# 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 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.

## Statement of the theoremEdit

For a topological space *X*, let *C*_{b}(*X*; **R**) denote the normed vector space of continuous, real-valued, bounded functions *f* : *X* → **R** equipped with the supremum norm ‖·‖_{∞}. This is an algebra, called the *algebra of scalars*, under pointwise multiplication of functions. For a compact space *X*, *C*_{b}(*X*; **R**) is the same as *C*(*X*; **R**), the space of all continuous functions *f* : *X* → **R**. The algebra of scalars is a functional analysis analog of the ring of regular functions in algebraic geometry, there denoted .

Let *X* and *Y* be compact, Hausdorff spaces and let *T* : *C*(*X*; **R**) → *C*(*Y*; **R**) be a surjective linear isometry. Then there exists a homeomorphism *φ* : *Y* → *X* and *g* ∈ *C*(*Y*; **R**) with

and

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 is a linear isometry.

## GeneralizationsEdit

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.

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.

## ReferencesEdit

- Araujo, Jesús (2006). "The noncompact Banach–Stone theorem".
*Journal of Operator Theory*.**55**(2): 285–294. ISSN 0379-4024. MR 2242851.

**^**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.