Uniformly continuous and weakly almost periodic functions on some topological semigroups

Abstract:

Let $\operatorname {LUC} (S)$ be the space of bounded left uniformly continuous functions and $\operatorname {WAP} (S)$ the space of bounded weakly almost periodic functions on a topological semigroup $S$. For a large class of topological semigroups called stips that are not relatively neocompact, we show that the quotient space $\operatorname {LUC} (S)/\operatorname {WAP} (S)$ contains an isometric linear copy of ${l^\infty }$. The result is particularly interesting on noting that the underlying topological semigroup is not assumed to support an absolutely continuous measure—a theme studied by the author in a previous paper.