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 ${X_A}$ and $\pi$ the canonical embedding of A into its second conjugate space ${A^{ \ast \ast }}$ (with the Arens product). We show that if A is a semisimple annihilator algebra, then $\pi (A)$ is a two-sided ideal of ${A^{ \ast \ast }}$. We also obtain that if A is a dense two-sided ideal of ${C_0}({X_A})$, then $\pi (A)$ is a two-sided ideal of ${A^{ \ast \ast }}$ if and only if A is a modular annihilator algebra.