Biseparating maps and homeomorphic real-compactifications
If \(X\) and \(Y\) are realcompact and \(T : C(X) \to C(Y)\) is a ring isomorphism then two things happen: (1) there is a homeomorphism \(h\) of \(X\) onto \(Y\) and (2) \(T\) is a composition map: for all \(f\) in \(C(X)\), \(Tf = f \circ h\). The authors strengthen that result. They discard the multiplicative property of \(T\) and assume only that \(T\) is biseparating in the sense that \(fg = 0 \Leftrightarrow Tf Tg = 0\). It remains true that \(X\) is homeomorphic to \(Y\) but this time, if \(T\) is also linear, it is a weighted composition map: for some weight function \(a\) in \(C(Y)\), \(Tf = a(f \circ h)\).
scientific article; zbMATH DE number 786129