On amalgamated Banach algebras

Let A and B be Banach algebras, $$\theta : A\rightarrow B$$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 $$\theta $$θ, denoted $$A\bowtie ^{\, \theta }I$$A⋈θI, 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 $$A\bowtie ^{\, \theta }I$$A⋈θI in terms of that of A and I. Using this, we provide a characterization of the Arens regularity of $$A\bowtie ^{\, \theta }I$$A⋈θI. Then we determine the character space of $$A\bowtie ^{\, \theta }I$$A⋈θI in terms of that of A and I. Moreover, we study the weak amenability of $$A\bowtie ^{\, \theta }I$$A⋈θI.