Extensions of left uniformly continuous functions on a topological semigroup

Abstract:

For any topological semigroup S with separately continuous operation, let $C(S)$ denote the set of all bounded continuous real valued functions on S with the supremum norm and let ${\text {LUC}}(S)$ denote the set of all f in $C(S)$ such that whenever $\{ s(\gamma )\}$ is a net in S which converges to some s in S, then $\sup \{ |f(s(\gamma )t) - f(st)|:t \in S\}$ converges to 0. In this paper we prove that if S is an abelian subsemigroup of a compact topological group and $f \in {\text {LUC}}(S)$, then there is an $F \in {\text {LUC}}(G)$ where $F(s) = f(s)$ for all $s \in S$. We also show whenever there is an extension of the type indicated above, there is a norm preserving extension.