Amenable Banach algebra

A Banach algebra, "A", is amenable if all bounded derivations from "A" into dual Banach "A"-bimodules are inner (that is of the form $amapsto\; a.x-x.a$ 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 $heta$ from "A" to another Banach algebra, then the closure of $heta(A)$ is amenable.

References

* F.F. Bonsall, J. Duncan, "Complete normed algebras"

* H.G. Dales, "Banach algebras and automatic continuity"

* B.E. Johnson, "Cohomology in Banach algebras", Memoirs of the AMS 127 (1972).