Certain dense subalgebras of continuous vector-valued operator algebras

Let X be a compact metric space with at least two elements, B be a unital commutative Banach algebra over the scalar field F=R or C, and α in R with 0<α≤1. Suppose that C(X,B) be the continuous, A(X,B) be the analytic, and Lipα(X,B) be the α-Lipschitz B-valued operator algebras on X. In this paper, we prove that the algebras Lip α(X,B) and A(X,B) are dense in C(X,B) under sup-norm. Also, we study the relationship between elements of the algebras Lip α(X,B) and A(X,B).

Keywords:

Dense , Banach algebra , Lipschitz algebra , Vector-valued operator