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 Y₀ of ∂B and a surjective continuous map τ : Y₀ → ∂A such that

for all f ∈ A⁻¹ and all y ∈ Y₀. Moreover we give an example which shows that the multiple α²β in the above inequality is the best possible.

中文翻译：

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论非满射复合算子

摘要

假设A, B分别是紧 Hausdorff 空间X和Y上的一致代数，∂A, ∂B是A, B的 Šilov 边界。令T : A ^-1 → B ^{- 1}是T 1 = 1 的映射。我们证明，如果存在常数α，β ≥ 1 使得β ^{- 1} ‖ f · g ^{- 1} ‖ ≤ ‖ Tf· ( Tg ) ^{− 1} ∥ ≤ α ∥f·g ^{− 1} ∥ 对于所有f , g ∈ A ^{− 1} ，则存在∂B的非空闭子集Y ₀和满射连续映射τ : Y ₀ → ∂A使得

对于所有f ∈ A ^{− 1}和所有y ∈ Y ₀。此外，我们举一个例子，表明上述不等式中的倍数α ² β是最好的。