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A [[Banach algebra]], ''A'', is '''amenable''' if all [[bounded linear map|bounded]] [[Differential algebra|derivation]]s from ''A'' into [[dual module|dual]] [[Banach module|Banach ''A''-bimodules]] are [[inner derivation|inner]] (that is of the form <math>a\mapsto a.x-x.a</math> for some <math>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>L^1(G)</math> for some [[locally compact group]] ''G'' then ''A'' is amenable if and only if ''G'' is [[amenable group|amenable]]. | |||
* If ''A'' is a [[C*-algebra]] then ''A'' is amenable if and only if it is [[nuclear C*-algebra|nuclear]]. | |||
* If ''A'' is a [[uniform algebra]] on a [[compact space|compact]] [[Hausdorff space]] then ''A'' is amenable if and only if it is trivial (i.e. the algebra ''C(X)'' of all [[continuous function|continuous]] [[complex number|complex]] [[function (mathematics)|functions]] on ''X''). | |||
* If ''A'' is amenable and there is a continuous algebra homomorphism <math>\theta</math> from ''A'' to another Banach algebra, then the closure of <math>\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). | |||
{{Mathanalysis-stub}} | |||
[[Category:Banach algebras]] | |||
Revision as of 16:43, 21 January 2014
A Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form for some 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 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 from A to another Banach algebra, then the closure of 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).