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

Weakly almost periodic functions and almost convergent functions on a group


Author: Ching Chou
Journal: Trans. Amer. Math. Soc. 206 (1975), 175-200
MSC: Primary 43A60; Secondary 43A07
MathSciNet review: 0394062
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Abstract: Let $ G$ be a locally compact group, $ UC(G)$ the space of bounded uniformly continuous complex functions on $ G,{C_0}(G)$ the subspace of $ UC(G)$ consisting of functions vanishing at infinity. Let $ W(G)$ be the space of weakly almost periodic functions on $ G$ and $ {W_0}(G)$ the space of functions in $ W(G)$ such that their absolute values have zero invariant mean. If $ G$ is amenable let $ F(G)$ be the space of almost convergent functions in $ UC(G)$ and $ {F_0}(G)$ the space of functions in $ F(G)$ such that their absolute values are almost convergent to zero. The inclusive relations among the above-mentioned spaces are studied. It is shown that if $ G$ is noncompact and satisfies certain conditions, e.g. $ G$ is nilpotent, then each of the quotient Banach spaces $ UC(G)/W(G),{W_0}(G)/{C_0}(G),{F_0}(G)/{W_0}(G)$ contains a linear isometric copy of $ {l^\infty }$. On the other hand, an example of a noncompact group $ G$ is given which satisfies the condition that $ {C_0}(G) = {W_0}(G)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0394062-8
PII: S 0002-9947(1975)0394062-8
Article copyright: © Copyright 1975 American Mathematical Society