## Approximate amenability of Banach category algebras with application to semigroup algebras.(English)Zbl 1222.43006

Summary: Let $$C$$ be a small category. Then we consider $$\ell ^{1}(C)$$ as the $$\ell ^{1}$$ algebra over the morphisms of $$C$$, with convolution product and also consider $$\ell^{1}(\widehat{C})$$ as the $$\ell ^{1}$$ algebra over the objects of $$C$$, with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of $$\ell ^{1}(C)$$ implies of $$\ell^{1}(\widehat{C})$$ and clearly this implies that $$C$$ has only finitely many objects. Some applications are given, the main one is the characterization of approximate amenability for $$\ell ^{1}(S)$$, where $$S$$ is a Brandt semigroup, which corrects a result of M. Lashkarizadeh Bami and H. Samea [Semigroup Forum 71, 312–322 (2005; Zbl 1086.43002)].

### MSC:

 43A20 $$L^1$$-algebras on groups, semigroups, etc.

Zbl 1086.43002
Full Text:

### References:

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