On Spectral Properties in Vector-Valued Beurling Algebra

Abstract

Let $G$ be an LCA group, $\om$ be a weight function on $G$, and $\A$ be a semisimple, commutative Banach algebra. We characterize some Banach algebra properties of vector-valued Beurling algebra $\Lgwa$ in terms of $\Lgw$ and $\A$ using the Bochner integration theory. These properties are unique uniform norm property (UUNP), unique $C^{\ast}$-norm property (UC$^{\ast}$NP), quasi divisor of zero property (QDZP), weak regularity (WR), regularity, and complete regularity (CR).