A bounded difference property for classes of Banach-valued functions

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.