## Algebras of functions on semitopological left-groups

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- by John F. Berglund and Paul Milnes
- Trans. Amer. Math. Soc.
**222**(1976), 157-178 - DOI: https://doi.org/10.1090/S0002-9947-1976-0422998-9
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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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## Bibliographic Information

- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**222**(1976), 157-178 - MSC: Primary 43A60; Secondary 22A20
- DOI: https://doi.org/10.1090/S0002-9947-1976-0422998-9
- MathSciNet review: 0422998