On the structure of into isomorphisms between spaces of continuous functions

Abstract

Let K K and S S be locally compact Hausdorff spaces and let T T be a real linear isomorphism of an extremely regular subspace A A of C 0 ( K ) C_{0}(K) into C 0 ( S ) C_{0}(S) satisfying ‖ T ‖ ‖ T − 1 ‖ > 2 \|T \| \ \|T^{-1}\|>2 . Put L = T ‖ T − 1 ‖ L=T \ \|T^{-1}\| and pick 0 ≤ ε > 1 0 \leq \varepsilon >1 such that ‖ T ‖ ‖ T − 1 ‖ = 1 + ε \|T \| \ \|T^{-1}\|=1+\varepsilon . We prove that there exist a subset S 0 S_0 of S S , a proper map φ \varphi of S 0 S_{0} onto K K and a map λ : S 0 → { − 1 , 1 } \lambda :S_{0} \to \{-1, 1\} such that | L f ( s ) − λ ( s ) f ( φ ( s ) ) | ≤ ε ‖ f ‖ , ∀ s ∈ S 0 and f ∈ A . \begin{equation*} |Lf(s)-\lambda (s) f(\varphi (s))| \leq \varepsilon \|f\|, \ \forall s \in S_{0} \ \text {and} \ f \in A. \end{equation*} For ε > 1 / 2 \varepsilon > 1/2 the maps φ \varphi and λ \lambda are continuous, and this approximation of L L by such a weighted composition operator improves the well-known Holsztyński theorem [Studia Math. 26 (1966), pp. 133–136], the case where ε = 0 \varepsilon =0 , K K and S S are compact and A = C 0 ( K ) A=C_{0}(K) . The approximation of L L in the general case where 0 ≤ ε > 1 0 \leq \varepsilon >1 is an extension of a Benyamini’s result [Proc. Amer. Math. Soc. 83 (1981), pp. 479–485] which allowed us to strengthen a classical Jarosz’s theorem [Proc. Amer. Math. Soc. 90 (1984), pp. 373–377] by proving that K K is a continuous image not just of a subset of S S but of a locally compact subset of S S . Moreover, when K K is compact, it is a continuous image not just a closed subset of S S but of a compact subset of S S , namely the set S 0 S_0 having the above properties.