On certain product of Banach modules

Mohammad Ramezanpour

paper

On certain product of Banach modules

arXiv:1606.04874

Abstract

Let $A$ and $B$ be Banach algebras and let $B$ be an algebraic Banach $A -$ bimodule. Then the $ℓ^{1} -$ direct sum $A \times B$ equipped with the multiplication $(a_{1}, b_{1}) (a_{2}, b_{2}) = (a_{1} a_{2}, a_{1} \cdot b_{2} + b_{1} \cdot a_{2} + b_{1} b_{2}), (a_{1}, a_{2} \in A, b_{1}, b_{2} \in B)$ is a Banach algebra denoted by $A ⋈ B$ . Module extension algebras, Lau product and also the direct sum of Banach algebras are the main examples satisfying this framework. We obtain characterizations of bounded approximate identities, spectrum, and topological center of this product. This provides a unified approach for obtaining some known results of both module extensions and Lau product of Banach algebras.

4 pages, The 4th Seminar on Functional Analysis and its Applications, 2 March 2016, Ferdowsi University of Mashhad, Iran