## 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.

### MSC:

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

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