Vol. 66, No. 1, 1976

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Counterexample in the theory of continuous functions on topological groups

Paul Milnes and John Sydney Pym

Vol. 66 (1976), No. 1, 205–209

If G is a topological group and τ is the topology on C(G) of pointwise convergence on G, a function space (G) of almost periodic type is defined by (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 (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 (G) (which are known always to extend to functions in C(G)) need not extend to functions in (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.)

Mathematical Subject Classification 2000
Primary: 43A15
Secondary: 22A99
Received: 12 March 1976
Published: 1 September 1976
Paul Milnes
John Sydney Pym