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

Font, J. J.; Hernandez, S.

Arch. Math. 63, No. 2, 158-165 (1994).

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.

Reviewer: J. J. Font (Castellón)

MSC:

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

Keywords:

disjointness preserving or separating map; Banach-Stone theorem; separating bijection; automatically continuous; weighted composition map

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