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(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, Cb(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 .
The case where X and Y are compact metric spaces is due to Banach, while the extension to compact Hausdorff spaces is due to Stone. 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.
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 (and their Spec) as non-commutative spaces. This is the basis of the field of noncommutative geometry.
- Araujo, Jesús (2006). "The noncompact Banach–Stone theorem". Journal of Operator Theory. 55 (2): 285–294. ISSN 0379-4024. MR 2242851.