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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
Citations:
Zbl 0086.31602
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