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

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.

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