Amenability-like properties of $C(X, A)$

Abstract

Let $A$ be a Banach algebra and $X$ be a compact Hausdorff space.
For given homomorphisms $ \sigma \in Hom(A)$ and $\tau \in Hom(C(X,
A))$, we introduce homomorphisms $\tilde{\sigma}\in Hom(C(X, A)) $
and $\tilde{\tau}_x \in Hom(A)$, where $x \in X$. We then study both
$\tilde{\sigma}$-(weak) amenability of $C(X, A)$, and
$\tilde{\tau}_x$-(weak) amenability of $A$.