Banach-Stone Theorems

Abstract:

Banach-Stone Theorems describe the relationship between a compact Hausdorff space K and the corresponding function space C(K). The surprising direction says that if C(K1) is the same as C(K2), then K1 is the same as K2. How surprising depends rather on what you mean by 'is the same as': one can consider these spaces as Banach spaces, Banach algebras or C*-algebras and you get different theorems in each case.