The Maximal Ideal Space of <i>H</i>^∞ + <i>C</i> on the ball in C^<i>n</i>

Abstract

Let S denote the unit sphere in C n , B the (open) unit ball in C n and H ∞ (B) the collection of all bounded holomorphic functions on B. For f ∈ H ∞ (B) the limits exist for almost every ζ in S, and the map ƒ → ƒ * defines an isometric isomorphism from H ∞ (B) onto a closed subalgebra of L ∞ (S), denoted H ∞ (S). (The only measure on S we will refer to in this paper is the Lebesgue measure, dσ, generated by Euclidean surface area.) Rudin has shown in [ 4 ] that the spaces H ∞ (B) + C(B) and H ∞ (S) + C(S) are Banach algebras in the sup norm. In this paper we will show that the maximal ideal space of H ∞ (B) + C(B), Σ (H ∞ (B) + C(B)), is naturally homeomorphic to Σ (H ∞ (B)) and that Z (H ∞ (S) + C(S)) is naturally homeomorphic to Σ (H ∞ (S))\B.