The Second Conjugates of Certain Banach Algebras

Abstract

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.