Vol. 58, No. 2, 1975

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Extension of continuous functions on topological semigroups

Paul Milnes

Vol. 58 (1975), No. 2, 553–562

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.

Mathematical Subject Classification 2000
Primary: 22A20
Secondary: 43A60
Received: 23 March 1974
Published: 1 June 1975
Paul Milnes