Regularity and amenability of weighted Banach algebras and their second dual on locally compact groups

Abstract

Let $\omega$ be a weight function on a locally compact group G mand let $M_{\ast }(G,\omega )$ be the subspace of $M(G,\omega {)}^{\ast }$ consisting of all functionals that vanish at infinity. In this paper, we first investigate the Arens regularity of $M_{\ast }(G,\omega {)}^{\ast }$ and show that $M_{\ast }(G,\omega {)}^{\ast }$ is Arnes regular if and only if G is finite or $\omega$ is zero cluster. This result is an answer to the question posed and it improves some well-known results. We also give necessary and sufficient criteria for the weight function spaces $Wap(G,1/\omega )$ and $Wap(G,1/\omega )$ to be equal to $C_b(G,1/\omega )$. We prove that for non-compact group G, the Banach algebra $M_{\ast }(G,\omega {)}^{\ast }$ is Arnes regular if and only if $Wap(G,1/\omega )=C_b(G,1/\omega )$. We then investigate amenability of $M_{\ast }(G,\omega {)}^{\ast }$ and prove that $M_{\ast }(G,\omega {)}^{\ast }$ is amenable and Arnes regular if and only if G is finite.