Z. Hu^{1}, M. Sangani Monfared^{2}, T. Traynor^{3} Studia Math. 193 (2009), 53-78
doi:10.4064/sm193-1-3 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.