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 ∈ 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.