Small isomorphisms of $C_{0}(K)$ onto $C_{0}(S)$ generate a unique homeomorphism of $K$ onto $S$ similar to that of isometries

Elói Medina Galego and André Luis Porto da Silva

Vol. 310 (2021), No. 1, 23–48

DOI: 10.2140/pjm.2021.310.23

Abstract

Let $K$ and $S$ be locally compact Hausdorff spaces with $S$ being a normal space. We prove that if $T$ is a linear isomorphism from an extremely regular subspace $A$ of $C_0\left(K\right)$ onto an extremely regular subspace $B$ of $C_0\left(S\right)$ with $\parallel T\parallel \parallel T^{-1}\parallel <\kappa$, then there exists exactly one homeomorphism $\phi$ from $S$ onto $K$ such that for some continuous function $a:S\to \mathbb{𝕂}$ we have

$$\left|T\left(f\right)\left(s\right)-a\left(s\right)f\left(\phi \left(s\right)\right)\right|\le r\left(\parallel T\parallel -1∕\parallel T^{-1}\parallel \right),$$

for every $f\in C_0\left(K\right)$ with $\parallel f\parallel \le 1$, and $s\in S$, where $\kappa =2$ and $r=1$ if $\mathbb{𝕂}$ is the real numbers, and $\kappa =\frac{3}{2}$ and $r=2$ if $\mathbb{𝕂}$ is the complex numbers.

This result provides a full extension of Banach–Stone theorem for small isomorphisms between extremely regular subspaces of $C_0\left(K\right)$ spaces that unifies and improves some classical results of Amir (1966), Cambern (1967), Cengiz (1973), and Cutland and Zimmer (2005).

Moreover, for the real scalar case, our result is optimal in the sense that $r=1$ cannot be replaced by any other number.

Keywords

Banach–Stone theorem, Amir–Cambern theorem, Cengiz theorem, weighted composition operators, extremely regular subspaces of $C_{0}(K)$ space, small isomorphisms, uniqueness of homeomorphisms

Mathematical Subject Classification

Primary: 46B03, 46E15

Secondary: 46B25

Milestones

Received: 12 September 2020

Accepted: 18 November 2020

Published: 26 January 2021