On extending continuous functions on dense subsemigroups (Q1814145)

Let \(S\) be a dense subsemigroup of a semitopological semigroup \(T\). A problem that has arisen in several contexts is that of extending a function \(f\in C(S)\) with a certain property (say, \(f\in P\)) to a function in \(C(T)\) with the same property. Examples of the kind of property the authors have in mind are strong and weak almost periodicity. Here the authors address this problem in two parts. (a) When is \(C(S)\cap P\subset C(T)\mid_ S\)? (b) When is \((C(S)\cap P)\cap C(T)\mid_ S\subset(C(T)\cap P)\mid_ S\)? The authors' main result states that the containment in (b) is an equality if and only if \[ \{g\in C(T)\mid g|_ S \hbox{ has property} P \hbox{ on} S\} \] is an \(m\)-admissible subalgebra of \(C(S)\). From it they derive a number of corollaries, some new and some already known.