We demonstrate that any surjective isometry not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form where and are continuous and unimodular, there exists a clopen set with on and on , and and are homeomorphisms.
"Nonlinear isometries between function spaces." Ann. Funct. Anal. 8 (4) 460 - 472, November 2017. https://doi.org/10.1215/20088752-2017-0010