Let A be a commutative Banach algebra and (K, d) be a compact metric space. In this paper, we examine various spectral properties, including the unique uniform norm property, weak regularity, Ditkin's condition and local operators, within the context of vector-valued Lipschitz algebras denoted as Lip (K, A). We demonstrate that the aforementioned concepts exhibit stability between A and Lip (K, A). Furthermore, we establish that a commutative Banach algebra A is classified as a Tauberian algebra if and only if Lip (K, A) is also a Tauberian algebra.
Aboubakri et al. (Tue,) studied this question.