Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\Vert\Phi(f) \Phi(g)\Vert\le\varepsilon\Vert f\Vert \Vert g\Vert$. 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?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46J10, 46E05

Retrieve articles in all journals with MSC (2000): 46J10, 46E05


Additional Information

Gregor Dolinar
Affiliation: Faculty of Electrical Engineering, University of Ljubljana, Slovenia
Email: gregor.dolinar@fe.uni-lj.si

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06023-3
PII: S 0002-9939(01)06023-3
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