A generalization of the Arens extension for Banach algebras

We construct an extension of the algebra product of a Banach algebra AA to Hb∗(A), where Hb(A)Hb(A) is the algebra of all analytic functions of bounded type on AA and Hb∗(A) is its dual space. The extension generalizes the Arens product. Some applications to the spectrum of Hb(A)Hb(A) are obtained.