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) $f \cdot FL(S) \subset FL(S)$; (ii) there is a constant $\alpha$ such that for each $\varepsilon > 0$ there exists a set $A \subset S$ satisfying (a) $\varphi ({X_A}) = 0$ for each left invariant mean $\varphi$ on $S$ and (b) $|f(x) - \alpha | < \varepsilon$ if $x \in S\backslash A$.