Banach valued algebras defined by a family of Banach algebras

Abstract

Let [Formula: see text] be a direct set, [Formula: see text] be a family of Banach algebras with bounded approximate identity (or unital) and [Formula: see text] be a set. We consider the Banach algebra [Formula: see text]. We show that this algebra has a bounded approximate identity (or is unital) if and only if [Formula: see text] is finite. We also characterize the left multipliers of these algebras and investigate their amenability of them. Moreover, we characterize the character spaces (Gelfand spaces) of these algebras in a special case.