On separating maps between locally compact spaces. (English) Zbl 0805.46049

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.


46H40 Automatic continuity
46J10 Banach algebras of continuous functions, function algebras
47B38 Linear operators on function spaces (general)
Full Text: DOI


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