Uniformly continuous and weakly almost periodic functions on some topological semigroups

Abstract

Let LUC ⁡ ( S ) \operatorname {LUC} (S) be the space of bounded left uniformly continuous functions and WAP ⁡ ( S ) \operatorname {WAP} (S) the space of bounded weakly almost periodic functions on a topological semigroup S S . For a large class of topological semigroups called stips that are not relatively neocompact, we show that the quotient space LUC ⁡ ( S ) / WAP ⁡ ( S ) \operatorname {LUC} (S)/\operatorname {WAP} (S) contains an isometric linear copy of l ∞ {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.