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Extensions of left uniformly continuous functions on a topological semigroup

Author: Samuel J. Wiley
Journal: Proc. Amer. Math. Soc. 33 (1972), 572-575
MSC: Primary 46E10; Secondary 22A20
MathSciNet review: 0296672
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Abstract: For any topological semigroup S with separately continuous operation, let $ C(S)$ denote the set of all bounded continuous real valued functions on S with the supremum norm and let $ {\text{LUC}}(S)$ denote the set of all f in $ C(S)$ such that whenever $ \{ s(\gamma )\} $ is a net in S which converges to some s in S, then $ \sup \{ \vert f(s(\gamma )t) - f(st)\vert:t \in S\} $ converges to 0. In this paper we prove that if S is an abelian subsemigroup of a compact topological group and $ f \in {\text{LUC}}(S)$, then there is an $ F \in {\text{LUC}}(G)$ where $ F(s) = f(s)$ for all $ s \in S$. We also show whenever there is an extension of the type indicated above, there is a norm preserving extension.

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Keywords: Extensions, topological semigroups, compact groups, norm preserving extensions, uniformly continuous functions
Article copyright: © Copyright 1972 American Mathematical Society

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