Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1972-0296672-8
MathSciNet review: 0296672
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46E10, 22A20

Retrieve articles in all journals with MSC: 46E10, 22A20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0296672-8
Keywords: Extensions, topological semigroups, compact groups, norm preserving extensions, uniformly continuous functions
Article copyright: © Copyright 1972 American Mathematical Society