Prakash Dabhi - Department of Mathematics, Institute of Infrastructure Technology Research and Management (IITRAM), Ahmedabad - ۳۸۰۰۲۶, Gujarat, India
Yuvraj Pipaliya - Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar- ۳۸۸۱۲۰, Gujarat, India

Let \mathcal{A}, \mathcal{B}, and \mathcal{X} be complex algebras, \theta : \mathcal{B} \longrightarrow \mathcal{X} be an algebra homomorphism, and let \mathcal{A} be an \mathcal{X}-bimodule. We define a product on \mathcal{A} \times \mathcal{B} as (a_۱, b_۱)(a_۲,b_۲) = (a_۱ a_۲ + a_۱ \cdot \theta(b_۲) + \theta(b_۱) \circ a_۲,b_۱ b_۲) for all (a_۱, b_۱), (a_۲,b_۲) \in \mathcal{A} \times \mathcal{B} and write \mathcal{A} \times \mathcal{B} with this product by \mathcal{A} \times_\theta\mathcal{B}. We shall study some basic properties of \mathcal{A} \times_\theta \mathcal{B}. When \mathcal{A}, \mathcal{B} and \mathcal{X} are Banach algebras, \mathcal{A} is a Banach \mathcal{X}-bimodule, and \theta is a continuous homomorphism with the norm at most ۱, we determine the ideals of \mathcal{A} \times_\theta \mathcal{B} of a certain type, the Gelfand space of this Banach algebra, and the module multipliers of this Banach algebra.

Module Lau - product of Banach algebras, ideals, Gelfand space, module multipliers