On separating maps between locally compact spaces (Q1337794)

A linear map \(H\) defined from a subalgebra \(A\) of \(C_ 0(T)\) into a subalgebra \(B\) of \(C_ 0(S)\) is said to be separating or disjointness preserving if \(x\cdot y\equiv 0\) implies \(Hx\cdot Hy\equiv 0\) for all \(x,y\in A\). The authors show that a separating bijection \(H\) is automatically continuous (indeed, a weighted composition map) and induces a homeomorphism between the locally compact spaces \(T\) and \(S\). If \(A\) and \(B\) are the continuous functions on \(T\) and \(S\), respectively, with compact support, then a similar result for a separating injection is obtained. This result is applied to generalize to functions with compact support a well-kown theorem by Holsztyński about linear into isometries between \(C(T)\) and \(C(S)\) with \(T\) and \(S\) compact spaces.