Banach–Stone theorem - Misplaced Pages

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

(Tf)(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).

References

Théorème 3 of Banach, Stefan (1932). Théorie des opérations linéaires. Warszawa: Instytut Matematyczny Polskiej Akademii Nauk. p. 170.
Theorem 83 of Stone, Marshall (1937). "Applications of the Theory of Boolean Rings to General Topology". Transactions of the American Mathematical Society. 41 (3): 375–481. doi:10.2307/1989788. JSTOR 1989788.

Araujo, Jesús (2006). "The noncompact Banach–Stone theorem". Journal of Operator Theory. 55 (2): 285–294. ISSN 0379-4024. MR 2242851.
Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.

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