## A bounded difference property for classes of Banach-valued functions

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- by Wilbur P. Veith PDF
- Trans. Amer. Math. Soc.
**190**(1974), 49-56 Request permission

## 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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## Additional Information

- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**190**(1974), 49-56 - MSC: Primary 43A60
- DOI: https://doi.org/10.1090/S0002-9947-1974-0387969-8
- MathSciNet review: 0387969