THE EXISTENCE OF NONZERO COMPACT RIGHT MULTIPLIERS AND ARENS REGULARITY OF WEIGHTED BANACH ALGEBRAS

Abstract

Let $\omega$ be a weight function on a locally compact group $\mathcal{𝒢}$ and let $M{(\mathcal{𝒢},\omega )}_0^{\ast }$ be the subspace of $M{(\mathcal{𝒢},\omega )}^{\ast }$ consisting of all functionals that vanish at infinity. We first introduce an Arens product on ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ under which it is a Banach algebra. We then show that the existence of a nonzero compact right multiplier on ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ is equivalent to compactness of $\mathcal{𝒢}$. We also prove that if ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ is Arens regular, then $\mathcal{𝒢}$ is discrete.