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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Nonseparability of quotient spaces of function algebras on topological semigroups

Author: Heneri A. M. Dzinotyiweyi
Journal: Trans. Amer. Math. Soc. 272 (1982), 223-235
MSC: Primary 43A60; Secondary 22A20
MathSciNet review: 656487
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Abstract: Let $S$ be a topological semigroup, $C(S)$ the space of all bounded real-valued continuous functions on $S$. We define $WUC(S)$ the subspace of $C(S)$ consisting of all weakly uniformly continuous functions and $WAP(S)$ the space of all weakly almost periodic functions in $C(S)$. Among other results, for a large class of topological semigroups $S$, for which noncompact locally compact topological groups are a very special case, we prove that the quotient spaces $WUC(S)/WAP(S)$ and, for nondiscrete $S$, $C(S)/WUC(S)$ are nonseparable. (The actual setting of these results is more general.) For locally compact topological groups, parts of our results answer affirmatively certain questions raised earlier by Ching Chou and E. E. Granirer.

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Keywords: Topological semigroup, uniformly continuous functions, weakly almost periodic functions
Article copyright: © Copyright 1982 American Mathematical Society