The Regularity of the Vector-Valued Lipschitz Algebras

Let (X, d) be a metric space and E be a unital commutative Banach algebra. In this paper, we prove that the regularity of unital commutative Banach algebra E is a necessary and sufficient condition for regularity of Lipd(X,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{d}(X,E)$$\end{document}, where (X, d) is a compact metric space. Moreover, we show that the vector-valued Lipschitz algebra Lipd(X,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{d}(X,E)$$\end{document} is regular, where (X, d) is any metric space and E is a certain unital semisimple commutative ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-Banach algebra. Furthermore, we study the regularity of some vector-valued function algebras.