The Maximal Ideal Space of lip A (X, α)

Abstract

Let X be a compact subset of the complex plane C, and let 0 < a < 1 . We show that the maximal ideal space of liPA (X, a) is X. Let F be a closed nonempty set in RI, and take ca with 0 < ca < 1. Then Lip(F, ca) is the algebra of bounded complex-valued functions f on F such that pa(f) = SUp {I f(x) jf(Y) I: XYEF.x Y} is finite, and lip(F, ca) is the subalgebra of functions f such that IfPx) f(Y) I,0 as Ix yJ '?. For f E Lip(X, c), set lIfila = i1f lIF +Pa(f), where I lF is the uniform norm on the closed set F . Then (Lip(F, ca), I I *a) is a Banach function algebra on F, and lip(F, ca) is a closed subalgebra of Lip(F, ca). These Lipschitz algebras were first studied by Sherbert [3]. Let X be a compact subset of the complex plane C. We follow established custom in denoting by C(X) the algebra of all continuous complex-valued functions on X and denoting by A(X) the subalgebra of functions analytic on int(X). We further define LiPA(X, ca) = Lip(X, ca) n A(X) and lipA(X, ca) = lip(X, ca) n A(X), so that LiPA(X, ca) and lipA(X, ca) are closed subalgebras of Lip(X, ca). This paper concerns the maximal ideal space of the algebra liPA(X, ca). It is clear that when the interior of X is empty, lipA(X, ca) is the algebra lip(X, ca) and its maximal ideal space is X [3]. So suppose that the interior of X is nonempty. We shall show that the maximal ideal space of lipA(X, ca) is X. The proof of this result depends on an extension theorem and some approximation lemmas. An interesting extension theorem for the Lipschitz algebra Lip(X, ca) is given in [4, Chapter VI, Theorem 3]. Here we modify the proof of that result to prove the following extension theorem for lip(X, ca). Received by the editors December 9, 1992. 1991 Mathematics Subject Classification. Primary 46J 10; Secondary 46J1 5, 46J20.