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 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 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).