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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Algebras of functions on semitopological left-groups


Authors: John F. Berglund and Paul Milnes
Journal: Trans. Amer. Math. Soc. 222 (1976), 157-178
MSC: Primary 43A60; Secondary 22A20
MathSciNet review: 0422998
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Abstract: We consider various algebras of functions on a semitopological left-group $ S = X \times G$, the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products $ S \times T$ of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0422998-9
PII: S 0002-9947(1976)0422998-9
Keywords: Semitopological semigroup, left-group, weakly almost periodic function, compactification, analytic function
Article copyright: © Copyright 1976 American Mathematical Society