Multipliers for the space of almost-convergent functions on a semigroup

Abstract

Let S be a countably infinite left amenable cancellative semigroup, FL(S) the space of left almost-convergent functions on S. The purpose of this paper is to show that the following two statements concerning a bounded real function f on S are equivalent: (i) f -FL(S) c FL(S); (ii) there is a constant cx such that for each ->0 there exists a set A (S satisfying (a) p(XA)=0 for each left invariant mean 9 on S and (b) If(x)cx 0 and qp(lf)=qp(f) for seS and fenm(S), where l,fe m(S) is defined by (lsf)(t)=f(st), t E S. The set of left invariant means on S is denoted by ML(S). If ML(S) is nonempty, then S is said to be left amenable [2]. A bounded real function f on a left amenable semigroup is called left almost-convergent if (f) equals a fixed constant d(f) as 79 runs through ML(S) [2]. The set of all left almost-convergent functions, denoted by FL(S), is a vector subspace of m(S) and it contains constant functions. But, in general, it is not closed under multiplication. The purpose of this paper is to study this aspect of FL(S) and our main result is the following. THEOREM. Let S be a countable left-cancellative left amenable semigroup withoutfinite left ideals. Then the following two statements concerning a function f E m(S) are equivalent: (i) f is a multiplier of FL(S), i.e., f * FL(S) cFL(S); (ii) f is S-convergent to a constant x, i.e.,for a given e>0 there exists a set A c S such that (a) 79(XA)=0 for each 9p E ML(S), and (b) If(x)-xl <e if x E S\A. 2. From now on S will always denote a left-cancellative left amenable semigroup without finite left ideals. Presented to the Society, January 17, 1972; received by the editors May 31, 1972. AMS (MOS) subject class/ifcations (1970). Primary 43A07; Secondary 46N05.