THE BSE CONCEPTS FOR VECTOR-VALUED LIPSCHITZ ALGEBRAS

Let (K, d) be a compact metric space, A be a commutative semi-simple Banach algebra and 0 < alpha <= 1. The overall purpose of the present paper is to demonstrate that all BSE concepts of Lip(alpha)(K, A) are inherited from A and vice versa. Recently, the authors proved in the case that A is unital, Lip(alpha)(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 Lip(alpha)(K, A) and prove that Lip(alpha)(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 Delta(Lip(alpha)(K,A)) are n-BSE-functions, then K is finite. As a result, we obtain that Lip(alpha)(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 lip(alpha)K, we show that for any commutative semisimple Banach algebra A, lip(alpha)(K, A)( ) fails to be a BSE-algebra, as well. Finally, we concentrate on the classical Lipschitz algebra Lip(alpha)X, for an arbitrary metric space (not necessarily compact) (X, d) and alpha > 0, when Lip(alpha)X separates the points of X. In particular, we show that Lip(alpha)X is a BSE-algebra, as well as a BSE-norm algebra.