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

TL;DR

This paper studies the regularity and amenability of weighted Banach algebras on locally compact groups, providing conditions under which these properties hold, especially focusing on the finiteness of the group G.

Contribution

It establishes new criteria for Arens regularity and amenability of weighted Banach algebras and their duals, improving existing results and answering open questions.

Findings

Arens regularity of $ M_* (G, ext{omega})^* $ iff G is finite or omega is zero cluster

Equality of $ Wap(G, 1/ ext{omega}) $ and $ C_b(G, 1/ ext{omega}) $ characterizes regularity for non-compact G

Amenability of $ M_* (G, ext{omega})^* $ iff G is finite

Abstract

Let $\omega $ be a weight function on a locally compact group G mand let $ M_* (G, \omega ) $ be the subspace of $ M(G , \omega )^* $ consisting of all functionals that vanish at infinity. In this paper, we first investigate the Arens regularity of $ M_* (G, \omega )^* $ and show that $ M_* (G, \omega )^* $ 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_* (G, \omega )^* $ is Arnes regular if and only if $ Wap(G , 1/ \omega ) = C_b (G , 1/ \omega ) $. We then investigate amenability of $ M_* (G, \omega )^* $ and prove that $ M_* (G, \omega )^* $ is amenable and Arnes regular if and only if G is finite.