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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).
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| An equivalent characterization is that ''A'' is amenable if and only if it has a [[virtual diagonal]].
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| ==Examples==
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| * 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]].
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| * If ''A'' is a [[C*-algebra]] then ''A'' is amenable if and only if it is [[nuclear C*-algebra|nuclear]].
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| * 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'').
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| * 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.
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| ==References==
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| * F.F. Bonsall, J. Duncan, "Complete normed algebras", Springer-Verlag (1973).
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| * H.G. Dales, "Banach algebras and automatic continuity", Oxford University Press (2001).
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| * B.E. Johnson, "Cohomology in Banach algebras", Memoirs of the AMS '''127''' (1972).
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| * J.-P. Pier, "Amenable Banach algebras", Longman Scientific and Technical (1988).
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| {{Mathanalysis-stub}}
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| [[Category:Banach algebras]]
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Latest revision as of 05:56, 25 June 2014
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