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

Abstract

Let ω be a weight function on a locally compact group 𝒢 and let M(𝒢,ω)0∗ be the subspace of M(𝒢,ω)∗ consisting of all functionals that vanish at infinity. We first introduce an Arens product on (M(𝒢,ω)0∗)∗ under which it is a Banach algebra. We then show that the existence of a nonzero compact right multiplier on (M(𝒢,ω)0∗)∗ is equivalent to compactness of 𝒢. We also prove that if (M(𝒢,ω)0∗)∗ is Arens regular, then 𝒢 is discrete.