On separating maps between locally compact spaces
type: Item
instance of: scholarly article
title: On separating maps between locally compact spaces
zbMATH Open document ID: 0805.46049
DOI: 10.1007/BF01189890
author: Salvador Hernández
author: Juan J. Font
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46H40
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 687058
zbMATH Keywords: disjointness preserving or separating map
zbMATH Keywords: Banach-Stone theorem
zbMATH Keywords: separating bijection
zbMATH Keywords: automatically continuous
zbMATH Keywords: weighted composition map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 687058