On peak sets for certain function spaces
The authors consider some generalizations to function spaces, of results on function algebras. Let \(X\) be a compact Hausdorff space and let \(C(X)\) denote the Banach space of complex-valued continuous function on \(X\). A function space on \(X\) is a closed subspace of \(C(X)\) containing the constant function and separating the points of \(X\). For a function space \(A\) on \(X\) let \(A(X)=\{f\in C(X): fa\in A\) for all \(a\in A\}\). The authors consider spaces satisfying the following condition: (A) Any peak set for \(A\) is a peak set for \(A(X)\). Any function algebra on \(X\) satisfied of course condition (A); the authors give several examples of function spaces (which are not algebras) which satisfy condition (A). Among results obtained are the following: Suppose \(A\) satisfies condition (A). If Every peak set for \(\Re A\) is a peak set for \(A\) then \(A=C(X)\). Also, if \(X\) is the union of countably many interpolation sets for \(A\), then \(A=C(X)\).
scientific article; zbMATH DE number 4113299