The BSE concepts for vector-valued Lipschitz algebras

Let $ (K,d) $ be a compact metric space, $ \mathcal A $ be a commutative semisimple Banach algebra and $ 0<\alpha\leq 1 $. The overall purpose of the present paper is to demonstrate that all BSE concepts of $ {\rm Lip}_\alpha(K,\mathcal A) $ are inherited from $ \mathcal A $ and vice versa. Recently, the authors proved in the case that $ \mathcal A $ is unital, $ {\rm Lip}_\alpha(K,\mathcal A) $ is a BSE-algebra if and only if $ \mathcal A $ is so. In this paper, we generalize this result for an arbitrary commutative semisimple Banach algebra $ \mathcal A $. Furthermore, we investigate the BSE-norm property for $ {\rm Lip}_\alpha(K,\mathcal A) $ and prove that $ {\rm Lip}_\alpha(K,\mathcal A) $ belongs to the class of BSE-norm algebras if and only if $ \mathcal A $ is owned by this class. Moreover, we prove that for any natural number $ n $ with $ n\geq 2 $, if all continuous bounded functions on $ \Delta({\rm Lip}_\alpha(K,\mathcal A)) $ are $ n $-BSE-functions, then $ K $ is finite. As a result, we obtain that $ {\rm Lip}_{\alpha}(K,\mathcal A) $ is a BSE-algebra of type I if and only if $ \mathcal A $ is a BSE-algebra of type I and $ K $ is finite. Furthermore, in according to a result of Kaniuth and Ülger, which disapproves the BSE-property for $ {\rm lip}_{\alpha}K $, we show that for any commutative semisimple Banach algebra $ \mathcal A $, $ {\rm lip}_{\alpha}(K,\mathcal A) $ fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra $ {\rm Lip}_\alpha X $, for an arbitrary metric space (not necessarily compact) $ (X,d) $ and $ \alpha>0 $, when $ {\rm Lip}_\alpha X $ separates the points of $ X $. In particular, we show that $ {\rm Lip}_\alpha X $ is a BSE-algebra, as well as a BSE-norm algebra.