On amalgamated Banach algebras

Abstract

Abstract. Let A and B be Banach algebras, θ : A → B be a continuous Banach algebra homomorphism and I be a closed ideal in B. Then the direct sum of A and I with respect to θ, denoted A ⊲⊳ θ I, with a special product becomes a Banach algebra which is called the amalgamated Banach algebra. In this paper, among other things, we compute the topological centre of A ⊲⊳ θ I in terms of that of A and I. Using this, we provide a characterization of the Arens regularity of A ⊲⊳ θ I. Then we determine the character space of A ⊲⊳ θ I in terms of that of A and I. Moreover, we study the weak amenability of A ⊲⊳ θ I.