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

A linear map $H$ defined from a subalgebra $A$ of ${C}_{0}\left(T\right)$ into a subalgebra $B$ of ${C}_{0}\left(S\right)$ is said to be separating or disjointness preserving if $x·y\equiv 0$ implies $Hx·Hy\equiv 0$ 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 $C\left(T\right)$ and $C\left(S\right)$ with $T$ and $S$ compact spaces.

##### MSC:
 46H40 Automatic continuity 46J10 Banach algebras of continuous functions, function algebras 47B38 Operators on function spaces (general)
##### References:
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