Compact contractive projections in continuous function spaces (Q1191401)

Suppose that \(X\) is a completely regular Hausdorff space, \(C(X)\) is the space of continuous, real-valued functions on \(X\) with the compact open topology, and \(E\) is a linear subspace of \(C(X)\). The authors seek conditions which ensure that there is some compact, contractive projection from \(C(X)\) onto \(E\). A point \(x\) in \(X\) is said to be a double point in case there is a second point \(x'\) in \(X\) such that \(f(x)+f(x')=0\) for all \(f\) in \(E\); if \(E\) separates points, then \(x'\) is unique. Otherwise, \(x\) is said to be a single point. Let \(d(E)\) denote the set of double points in \(X\), \(D\) the set of all extreme double points, and \(S\) the set of all extreme single points. The authors' main result is that if \(E\) separates points and closed sets of \(X\), \(d(E)\) is closed, and some compact, contractive projection maps \(C(X)\) onto \(E\), then \(E\) is compact isometric to \(F=\{f\in C(\bar S\cup\bar D): f(x)+f(x')=0\text{ for all } x\in \bar D\}\).