拟保值域算子在一致代数两个问题中的应用

The Application of Quasi-keep-range Operator to the two Problems of Uniform Algebra

本文给出了判定一个复值连续函数和它的复共轭函数同时属于指定的一致代数的一个充要条件，给出了判定指定的一致代数等于C（X）的一个充要条件，X上的一致代数A等距线性同构于某个C（K）的充要条件是A=C（X）。

Let X be a compact Hausdorff space. Y be a topological space. C be the complex plane,A be a uniform algebra on X,K be a compact space.By M（Y→C）we denote the normed linear space of all the bounded complex functio ns with the norm ｜｜f｜｜= sup y∈Y^sup｜f （y） ｜. This paper proves the following results by using quasi-keep-range operator. （ 1 ） Let f∈C M （X）, then f, f^-∈A if and only if for all x,y y∈X. f（x）≠f （y）. there exists a linear subspace B of M（Y→C）which contains the constant functions and f∈B←→ f ^-∈B,and a linear operator T from B into .A such thatT1=1. ｜｜T｜｜ =1 andTB separates the points x and y. （ 2 ）A=C（X） if and only if for all x, y∈X, x≠y, there exists a linear subspace B of M （Y→C）which contains the constant functions and f∈B←→f^-∈B, and a linear operator T from B into A such that T1=1,｜｜T｜｜ =1 and TB separates the points x and y. （ 9 ）If A as a Banach space is isometric isomorphism to C（K） ,then A=C（X）.