Stability of disjointness preserving mappings
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- by Gregor Dolinar PDF
- Proc. Amer. Math. Soc. 130 (2002), 129-138 Request permission
Abstract:
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 $\|\Phi (f) \Phi (g)\|\le \varepsilon \|f\| \|g\|$. 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 $\|\Phi (f)-\Psi (f)\|\le 20\sqrt {\varepsilon }\|f\|$. We also prove that every unbounded $\varepsilon$-disjointness preserving linear functional on $\mathcal {C}(X)$ is disjointness preserving.References
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Additional Information
- Gregor Dolinar
- Affiliation: Faculty of Electrical Engineering, University of Ljubljana, Slovenia
- Email: gregor.dolinar@fe.uni-lj.si
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 129-138
- MSC (2000): Primary 46J10; Secondary 46E05
- DOI: https://doi.org/10.1090/S0002-9939-01-06023-3
- MathSciNet review: 1855629