Girish Mahajan (Editor)

Amenable Banach algebra

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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 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 θ from A to another Banach algebra, then the closure of θ ( A ) ¯ is amenable.
  • References

    Amenable Banach algebra Wikipedia