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Zbl 0805.46049
Font, J.J.; Hernandez, S.
On separating maps between locally compact spaces.
(English)
[J] Arch. Math. 63, No.2, 158-165 (1994). ISSN 0003-889X; ISSN 1420-8938/e

A linear map $H$ defined from a subalgebra $A$ of $C\sb 0(T)$ into a subalgebra $B$ of $C\sb 0(S)$ is said to be separating or disjointness preserving if $x\cdot y\equiv 0$ implies $Hx\cdot Hy\equiv 0$ for all $x,y\in A$.\par 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 $S$.\par 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.
[J.J.Font (Castellón)]
MSC 2000:
*46H40 Automatic continuity
46J10 Banach algebras of continuous functions
47B38 Operators on function spaces

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

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