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 Lip d ( X , E ) , where ( X , d ) is a compact metric space. Moreover, we show that the vector-valued Lipschitz algebra Lip d ( X , E ) is regular, where ( X , d ) is any metric space and E is a certain unital semisimple commutative ∗ -Banach algebra. Furthermore, we study the regularity of some vector-valued function algebras.