Normed algebras of infinitely differentiable Lipschitz functions

Abstract

Let X be a perfect compact plane set, ω ={w n } be a weight sequence of positive numbers, and 0 < α ≤ 1. Then Lip(X, α, ω) denotes the algebra of infinitely differentiable functions f on X such that f (n), for n = 0, 1, …, satisfies the Lipschitz condition of order α, and such that ,where ||·||α is the Lipschitz norm on X. Using some formulae from combinatorial analysis, we show that, under certain conditions on ω, if f ∈ Lip(X, α, ω) and f (z)≠0 for every z ∈ X, then 1/f ∈ Lip(X, α, ω). We then conclude that if, moreover, the maximal ideal space of (X, α, ω), the uniform closure of Lip(X, α, ω), equals X then every non-zero continuous complex homomorphism on Lip(X, α, ω) is an evaluation character at some point of X.