Nonseparability of quotient spaces of function algebras on topological semigroups

Dzinotyiweyi, Heneri A. M.

doi:10.1090/S0002-9947-1982-0656487-6

Nonseparability of quotient spaces of function algebras on topological semigroups
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Trans. Amer. Math. Soc. 272 (1982), 223-235 Request permission

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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