Algebras of functions on semitopological leftgroups
Authors:
John F. Berglund and Paul Milnes
Journal:
Trans. Amer. Math. Soc. 222 (1976), 157178
MSC:
Primary 43A60; Secondary 22A20
MathSciNet review:
0422998
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Abstract: We consider various algebras of functions on a semitopological leftgroup , the direct product of a leftzero 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 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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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197604229989
PII:
S 00029947(1976)04229989
Keywords:
Semitopological semigroup,
leftgroup,
weakly almost periodic function,
compactification,
analytic function
Article copyright:
© Copyright 1976 American Mathematical Society
