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 ${\displaystyle a\mapsto a.x-x.a}$ for some ${\displaystyle x}$ 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 ${\displaystyle L^{1}(G)}$ 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 ${\displaystyle \theta }$ from A to another Banach algebra, then the closure of ${\displaystyle {\overline {\theta (A)}}}$ is amenable.