Amenable Banach algebra

In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form $$a\mapsto a.x-x.a$$ for some $$x$$ in the dual module).

An equivalent characterization is that A is amenable if and only if it has a virtual diagonal.

Examples

 * If A is a group algebra $$L^1(G)$$ for some locally compact group G then A is amenable if and only if G is amenable.
 * If A is a C*-algebra then A is amenable if and only if it is nuclear.
 * If A is a uniform algebra on a compact Hausdorff space then A is amenable if and only if it is trivial (i.e. the algebra C(X) of all continuous complex functions on X).
 * If A is amenable and there is a continuous algebra homomorphism $$\theta$$ from A to another Banach algebra, then the closure of $$\overline{\theta(A)}$$ is amenable.