Deleeuw and Glicksberg have
shown that every weakly almost periodic function on a subgroup (respectively, a
dense subgroup) of a locally compact abelian topological group (respectively, an
abelian topological group) extends to the whole group as a weakly almost periodic
function. By way of a theorem about extending strongly almost periodic functions on
subsemigroups of semitopological semigroups, we show that every almost periodic
function on a subgroup (respectively, a dense subgroup) of a locally compact abelian
topological group (respectively a topological group) extends to the whole group as an
almost periodic function. Furthermore, for the algebra of weakly almost periodic
functions on a dense subsemigroup of a semitopological semigroup to consist of
restrictions of weakly almost periodic functions on the larger semigroup, we need
only require that each weakly almost periodic function on the subsemigroup
extend continuously. Analogous statements hold for almost periodic and
strongly almost periodic functions on dense subsemigroups of semitopological
semigroups.
