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 ↦ a . x − x . a {\displaystyle a\mapsto a.x-x.a} for some x {\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 L 1 ( G ) {\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 θ {\displaystyle \theta } from A to another Banach algebra, then the closure of θ ( A ) ¯ {\displaystyle {\overline {\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). Volker Runde, "Amenable Banach Algebras. A Panorama", Springer Verlag (2020).