## 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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