Almost convergent and weakly almost periodic functions on a semigroup

Abstract:

Let $S$ be a topological semigroup, ${\text {US}}(S)$ the set of all bounded uniformly continuous functions on $S,{\text {WAP(}}S)$ the set of all (bounded) weakly almost periodic functions on $S,{E_0}(S): = \{ f \in {\text {UC(}}S):m(|f|) = 0$ for each left and right invariant mean $m$ on ${\text {UC(}}S)\}$ and ${W_0}(S): = \{ f \in {\text {WAP}}(S):\:m(|f|) = 0$ for each left and right invariant mean $m$ on ${\text {WAP(}}S)\}$. Among other results, for a large class of noncompact locally compact topological semigroups $S$, we show that the quotient space ${E_0}(S)/{W_0}(S)$ contains a linear isometric copy of ${l^\infty }$ and so is nonseparable.