Best constants of Banach-Stone type theorems in subgroups of C(K).

Assume that X, Y are compact Hausdorff spaces. Let C(X)_+, C(Y)_+ be the sets of all positive real functions on X,Y respectively. Let T be a surjective map from a point separating multiplicative subgroup G\subset C(X)_+ onto a point separating multiplicative subgroup H\subset C(Y)_+ satisfying T1=1 and \begin{equation*} \frac {1}{M}\|f\cdot g^{-1}\|\leq \|Tf\cdot (Tg)^{-1}\|\leq M \|f\cdot g^{-1}\| \end{equation*} for all f, g\in G and for some constant M\geq 1. Then there exists a homeomorphism \tau from the Šilov boundary \partial H of H onto the Šilov boundary \partial G of G such that \begin{equation*} \frac {f(\tau (y))}{M^2}\leq (Tf)(y)\leq M^2\cdot f(\tau (y)) \end{equation*} for all f\in G and all y\in \partial H. The constant M^2 in the inequality above is optimal. A similar argument yields a generalization of the classical Banach-Stone Theorem to isometries defined from an additive subgroup of C_0(X) into C_0(Y) for locally compact Hausdorff spaces X and Y. [ABSTRACT FROM AUTHOR]