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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 (T) into a subalgebra B of C 0 (S) is said to be separating or disjointness preserving if x·y0 implies Hx·Hy0 for all x,yA.

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:
46H40Automatic continuity
46J10Banach algebras of continuous functions, function algebras
47B38Operators on function spaces (general)
References:
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