Stability of disjointness preserving mappings

Dolinar, Gregor

doi:10.1090/S0002-9939-01-06023-3

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.

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