If G is a topological group and τ is
the topology on C(G) of pointwise convergence on G, a function space ℳ(G) of almost
periodic type is defined by ℳ(G) = {f ∈ C(G)|{rsf|s ∈ G} is relatively τ-compact}.
Generalizing results of T. Mitchell, C. R. Rao, and P. Milnes, we show here that
ℳ(G) is just the left uniformly continuous subspace, LUC(G), of C(G) for groups
satisfying a completeness condition and give an example on the rational numbers
which shows that some completeness condition is necessary for this conclusion to
hold. The example also shows that, if G is a dense subgroup of a topological group
G′, functions in ℳ(G) (which are known always to extend to functions in
C(G′)) need not extend to functions in ℳ(G′); this result is at variance with
what happens in the case of the familiar almost periodic or weakly almost
periodic functions, where a functlon always extends to a function of the same
type.
(The conclusions of the theorems of this paper hold in more general settings than
have been described above.)
