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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

(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,^[1] while the extension to compact Hausdorff spaces is due to Stone.^[2] 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.

[1] Théorème 3 of Banach, Stefan (1932). Théorie des opérations linéaires. Warszawa: Instytut Matematyczny Polskiej Akademii Nauk. p. 170.

[2] 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.

[1]

[2]

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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 Harvey Stone|Marshall Stone]].
+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 from the algebra of scalars (the bounded continuous functions on the space). In modern language, this is the commutative case of the [[spectrum of a C*-algebra]], and the Banach–Stone theorem can be seen as a functional analysis analog of the connection between a ring ''R'' and the [[spectrum of a ring]] Spec(''R'') in [[algebraic geometry]].
+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 of the theorem==
+==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>.
-For a topological space ''X'', let ''C''<sub>b</sub>(''X'';&nbsp;'''R''') denote the [[normed space|normed vector space]] of continuous, [[real number|real-valued]], [[bounded function]]s ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;'''R''' equipped with the [[supremum norm]] ‖·‖<sub>∞</sub>. This is an algebra, called the ''algebra of scalars'', under pointwise multiplication of functions. For a [[compact space]] ''X'', ''C''<sub>b</sub>(''X'';&nbsp;'''R''') is the same as ''C''(''X'';&nbsp;'''R'''), the space of all continuous functions ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;'''R'''. The algebra of scalars is a functional analysis analog of the ring of [[regular function]]s in algebraic geometry, there denoted <math>\mathcal{O}_X</math>.
-Let ''X'' and ''Y'' be [[compact space|compact]], [[Hausdorff space]]s and let ''T''&nbsp;:&nbsp;''C''(''X'';&nbsp;'''R''')&nbsp;→&nbsp;''C''(''Y'';&nbsp;'''R''') be a [[surjective function|surjective]] [[linear isometry]]. Then there exists a [[homeomorphism]] ''&phi;''&nbsp;:&nbsp;''Y''&nbsp;→&nbsp;''X'' and ''g''&nbsp;∈&nbsp;''C''(''Y'';&nbsp;'''R''') with
+Given compact Hausdorff spaces ''X'' and ''Y'', suppose ''T''&nbsp;:&nbsp;''C''(''X'')&nbsp;→&nbsp;''C''(''Y'') is a [[surjective function|surjective]] [[linear isometry]]. Then there exists a [[homeomorphism]] ''&phi;''&nbsp;:&nbsp;''Y''&nbsp;→&nbsp;''X'' and a function ''g''&nbsp;∈&nbsp;''C''(''Y'') with
 :<math>| g(y) | = 1 \mbox{ for all } y \in Y</math>
+such that
-and
-:<math>(T f) (y) = g(y) f(\varphi(y)) \mbox{ for all } y \in Y, f \in C(X; \mathbf{R}).</math>
+:<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}}</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.
+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'';&nbsp;''E'') onto ''C''(''Y'';&nbsp;''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''.
-More significantly, the Banach–Stone theorem suggests the philosophy that one can replace a ''space'' (a geometric notion) by an ''algebra'', with no loss. Reversing this, it suggests that one can consider algebraic objects, even if they do not come from a geometric object, as a kind of "algebra of scalars". In this vein, any ''commutative'' [[C*-algebra]] is the algebra of scalars on a Hausdorff space. Thus one may consider ''non''commutative C*-algebras (and their Spec) as non-commutative spaces. This is the basis of the field of [[noncommutative geometry]].
+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}}
-==References==
 * {{cite journal
 | last = Araujo
@@ Line 34: / Line 42: @@
 | mr = 2242851
 }}
+* {{Banach Théorie des Opérations Linéaires}} <!-- {{sfn | Banach | 1932 | p=}} -->
+{{Functional analysis}}
+{{Banach spaces}}
 {{DEFAULTSORT:Banach-Stone theorem}}
-[[Category:Continuous mappings]]
+[[Category:Theory of continuous functions]]
 [[Category:Operator theory]]
 [[Category:Theorems in functional analysis]]

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