On the Arens product and commutative Banach algebras

Abstract

The purpose of this note is to generalize two recent results by the author for commutative Banach algebras. Let A be a commutative Banach algebra with carrier space X A {X_A} and π \pi the canonical embedding of A into its second conjugate space A ∗ ∗ {A^{ \ast \ast }} (with the Arens product). We show that if A is a semisimple annihilator algebra, then π ( A ) \pi (A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} . We also obtain that if A is a dense two-sided ideal of C 0 ( X A ) {C_0}({X_A}) , then π ( A ) \pi (A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} if and only if A is a modular annihilator algebra.