Quasi-isometries on subsets of 𝐶₀(𝐾) and 𝐶₀⁽¹⁾(𝐾) spaces which determine 𝐾

Abstract

We introduce the concept of Banach-Stone subsets of C 0 ( K ) C_{0}(K) spaces. This allows us to unify and improve several extensions of the classical theorem due to Banach (1933) and Stone (1937). More precisely, we prove that if K K and S S are locally compact Hausdorff spaces, A A and B B are Banach-Stone subsets of C 0 ( K ) C_{0}(K) and C 0 ( S ) C_{0}(S) , respectively, and there exists a map T T from A A to B B (not necessarily injective) with image θ \theta -dense in B B for some θ > 0 \theta >0 such that 1 M ‖ f − g ‖ − L ≤ ‖ T ( f ) − T ( g ) ‖ ≤ M ‖ f − g ‖ + L , \begin{equation*} \frac {1}{M} \|f-g\|-L \leq \|T(f)-T(g)\|\leq M \|f-g\|+L, \end{equation*} for every f , g ∈ A f, g \in A , then K K and S S are homeomorphic whenever L ≥ 0 L \geq 0 and M > 2 M> \sqrt {2} . As an application of this more general theorem concerning the quasi-isometries T T on subsets of C 0 ( K ) C_{0}(K) spaces, we show that certain quasi-isometries on C 0 ( 1 ) ( K ) C_0^{(1)}(K) spaces also determine the locally compact subspaces K K of the real line R \mathbb R with no isolated points. In turn, this result enables us to prove a unification and improvement of some theorems of Cambern, Pathak, and Vasavada for the first time to the nonlinear case.