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Stability of disjointness preserving mappings

Author: Gregor Dolinar
Journal: Proc. Amer. Math. Soc. 130 (2002), 129-138
MSC (2000): Primary 46J10; Secondary 46E05
Published electronically: May 25, 2001
MathSciNet review: 1855629
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Let $X$ and $Y$ be compact Hausdorff spaces and let $\varepsilon \ge 0$. A linear mapping $\Phi\colon\mathcal{C}(X) \to \mathcal{C}(Y)$ is called $\varepsilon $-disjointness preserving if $f g =0$ implies that $\Vert\Phi(f) \Phi(g)\Vert\le\varepsilon\Vert f\Vert \Vert g\Vert$. If $\Phi \colon \mathcal{C}(X) \to \mathcal{C}(Y)$ is a continuous or surjective $\varepsilon$-disjointness preserving linear mapping, we prove that there exists a disjointness preserving linear mapping $\Psi \colon \mathcal{C}(X) \to \mathcal{C}(Y)$ satisfying $\Vert\Phi(f)-\Psi(f)\Vert\le 20\sqrt{\varepsilon}\Vert f\Vert$. We also prove that every unbounded $\varepsilon$-disjointness preserving linear functional on $\mathcal{C}(X)$ is disjointness preserving.

References [Enhancements On Off] (What's this?)

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Additional Information

Gregor Dolinar
Affiliation: Faculty of Electrical Engineering, University of Ljubljana, Slovenia

Keywords: $\varepsilon$-disjointness preserving mapping, stability of disjointness preserving mappings
Received by editor(s): November 19, 1999
Received by editor(s) in revised form: June 9, 2000
Published electronically: May 25, 2001
Communicated by: Dale Alspach
Article copyright: © Copyright 2001 American Mathematical Society