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Amenability for discrete convolution semigroup algebras. (English) Zbl 0748.46027

Summary: For any semigroup \(S\) we show that if the convolution algebra \(\ell^ 1(S,\omega)\) is amenable for some weight \(\omega\) then \(S\) is a regular semigroup with a finite number of idempotents: in particular, for the case of an inverse semigroup \(S\), we have \(\ell^ 1(S)\) amenable if and only if \(S\) has a finite number of idempotents and every subgroup of \(S\) is amenable. Various known results on amenability are shown to be easy corollaries of our results.

MSC:

46H15 Representations of topological algebras
46B45 Banach sequence spaces
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46M05 Tensor products in functional analysis
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