# 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 $a\mapsto a.x-x.a$ 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 $\theta(A)$ is amenable.

## References

• F.F. Bonsall, J. Duncan, "Complete normed algebras", Springer-Verlag (1973).
• H.G. Dales, "Banach algebras and automatic continuity", Oxford University Press (2001).
• B.E. Johnson, "Cohomology in Banach algebras", Memoirs of the AMS 127 (1972).
• J.-P. Pier, "Amenable Banach algebras", Longman Scientific and Technical (1988).