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Banach–Stone theorem

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{{context|date=May 2025}}

{{technical|date=May 2025}}

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In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.

In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space X from the Banach space structure of the space C(X) of continuous real- or complex-valued functions on X. If one is allowed to invoke the algebra structure of C(X) this is easy – we can identify X with the spectrum of C(X), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space C(X)*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering X from the extreme points of the unit ball of C(X)*.

Statement

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,Théorème 3 of {{cite book |last1=Banach |first1=Stefan |title=Théorie des opérations linéaires |date=1932 |publisher=Instytut Matematyczny Polskiej Akademii Nauk |location=Warszawa |page=170}} while the extension to compact Hausdorff spaces is due to Stone.Theorem 83 of {{cite journal |last1=Stone |first1=Marshall |title=Applications of the Theory of Boolean Rings to General Topology |journal=Transactions of the American Mathematical Society |date=1937 |volume=41 |issue=3 |pages=375–481 |doi=10.2307/1989788|doi-access=free |jstor=1989788 }} 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

  • {{annotated link|Banach space}}

References

{{reflist}}

  • {{cite journal

| last = Araujo

| first = Jesús

| title = The noncompact Banach–Stone theorem

| journal = Journal of Operator Theory

| volume = 55

| year = 2006

| issue = 2

| pages = 285–294

| issn = 0379-4024

| mr = 2242851

}}

  • {{Banach Théorie des Opérations Linéaires}}

{{Functional analysis}}

{{Banach spaces}}

{{DEFAULTSORT:Banach-Stone theorem}}

Category:Theory of continuous functions

Category:Operator theory

Category:Theorems in functional analysis