On the maximal ideal space of extended analytic Lipschitz algebras

Abstract

Let X be a compact, plane set and let K be a compact subset of X. We introduce new classes of Lipschitz algebras Lip(X, K, α), lip(X, K, α), consisting of those continuous functions f on X such that f|K ε Lip(K, α), lip(K, α), and their analytic subalgebras Lip_A(X, K, α) = Lip(X, K, α) ∩ A(X, K) and lip_A(X, K, α) = lip(X, K, α) ∩ A(X, K), where 0 < α ≤ 1 and A(X, K) is the algebra of all continuous complex-valued functions on X, which are analytic on the interior of K. We show that the maximal ideal spaces of these extended Lipschitz algebras coincide with X.

Quaestiones Mathematicae 30(2007), 349–353