## On character amenable Banach algebras

### Volume 193 / 2009

#### Abstract

We obtain characterizations of left character amenable Banach algebras in terms of the existence of left $\phi$-approximate diagonals and left $\phi$-virtual diagonals. We introduce the left character amenability constant and find this constant for some Banach algebras. For all locally compact groups $G$, we show that the Fourier–Stieltjes algebra $B(G)$ is $C$-character amenable with $C<2$ if and only if $G$ is compact. We prove that if $A$ is a character amenable, reflexive, commutative Banach algebra, then $A\cong \mathbb C^n$ for some $n\in \mathbb N$. We show that the left character amenability of the double dual of a Banach algebra $A$ implies the left character amenability of $A$, but the converse statement is not true in general. In fact, we give characterizations of character amenability of $L^1(G)^{**}$ and $A(G)^{**}$. We show that a natural uniform algebra on a compact space $X$ is character amenable if and only if $X$ is the Choquet boundary of the algebra. We also introduce and study character contractibility of Banach algebras.