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

Abstract

Assume that X , Y X, Y are compact Hausdorff spaces. Let C ( X ) + C(X)_+ , C ( Y ) + C(Y)_+ be the sets of all positive real functions on X , Y X,Y respectively. Let T T be a surjective map from a point separating multiplicative subgroup G ⊂ C ( X ) + G\subset C(X)_+ onto a point separating multiplicative subgroup H ⊂ C ( Y ) + H\subset C(Y)_+ satisfying T 1 = 1 T1=1 and 1 M ‖ f ⋅ g − 1 ‖ ≤ ‖ T f ⋅ ( T g ) − 1 ‖ ≤ M ‖ f ⋅ g − 1 ‖ \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 ∈ G f, g\in G and for some constant M ≥ 1 M\geq 1 . Then there exists a homeomorphism τ \tau from the Šilov boundary ∂ H \partial H of H H onto the Šilov boundary ∂ G \partial G of G G such that f ( τ ( y ) ) M 2 ≤ ( T f ) ( y ) ≤ M 2 ⋅ f ( τ ( y ) ) \begin{equation*} \frac {f(\tau (y))}{M^2}\leq (Tf)(y)\leq M^2\cdot f(\tau (y)) \end{equation*} for all f ∈ G f\in G and all y ∈ ∂ H y\in \partial H . The constant M 2 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 ) C_0(X) into C 0 ( Y ) C_0(Y) for locally compact Hausdorff spaces X X and Y Y .