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In [[mathematics]], the '''Banach–Stone theorem''' is a classical result in the theory of [[continuous function]]s on [[topological space]]s, named after the [[mathematician]]s [[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 a C*-algebra|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 space|compact]] [[Hausdorff space]] ''X'', let ''C''(''X'') denote the [[Banach space]] of continuous real- or complex-valued [[function (mathematics)|functions]] on ''X'', equipped with the [[supremum norm]] ‖·‖<sub>∞</sub>.

Given compact Hausdorff spaces ''X'' and ''Y'', suppose ''T'' : ''C''(''X'') → ''C''(''Y'') is a [[surjective function|surjective]] [[linear isometry]]. Then there exists a [[homeomorphism]] ''φ'' : ''Y'' → ''X'' and a function ''g'' ∈ ''C''(''Y'') with

:<math>| g(y) | = 1 \mbox{ for all } y \in Y</math>

such that

:<math>(T f) (y) = g(y) f(\varphi(y)) \mbox{ for all } y \in Y, f \in C(X).</math>

The case where ''X'' and ''Y'' are compact [[metric spaces]] is due to Banach,<ref>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}}</ref> while the extension to compact Hausdorff spaces is due to Stone.<ref>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 }}</ref> 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 <math>T - T(0)</math> 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 [[Multipliers and centralizers (Banach spaces)|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]]

Banach–Stone theorem - Revision history

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