Examples show that functions
of various kinds on subsemigroups of topological semigroups do not always extend to
functions of the same kind on the containing semigroup. We show here that, if S is a
dense subsemigroup with identity of a topological group G, then there is
a fairly large subspace of C(S) whose members always extend at least to
members of C(G). As important applications of this theorem, we prove in this
setting that the weakly almost periodic functions on S extend to functions
weakly almost periodic on G and, in a somewhat more restricted setting, that
the weakly almost periodic functions on S are uniformly continuous. These
results broaden the scope of answers we gave recently to some questions
posed by R. Burckel. We also prove variants of some recent results of A.
T. Lau and of S. J. Wiley, results concerning the extension of functions
and the existence of invariant means on dense subsemigroups of topological
groups.
