Approximate amenability of Banach category algebras with application to semigroup algebras

Let C be a small category. Then we consider ℓ1(C) as the ℓ1 algebra over the morphisms of C, with convolution product and also consider \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell^{1}(\hat{C})$\end{document} as the ℓ1 algebra over the objects of C, with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of ℓ1(C) implies of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ell^{1}(\hat{C})$\end{document} 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 ℓ1(S), where S is a Brandt semigroup, which corrects a result of Lashkarizadeh Bami and Samea (Semigroup Forum 71:312–322, 2005).