# Amenability-Like properties of C(X,A)

## TL;DR

This paper explores how amenability-like properties transfer between a Banach algebra A and the algebra of continuous functions C(X, A) over a compact Hausdorff space, focusing on induced homomorphisms and their amenability implications.

## Contribution

It introduces induced homomorphisms between A and C(X, A) and analyzes conditions under which (weak) amenability properties are preserved or transferred.

## Key findings

- Conditions for $ au$-(weak) amenability transfer from $C(X, A)$ to $A$
- Conditions for $ ilde{ au}$-(weak) amenability transfer from $A$ to $C(X, A)$
- Characterization of when induced homomorphisms preserve amenability properties

## Abstract

Let $A$ be a Banach algebra and $X$ be a compact Hausdorff space. Given homomorphisms $ \sigma \in Hom(A)$ and $\tau \in Hom(C(X, A))$, we introduce induced homomorphisms $\tilde{\sigma}\in Hom(C(X, A)) $ and $\tilde{\tau}\in Hom(A)$, respectively. We study when $\tau$-(weak) amenability of $C(X, A)$ implies $\tilde{\tau}$-(weak) amenability of $A$. We also investigate where $ \sigma$-weak amenability of $A$ yields $\tilde{\sigma}$-weak amenability of $C(X, A)$.