Nonlinear isometries between function spaces

Abstract

We demonstrate that any surjective isometry $T:\mathcal{A}\to \mathcal{B}$ not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form $$T\left(f\right)=T\left(0\right)+Re[\mu \cdot (f\circ \tau \left)\right]+iIm[\nu \cdot (f\circ \rho \left)\right],$$ where $\mu$ and $\nu$ are continuous and unimodular, there exists a clopen set $K$ with $\nu =\mu$ on $K$ and $\nu =-\mu$ on $K^c$, and $\tau$ and $\rho$ are homeomorphisms.