Weak amenability of module extensions of Banach algebras
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Bibliographic record
Abstract
We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -weakly amenable, for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 0 comma 1 comma 2 comma midline-horizontal-ellipsis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo> ⋯ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n = 0,1,2,\cdots</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.
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