\(C_\infty(X)\) and related ideals
The intersection of all free maximal ideals of \(C^*(X)\) is denoted by \(C_\infty(X)\) and the smallest (possibly improper) \(z\)-ideal containing \(C_\infty(X)\) is denoted by \(C_{l\sigma}(X)\). Spaces for which \(C_{l\sigma}(X)\) is a prime ideal are characterized and it is shown that \(C_\infty(X)\) is a \(z\)-ideal in \(C(X)\) if and only if every zero-set contained in an open, locally compact and \(\sigma\)-compact subset of \(X\) is compact.
scientific article; zbMATH DE number 6032472