The bse concepts for vector-valued lipschitz algebras

Abstract

Let (K; d) be a compact metric space, A be a commutative semisimple Banach algebra and 0 < α ≤ 1. The overall purpose of the present paper is to demonstrate that all BSE concepts of Lipα(K;A) are inherited from A and vice versa. Recently, the authors proved in the case that A is unital, Lipα(K;A) is a BSE-algebra if and only if A is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra A. Furthermore, we investigate the BSE-norm property for Lipα(K;A) and prove that Lipα (K;A) belongs to the class of BSE-norm algebras if and only if A is owned by this class. Moreover, we prove that for any natural number n with n ≥ 2, if all continuous bounded functions on Δ(Lipα (K;A)) are n-BSE-functions, then K is finite. As a result, we obtain that Lipα (K;A) is a BSE-algebra of type I if and only if A is a BSE-algebra of type I and K is finite. Furthermore, in according to a result of Kaniuth and Ulger, which disapproves the BSE-property for lipαK, we show that for any commutative semisimple Banach algebra A, lipα (K;A) fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra LipαX, for an arbitrary metric space (not necessarily compact) (X; d) and α > 0, when LipαX separates the points of X. In particular, we show that LipαX is a BSE-algebra, as well as a BSE-norm algebra. © 2021 American Institute of Mathematical Sciences. All rights reserved.