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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.

Citations:

Zbl 1086.43002
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References:

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