s f i e d property (B) . Then JTJlCG) i s i n f i n i t e dimensional. I t i s our aim to extend the above theorem to c e r t a i n t o p o l o g i c a l semigroups (theorem 11.3). Our method i s more or l e s s of that used i n the proof of the above theorem. We w i l l give an example to show that • the above theorem cannot be extended to any semigroup. F i r s t , we need the f o l l o w i n g theorem of E. G r a n i r e r [7, p. 112, theorem 1 ] . For the sake of completeness, we w i l l g i v e i t s proof below.. 40 11.2.. THEOREM. Let S be. a LA semigroup and L C m(S) be a l e f t i n v a r i a n t subspace c o n t a i n i n g the constants. I f J £(S) = {<j> eL* : L d). = (j>. f o r each s £ S} , assume that there i s a Li S • R . 0 0 ] sequence {p > •• . (~ S such that ^n n=l w {d) £ L* : L <(> = d) , n = l , 2 ,...} = J £(S) P n I f J^£(S) i s n dimensional f o r some n < 0 0 then each LIM <!> £ L* i s a w*-sequential l i m i t of f i n i t e means on L PROOF. Let <j> £ L* be a LIM and l e t ij; £ m(S)* be the norm pres e r v i n g extension of <f> . Then 1 = ||d> II = <j> (1) = ip(l) • This o o o im p l i e s U> i s a mean on m(S) . Let v be any LIM on m(S) . Let <J>^  = v 0 IJJ , where v © 4> i s the f u n c t i o n a l defined on m(S) by v © 4>(£) = v(h) , where h i s defined by h(s) •- f ) , f z m(S) , s £ S . Then <j>^  i s a LIM on m(S) which i s an extension of d>Q [1,. p. 526-527 and p. 529, c o r o l l a r y 2 ] . In f a c t , i f f e L , then h(s) = U>(£gf) = <J>o(Jlsf) = <J>Q(f) , i . e . , h i s a constant on s • Therefore v © \ | i ( f ) = v(h> = v(d> o(f)l) = 4> o(f)v(l) = d>Q(f) , i . e . , v © u i s an extension of' <j> . o Let now {d>'} be a net of f i n i t e means i n m(S)* a such that l i m <j>'(f) = d)'(f) f o r every f £ m(S) and l i m ||L d/ - <j>'|| = 0 Q Or O g S Ot Ot f o r each s e S [6, p. 44, (5.8)*] . I f <j> i s the r e s t r i c t i o n of <b' to a a L then i t can e a s i l y be checked that l i m <J> ( f ) = <j> ( f ) f o r each f e L a a o and l i m ||L <$> - (f> || = 0 f o r each s e S (where the norm now i s that of S Ot (X L*) 41 Let S(<J>o,l/n) = {<j>eL* : || 9 - <jj || < l / n } . Since J £(S) i s f i n i t e dimensional, there i s a decreasing sequence V of Li n w*-closed convex neighbourhoods of <j>Q such that 9 £ V r\ J T 5,(S)C S(9 ,1/n) r\ J T£(S) . O n L O L Now f o r each n there i s a' such that i f a > a' n — n then IIL 9 - 9 \\^< l / n f o r 1 < i < n . Since 9 i s a w * - l i m i t of " a a" — — o {9 } there i s a > a' such that 9 £ V For convenience, w r i t e 9 a n — n a n n n f o r 9 . Since {9 } i s a net contained i n the w*-compact set of a n r 1 n I means, l e t ty^ be any w * - c l u s t e r p o i n t of • Let f £ L , f ^ 0 , \ and. p^ . be f i x e d . I f £ > 0 i s given, there i s such that .. l / n < e/3||f|| and f o r n > n , ||L 9 - 9 |l < e/3||f.||. Since i s a o " " — o " p. n n" " " o J w * - c l u s t e r point of {9 } there i s an n n > n such that n l — o - 9 ) A . f I <'e/3 and | (9 - * ) f | < e/3 . - Thus o n x p..' n x (L * - * Q ) f | < |L (*Q - 9 ) f | + |(L 9 - 9 ) f | j o * j 1 * j 1 1 + K * n i - v f l ' \0> - 9 H f 1 + 1L 9 - 9 1 1 f I 1 o n.. p. P. n n T n n 11 11 1 1 3 J 1 1 <• e/3 + e/3||f || • ||f || + E/3 = £ Therefore, f o r each p. , L ij> = and hence by hypothesis, 3 p j 0 CO e J T £ ( S ) . A l s o , s i n c e ^ i s a w*-cluster p o i n t of ' {d> } , C V, , To L o n n=k ^— k we have <j/ z V, O J T U S ) C S (<j> ,1/k) O JL U S ) . This shows O K. L o' L | | i ^ o - § q \ \ < 1/k f o r each, k and thus <J> =, <f>o . Summarizing, we have shown that the sequence ^* n^ has a unique w*-cluster p o i n t <)> • We--now show l i m 4>n(f) = $ Q(*} f° r e v e r y f £ L . I f not, there would be a f e L and a subsequence n_^  such that I (d> - d> ) f I > £ f o r some e > 0 . But {<f> } , as net i n the ' n. o o' — n. i . i w*-compact set of means, has a w*-cluster p o i n t y . But y , being a w-'-cluster p o i n t of <j> , has to be a w'"-cluster p o i n t of {<)>}• By i uniqueness y = <j>Q , which cannot be, s i n c e y £ {\p : | (\p - ( j ) 0 ) f | 2l This completes the proof. The f o l l o w i n g i s the main thoerem i n t h i s chapter: 11.3. THEOREM.. Let S be a d i s c r e t e amenable semigroup which i s a i separable t o p o l o g i c a l subsemigroup of a t o p o l o g i c a l group G * Let W be a symmetric neighbourhood of the i d e n t i t y e i n G and V = {(x,y) £ G x G : y £ Wx} such that f o r every n , V n* does not!; t o t a l l y cover S . Then J^Jl(S) i s i n f i n i t e dimensional. PROOF. Let (p } n be the countable dense subset of S and n n=l l e t cj> £ LUC(S)* be such that L <{> = <!> f o r every n . Let s £ S be P n a r b i t r a r y , f £ LUC(S) . Then s i n c e (L <fr - .40(f) = K H - f) s s = <K«. f - 4 f + S , f - f ) i t f o l l o w s that | (L 9 - 9) ( f ) | <_ ||9||||£ f - A f || 0 as p s . Hence n {* £LUC(S>* : L 9 = 9 ', n = 1, 2 , •...} = J £(S) . p u r n Suppose J £(S) i s n dimensional f o r some n < 00 u Let 9 be a f i x e d two sided i n v a r i a n t mean on m(S) . Since the r e s t r i c t i o n of 9 to LUC(S) i s a LIM on LUC(S) , by 11 .2 there i s a sequence {9 } of f i n i t e means on LUC(S) such that 9 ( f ) = l i m 9, ( f ) fo r every f e LUC(S) . Define now the f o l l o w i n g bounded sequence of uniform l y continuous f u n c t i o n on the r e a l l i n e : For n = 1, 2 , . . . f (x) = n 1 - 2| x - (n - 1/2) I i f n -1 <_ x <_ n 0 otherwise g n ( x ) = 1 - 2| x - n+11 i f T L - ^ < _ X < _ T L - J 0 otherwise Let f ( x ) = 5 ! f (x) . Then f i s w e l l defined s i n c e f o r i ^ j , f. n=l 00 and f . have d i s j o i n t c a r r i e r s . S i m i l a r l y , g(x) = £ g (x) i s w e l l 2 n=l n defined. The graphs of f and g are shown below. 1-rr 0 1/2 3 /2 •5/2 4 4 And i f {a } i s a bounded sequence of r e a l s then )a f and /a g are n u n n u n n n n bounded uniform l y continuous f u n c t i o n s on the r e a l l i n e . Let W be the symmetric neighbourhood of.the i d e n t i t y e i n G and V = {(x,y) : y e Wx} such that f o r every n V n* does not t o t a l l y cover S . By lemma 1 0 . 2 , V* n does not t o t a l l y cover S f o r every n . Let F be the unbounded l e f t u n i f o r m l y continuous f u n c t i o n obtained i n lemma 1 0 . 4 . Then f o r each s e S , [ ( f + g )oF](s) = 1 and so < K(f + g)°F) = 1 . Hence cj)(f°F) > 0 or <|>(g°F) > 0 . Without l o s s of g e n e r a l i t y assume <j>(f°F) > 0 . For the sequence <j> of f i n i t e means which converges w* to the LIM <j> i n K. LUC(S)* d e f i n e the f o l l o w i n g l i n e a r f u n c t i o n a l s 9' ,9' on the space m of bounded r e a l sequences ^ a n ^ DY 9 k ( ( a n } ) = 9 k ( I a n V F ) n 9 ' ( ( a n } ) = 9 ( I a n f n o F ) . n m Let 9 (f) = I 9, ( t . ) f ( t . ) . Then 1=1 i = i n m I 9 k ( t . ) ( I V F ) ( t . ) 1 n n 1 = . K ( f n ° F ) = l 9 k ( U n } ) , . n 45 where i s the sequence which i s 1 at the n-th place' and 0 everywhere e l s e . This shows '<j>' c QlJl-,] , where JL i s the space of l£ X X | ! a b s o l u t e l y convergent sequences and Q i s the n a t u r a l mapping of i n t o i t s second conjugate H ** . Since Jl i s weakly s e q u e n t i a l l y complete [1, p. 33, c o r o l l a r y 3 ] , QfJl-^] i s w* - s e q u e n t i a l l y complete i n m* . | Thus f o r any sequence { a n ^ •> \ n u n n n = l i m 4>. ( Ja f °F) v T k L n n = l i m cf>/ ({a }) k k n Thus fib' e Jl and i t f o l l o w s that <j>'({l}) = £<j>'({ln}). = I^ 'C^F) • This n n means cj>(fn°F) > 0 f o r at l e a s t one n si n c e . 0 < <Klfn°F) = <*>'({!}) = I+(fn»F) . n n Since {s e S : f <>F(s) > 0} C {s £ S : F ( s ) C [0,n]} C uV ( n + 2 )[ P .] , i = l i t f o l l o w s <j>(V* 2^ n + 2^ [p.. ]) > 0 f o r some p^ . By lemma 10.2 <Kv 2 ( n + 2 ) *[ P . ] ) > 0 . Now l e t ir : m(G) -»- m(S) be defined by irf (s) = f ( s ) v \ _ f o r f e m(G) , s e S . Then ir* : m(S)* -> m(G)* and i f i> i s a mean on m(S) then TT'VI|J i s a mean on m(G) . In p a r t i c u l a r , f o r our i n v a r i a n t mean d> , TT*<J> i s a mean on m(G) . We now show that ir*d>(r f ) = iT5'c<j>(f) 46 f o r a l l s e S and f e m(G) . F i r s t irr f = r irf s i n c e f o r t £ S , s s T r r g f ( t ) = ' r g f ( t ) = f ( t s ) = Trf(ts) = r i T f(t) . Hence Tr*9 ( r g f ) = 9 ( i r r s f ) = <j)(r Trf) = <j>(irf) = Tr*cj>(f) . Let U = v 2 ( n + 2 ^ > Since U does not t o t a l l y cover XS^, 2 l e t a 1 £ S u * [ p j . By lemma 10.3, U*[p ] H U * ^ ] = 0 . In ge n e r a l , i f U*[p • ] , U * [ a 1 ] , u * C a n _ 1 ] n a s been chosen to be p a i r w i s e d i s j o i n t sets l e t a e S -v. ( U 2 * { D ; ] ( J - - - U U 2 * [ a .]) . n J n-1 By lemma 10.3, U*[p ], U*[a n] are p a i r w i s e d i s j o i n t . Thus f o r any n, j i = 9(s) > <Ku*[P.]) + . . . + (Ku*[a ]) : — 3 n | = 9 ( W 2 ( n + 2 ) P . ^ S ) + ... + 9 ( W 2 ( n + 2 ) a H S ) 3 n = 9( 1TW 2 ( n + 2 )p.) + ... + 9 ( ^ W 2 ( n + 2 ) a ) J n _ , 2(n+2) ,,,,,2^+2) . = TT'«9(W p . ) + . . . + IT" cj>(W a ) 3 n = ( n + l ) T T * 9 ( W 2 ( N + 2 ) ) . I i This i m p l i e s T r * < f > ( W 2 ( N + 2 ) ) = 0 , which cannot be s i n c e 9 ( V 2 ( n + 2 ) * [ p . ] ) > (f im p l i e s IT*9(W ) > 0 . Hence J^JlCS) cannot be f i n i t e dimensional. 11.4. REMARK. (a) I t i s knbwn that i f S i s LA and L i s a l e f t i n v a r i a n t subspace of m(S) then each LIM on L can be extended to a LIM on m(S) Such an extension can be obtained by an extension theorem of 47 R. J . Silverman [16] or by a d i r e c t argument as given i n the proof of theorem 11.2. (b) If S i s LA and L i s a l e f t i n v a r i a n t subspace of m(S) l e t J T £(S) = (<j) e.L* : L <j> = <j> for each s e S} ,. I t i s shown i n [7, p. 114, remark 2] that J £(S) coincides with the l i n e a r subspace l-i spanned by the l e f t i n v a r i a n t means i n J £(S) . J-i In the following c o r o l l a r y C(S) denotes the space of bounded real-valued continuous function on S and J c£(S) = {<j> e C(S)* : L if = d> for each s e S} . 11.5. COROLLARY. If S i s the semigroup i n the theorem 11.3 then J^KS) i s i n f i n i t e dimensional. PROOF. Each LIM on LUC(S) can be extended to a LIM on C(S) . For, by remark 11.4(a) each LIM <j> on LUC(S) can be extended to LIM <j>" on m(S) and the required extension i s obtained by r e s t r i c t i n g d>" to C(S) . Suppose J^lCS) has dimension n for some n < °° . Let {d>^, b e n + 1 l i n e a r l y independent set of LIM on LUC(S) , and l e t {<j>^ , be the respective extensions to LIM on C(S) . n+1 1 Then £ a. <f>! = 0 for some a. 4 0 . But for every f e LUC(S) , , i= l ! I n+1 n+1.. . | T a.d>!(f) = T a .d)(f) = 0 implies a. = 0 for each i , which cannot be. i= l i = l j Hence J £(S) i s i n f i n i t e dimensional. 48 12. EXAMPLES. 12.1. Let S = {0,1,2,...} w i t h o r d i n a r y m u l t i p l i c a t i o n and the d i s c r e t e topology. This topology i s generated by the met r i c d defined by d(x,y) = 1 i f f x ^ y and 0 otherwise. I f LUC(S) i s the bounded r e a l - v a l u e d u n i f o r m l y continuous f u n c t i o n s on S then LUC(S) = m(S) . By [7, p. 34, theorem 3.1] J£(S) has dimension 1 . C l e a r l y , S i s separable and has property (B) . This example shows theorem 11.3 cannot be extended to every t o p o l o g i c a l semigroup. The f o l l o w i n g lemma i s e s s e n t i a l l y known: 12.2. LEMMA. I f S^ i s a LA subsemigroup of a t o p o l o g i c a l group and S i s a dense subsemigroup of S^ then there i s a p o s i t i v e l i n e a r isometry from J U ( S ) onto J^CS.^ . PROOF. • Let TT : L U C ^ ) -> LUC(S) be defined by Trf(s) = f ( s ) \ f o r f e LUC(S 1) and s e S .. I f f e LUC(S) then f has a unique extension ' f i n LUC(S^) by [9, p..195, theorem 26]. Moreover, t h i s extension preserves norm si n c e i f s e S n and s -> s s e S , then r 1 a a ' f o r every e > 0 there i s an such that | f ' ( s ) - f ( s a ) | < e i f a > a.. . Thus | f ' ( s ) | < l f ' ( s ) - f ( s )| + | f ( s )| < e + l l f l l and so I  f '111. I l f l l • 0 n t h e other hand, i t i s c l e a r that || f || f.||f'|| . Hence : t TT i s a map which sends the u n i t , b a l l i n LUC(S.,) onto the u n i t b a l l i n ! • i LUC(S) . I t f o l l o w s from t h i s that TT* : LUC(S)'* LUC(S n)* i s a l i n e a r | 1 i isometry. That i t i s p o s i t i v e i s easy to see. 49 Since S^ . i s LA , J^US^) i s non-empty. I f <j>' e'J £(S ) then d e f i n e <f> e LUC(S)* by d>(f) = <j>'(f') , where f the unique extension of f to S n . Then <j> £ J £(S) s i n c e i f s £ S i. u i and f £ LUC(S) then * ( A f ) = V ( ( A f ) ' ) '='* 1 ( A f') = * ' ( f ' ) = <Kf) : s s s To see the second e q u a l i t y above, we note that i f s £ S , t £ S^ and s a e S i s such that s a -> t then (£gf)'(t) = (£gf)'(lim s^) = l i m ( A f ) ' ( s J = l i m £ f ( s ) = l i m f ( s s ) = f ( l i m ss ) = f ' ( l i m ss ) „ . s a a s aJ a a . .a a - a a = f ' ( s t ) = £ f ' ( t ) . Thus J £(S) 4 0 and Tr*d> = <f>' . i s Then Now l e t f £ LUC(S n) , s £ S n , s £ S such that s •+ s. 1 ' 1 ' a a TT£ f - £ irf|| = sup |TT£ f ( t ) - £ i T f ( t ) | S S a teS S s a sup |A f ( t ) - f ( s t ) | teS S sup |A f ( t ) - £ f ( t ) | teS S S a = £ f - £ f + 0 as s s " s s a a Hence i f d> £ J"u£(S) then TT*<K£ f ) = <J>(TT£ f ) = d)(lim £ uf) S S a s a .= l i m <K£ -rrf) = l i m <J> (irf) a s a a ' = TT*<|> ( f ) . Consequently ir*-[J £(S)J = J £(S n) u u 1 50 12.3. THEOREM. Let S be a separable subsemigroup of a l o c a l l y compact group G such that , the c l o s u r e of S i n G , i s amenable and non-compact. Then J^£(S) i s i n f i n i t e dimensional. PROOF. Let W be a compact symmetric neighbourhood of the i d e n t i t y e i n G . I f V = {(x,y) e G x G : y £ Wx} then f o r every n , V n* does not t o t a l l y cover , s i n c e otherwise, there i s a p o s i t i v e i n t e g e r n and a f i n i t e subset {a^, a ^ } C such t h a t k k S-^CL U V *[a.] = LJ (W a . O S 1 ) , which i s a compact s e t . But t h i s i = l i = l 1 cannot be s i n c e i s non-compact. By theorem 11.3, J^US^) i s i n f i n i t e * 1 dimensional and hence by lemma 12.2, J £ (S) i s i n f i n i t e dimensional. u 12.4. EXAMPLES. Using theorem 12.3 we can see J"u£(S) i s i n f i n i t e dimensional i f S i s the f o l l o w i n g t o p o l o g i c a l semigroups: ( i ) S = [0,°°) w i t h o r d i n a r y a d d i t i o n and the induced topology from the usual topology on the r e a l l i n e R . ( i i ) S = [1,°°) w i t h o r d i n a r y m u l t i p l i c a t i o n and the induced topology from R . ^ ( i i i ) S = (0,1] w i t h o r d i n a r y m u l t i p l i c a t i o n and the induced topology from R . ( i v ) S i s any p o s i t i v e cone i n a Euclidean v e c t o r space E w i t h the usual a d d i t i o n of v e c t o r s and the induced topology from E . (S i s a p o s i t i v e cone i f S + S S and AS S f o r any non-negative s c a l a r A .) 1 51 (v) S = P O [1,°°) , where P i s the set of negative i r r a t i o n a l s , w i t h o r d i n a r y m u l t i p l i c a t i o n and the induced topology from R . \\ (v i ) Let S be the set of a l l r e a l 3x3 diagonal matrices whose determinant i s greater than or equal to 1. Then S w i t h the usua l m u l t i p l i c a t i o n of two matrices i s a commutative (thus amenable) subsemigroup of the group ^ ( R ) , the f u l l l i n e a r group. I t i s w e l l known that G^(R) can be considered as ais u b s e t of 9 R , and G^(R) becomes a t o p o l o g i c a l group w i t h the induced 9 I topology from R . The semigroup S i s a separable, c l o s e d •• non-compact subset i n ^^(R) • That S i s closed because the determinant f u n c t i o n D i s continuous and S = D . !j And si n c e f o r each A = (a..) e S , the norm of A i s i j 3 2 1/2 ||A|| = ( £ | a. . | ) , S i s an unbounded set and hence cannot i = l 1 1 be a compact s e t . ( v i i ) Let A be a r e a l nxn ma t r i x of the form XI 0 P XI where X i s any s c a l a r greater than or equal to 1 , I and I p q are i d e n t i t y matrices of f i x e d orders p and q r e s p e c t i v e l y , p + q = n , and B i s any qxp mat r i x . Let S be the set of a l l such matrices A . Then S i s a commutative (thus amenable) separable closed non-compact subsemigroup of ^ ( R ) > t n e f u l l l i n e a r group. That S i s closed because the determinant f u n c t i o n D i s continuous and S = D "*"([1,°°)) . A l s o , s i n c e f o r each m a t r i x A = (a ) , i t s norm i s ||A|| = ( £ | a . . | 2 ) 1 / 2 we see i , j = l , 1 J ji that S i s unbounded and hence cannot be compact. / 53 BIBLIOGRAPHY M. M. Day, Amenable semigroups, 111. J . Math. 1 (1957), 509-544. , Normal l i n e a r spaces, 2nd e d i t i o n , Springer, B e r l i n , 1962. E. E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. _, Extremely amenable, semigroups I I , Math. Scand. 20 (1967), 93-113. , Functional a n a l y t i c properties of extremely amenable semi-groups, Trans. Amer. Math. Soc. 137 (1969), 53-75. , On amenable semigroups with a f i n i t e dimensional set of invari a n t means I, I I , 111. J . Math. 7 (1963), 32-48 and 49-58. , On the invar i a n t mean on to p o l o g i c a l semigroups and on topo l o g i c a l groups, P a c i f i c J . Math. 15 (1965), 107-140. P. R. Halmos, Introduction to H i l b e r t space and the theory of s p e c t r a l m u l t i p l i c i t y , Chelsea, New York, 1951. J. L. Ke l l e y , General topology, Van Nostrand, New York, 1955. J. P. King, Almost summable sequences, Proc. Amer. Soc. 17 (1966), 1219-1225. G. G. Lorentz, A contribution of theory of divergent sequences, Acta Math. 80 (1948), 167-190. T. M i t c h e l l , Constant functions and l e f t i n v a r i a n t means on semigroups Trans. Amer. Math. Soc. 119 (1965), 224-261. , Fixed points and m u l t i p l i c a t i v e l e f t i n v a r i a n t means, Trans. Amer. Math. Soc. 122 (1966), 195-202. P. Schaefer, Almost convergent and almost summable sequences, Proc. Amer. Math. Soc. 20 (1969), 51-54. J. Schur, Uber l i n e a r e Transformation i n der Theorie der unendlichen. R^ihen, J . fur die reine und angewandte Math. 151 (1921), 79-111. [ R. J. Silverman, Invariant l i n e a r function, Trans. Amer. Soc. 81 ; (1956), 411-424. ti 0. T o e p l i t z , Uber allegemeine l i n e a r e Mittelbildungen, Prace Math.-Fiz 22 (1911), 113-119. j 54 [18] A. W e i l , Sur l e s espaces a S t r u c t u r e uniforme et sur l a t o p o l o g i e generale, Hermann, P a r i s , 1938. . 

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