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
$a:S\to \mathbb{\mathbb{K}}$
we have
$$\leftT\left(f\right)\left(s\right)a\left(s\right)f\left(\phi \left(s\right)\right)\right\le r\left(\parallel T\parallel 1\u2215\parallel {T}^{1}\parallel \right),$$
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
$\mathbb{\mathbb{K}}$ is the real
numbers, and
$\kappa =\frac{3}{2}$
and
$r=2$ if
$\mathbb{\mathbb{K}}$ 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.
