Banach–Stone theorem

For a compact Hausdorff space X, let C(X) denote the Banach space of continuous real- or complex-valued functions on X, equipped with the supremum norm ‖·‖∞. Given compact Hausdorff spaces X and Y, suppose T : C(X) → C(Y) is a surjective linear isometry. Then there exists a homeomorphism φ : Y → X and a function g ∈ C(Y) with :| g(y) | = 1 \mbox{ for all } y \in Y such that :(T f) (y) = g(y) f(\varphi(y)) \mbox{ for all } y \in Y, f \in C(X). The case where X and Y are compact metric spaces is due to Banach, while the extension to compact Hausdorff spaces is due to Stone. In fact, they both prove a slight generalization—they do not assume that T is linear, only that it is an isometry in the sense of metric spaces, and use the Mazur–Ulam theorem to show that T is affine, and so T - T(0) is a linear isometry. ==Generalizations==

The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if E is a Banach space with trivial centralizer and X and Y are compact, then every linear isometry of C(X; E) onto C(Y; E) is a strong Banach–Stone map. A similar technique has also been used to recover a space X from the extreme points of the duals of some other spaces of functions on X. The noncommutative analog of the Banach-Stone theorem is the folklore theorem that two unital C*-algebras are isomorphic if and only if they are completely isometric (i.e., isometric at all matrix levels). Mere isometry is not enough, as shown by the existence of a C*-algebra that is not isomorphic to its opposite algebra (which trivially has the same Banach space structure). == See also ==