Approximate amenability of Banach category algebras with application to semigroup algebras

Let C be a small category. Then we consider l(1)(C) as the l(1) algebra over the morphisms of C, with convolution product and also consider l(1)((C) over cap) as the l(1) algebra over the objects of C, with pointwise multiplication. The main purpose of this paper is to show that approximate amenability of l(1)(C) implies of l(1)((C) over cap) 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 l(1)(S), where S is a Brandt semigroup, which corrects a result of Lashkarizadeh Bami and Samea (Semigroup Forum 71:312-322, 2005).