# Reza Ghamarshoushtari

## Amenability Properties Of Banach Algebra-Valued Continuous Functions

Date Friday, February 28, 2014

In this thesis we discuss amenability properties of the Banach algebra-valued continuous functions on a compact Hausdorff space $X$. Let $A$ be a Banach algebra. The space of $A$-valued continuous functions on $X$, denoted by $C(X,A)$, form a new Banach algebra. We show that $C(X,A)$ has a bounded approximate diagonal (i.e. it is amenable) if and only if $A$ has a bounded approximate diagonal. We also show that if $A$ has a compactly central approximate diagonal then $C(X,A)$ has a compact approximate diagonal. We note that, unlike $C(X)$, in general $C(X,A)$ is not a $C^*$- algebra, and is no longer commutative if $A$ is not so. Our method is inspired by a work of M. Abtahi and Y. Zhang. In addition to the above investigation, we directly construct a bounded approximate diagonal for $C_0(X)$, the Banach algebra of the closure of compactly supported continuous functions on a locally compact Hausdorff space $X$.