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

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 ${\displaystyle {\mathcal {O}}_{X}}$.

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

${\displaystyle |g(y)|=1{\mbox{ for all }}y\in Y}$

and

${\displaystyle (Tf)(y)=g(y)f(\varphi (y)){\mbox{ for all }}y\in Y,f\in C(X;\mathbf {R} ).}$

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 ${\displaystyle T-T(0)}$ is a linear isometry.

Generalizations

The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. 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 noncommutative C*-algebras (or rather their Spec) as non-commutative spaces. This is the basis of the field of noncommutative geometry.

See also

• Banach space – Normed vector space that is complete

References

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.

• 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). 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11.CS1 maint: ref=harv (link)