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A characterization of essential sets of function algebras. (English) Zbl 1115.46041
Let $$X$$ be a compact topological space. A function algebra $$A$$ on $$X$$ is any closed subalgebra of the Banach algebra of continuous complex-valued functions on $$X$$. The following statement is the main result of the paper.
Let $$A$$ be a function algebra on $$X$$. Denote by $$E$$ its essential set. Let $$x\in X$$. Then $$x\in X\setminus E$$ if and only if there exists a closed neighbourhood $$V$$ of $$x$$ in $$X$$ such that the following two conditions are fulfilled:
(1) $$A/V=C(V)$$, where $$A/V$$ means the algebra of all restrictions of functions from $$A$$ to the set $$V$$;
(2) $$V$$ is an intersection of peak sets of $$A$$.
Note that the concept of essential set was introduced by H. S. Bear [Trans. Am. Math. Soc. 90, 383–393 (1959; Zbl 0086.31602)]. Roughly speaking, a set $$E\subset X$$ is the essential set of a function algebra $$A$$ if it is the maximal set with the property that $$A$$ consists of all continuous extensions of functions in the algebra of restrictions $$A/E$$.
##### MSC:
 46J10 Banach algebras of continuous functions, function algebras
##### Keywords:
function algebra; essential set
Zbl 0086.31602
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