On the Arens product and annihilator algebras

Abstract

The purpose of this paper is to generalize two results in a recent paper by B. J. Tomiuk and the author. Let A be a B ∗ {B^ \ast } -algebra and M ( A ) M(A) the algebra of double centralizers of A. We show that A is a dual algebra if and only if M ( A ) M(A) coincides with A ∗ ∗ {A^{ \ast \ast }} . We also obtain that if A is an annihilator A ∗ {A^\ast } -algebra, then π A ( A ) {\pi _A}(A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} .