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 theorem

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 \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

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

and

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

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 (and their Spec) as non-commutative spaces. This is the basis of the field of noncommutative geometry.

References