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