A remark on non-surjective composition operators

Abstract

Assume that A, B are uniform algebras on compact Hausdorff spaces X and Y, respectively, and ∂A, ∂B are the Šilov boundaries of A, B. Let T : A ⁻¹ → B^{− 1} be a map with T1 = 1. We show that, if there exist constants α, β ≥ 1 such that β^{− 1}‖f·g^{− 1}‖ ≤ ‖Tf·(Tg) ^{− 1}∥ ≤ α∥f·g^{− 1}∥ for all f, g ∈ A^{− 1}, then there is a non-empty closed subset Y ₀ of ∂B and a surjective continuous map τ : Y ₀ → ∂A such that