Arens Regularity of Module Extensions of Banach Algebras

Let A be a Banach algebra and A be its second dual equipped with the first or second Arens product, X be a Banach A-bimodule and let U=A ⊕ X as a module extensions of Banach algebra. In this paper we study the topological centres of U and show that under certain conditions Arens regularity of A implies that of U.