POINT DERIVATIONS ON BANACH ALGEBRAS OF α-LIPSCHITZ VECTOR-VALUED OPERATORS

The Lipschitz function algebras were first defined in the 1960s by some mathematicians, including Schubert. Initially, the Lipschitz real-value and complex-value functions are defined and quantitative properties of these algebras are investigated. Over time these algebras have been studied and generalized by many mathematicians such as Cao, Zhang, Xu, Weaver, and others. Let  be a non-empty compact metric space and  be a unital commutative Banach space over the scalar field , and . In this paper, we first introduce the Banach algebras of vector-valued (B-valued) -Lipschitz operators on ,  and , then we study the point derivations on them. In the main results of this paper, we prove that all continuous point derivatives on  are zero, and at any non-isolated point X, there is a non-zero continuous point derivation on .