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 Cb(XR) 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, Cb(XR) is the same as C(XR), 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(XR) → C(YR) be a surjective linear isometry. Then there exists a homeomorphism φ : Y → X and g ∈ C(YR) 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(XE) onto C(YE) 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 (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.