Banach–Stone theorem

Banach–Stone theorem 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)*.

Contenuti 1 Dichiarazione 2 generalizzazioni 3 Guarda anche 4 References 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) insieme a {stile di visualizzazione |g(y)|=1{mbox{ per tutti }}yin y} tale che {stile di visualizzazione (Tf)(y)=g(y)f(varfi (y)){mbox{ per tutti }}yin y,fin C(X).} The case where X and Y are compact metric spaces is due to Banach,[1] while the extension to compact Hausdorff spaces is due to Stone.[2] Infatti, 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, e così {displaystyle 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. Per esempio, 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 (cioè., 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 Banach space – Normed vector space that is complete References ^ Théorème 3 of Banach, Stefano (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". Transazioni dell'American Mathematical Society. 41 (3): 375–481. doi:10.2307/1989788. Araujo, Jesús (2006). "The noncompact Banach–Stone theorem". Journal of Operator Theory. 55 (2): 285–294. ISSN 0379-4024. SIG 2242851.* Banach, Stefano (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in francese). vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archiviato dall'originale (PDF) Su 2014-01-11. Recuperato 2020-07-11. show vte Functional analysis (argomenti – glossario) show vte Banach space topics Categories: Continuous mappingsOperator theoryTheorems in functional analysis