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

A bounded difference property for classes of Banach-valued functions


Author: Wilbur P. Veith
Journal: Trans. Amer. Math. Soc. 190 (1974), 49-56
MSC: Primary 43A60
MathSciNet review: 0387969
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Abstract: Let $ A(G,E)$ denote the set of functions f from a Hausdorff topological group G to a Banach space E such that the range of f is relatively compact in E and $ \phi \circ f$ is in $ A(G,C)$ for each $ \phi $ in the dual of E, where $ A(G,C)$ is a translation-invariant $ {C^\ast}$ algebra of bounded, continuous, complex-valued functions on G with respect to the supremum norm and complex conjugation. $ A(G,E)$ has the bounded difference property if whenever $ F:G \to E$ is a bounded function such that $ {\Delta _t}F(x) = F(tx) - F(x)$ is in $ A(G,E)$ for each t in G, then F is also an element of $ A(G,E)$. A condition on $ A(G,C)$ and a condition on E are given under which $ A(G,E)$ has the bounded difference property. The condition on $ A(G,C)$ is satisfied by both the class of almost periodic functions and the class of almost automorphic functions.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0387969-8
PII: S 0002-9947(1974)0387969-8
Keywords: Vector-valued almost periodic functions, vector-valued almost automorphic functions, bounded difference property, integration of almost periodic functions, Banach spaces with no copy of $ {c_0}$
Article copyright: © Copyright 1974 American Mathematical Society