Commutative C*-algebras and Gelfand duality

A character is a nonzero complex-linear multiplicative map from A to the complex numbers. In the unital C*-setting it sends the identity to one, respects adjoints, and is automatically continuous. Evaluation at a point is the basic example for continuous functions. A general state need not be multiplicative: averaging values at two points normally changes the value of a product differently from multiplying the two averages. That distinction separates a definite classical point from a probability distribution over points.

The character space X carries the weak-star topology inherited from the dual space, meaning that each function chi maps to chi(a) is continuous for fixed a. For a nonzero commutative unital C*-algebra, X is nonempty compact Hausdorff. These properties are theorems using algebraic maximal ideals, spectral theory, and compactness, not consequences of merely writing down a set of functionals. The analogous character space of a noncommutative algebra may fail to capture its information; full matrix algebras of size at least two have no characters.

Gelfand duality states that the transform is onto all continuous complex-valued functions on X, not merely an embedding into some arbitrary function collection. It preserves multiplication, addition, adjoints, and norm. For each a, its spectrum is exactly the set of values chi(a). A single element need not distinguish every character, but the complete algebra does. The topology is encoded by simultaneous continuity of these evaluations, so reconstructing only an untopologized point set would discard part of the result.

For C(X), every character is evaluation at one point of X. One route to this fact studies the common zero set of a maximal ideal and uses compactness to rule out an empty common zero set. In the nonunital version, every commutative C*-algebra is C₀(X) for a locally compact Hausdorff space. A constant-one function belongs exactly in the compact case. We restrict the following map discussion to compact spaces and unital homomorphisms to avoid hiding the additional properness and nondegeneracy conditions that arise in broader categories.

If a device turns an input x into an output q(x), an observable f on outputs becomes the observable f composed with q on inputs. Thus a continuous map from X to Y naturally produces a unital *-homomorphism from C(Y) to C(X). Conversely, every such homomorphism comes from one continuous map. Composition reverses in exactly the way required by this pullback rule. This is the categorical meaning of duality, beyond the reconstruction of individual objects.

If q is surjective, distinct output functions remain distinct after pullback, so the algebra map is injective. If q identifies two input points, every pulled-back function has equal values at them, so it cannot describe all possible input observables. For a finite example these statements are transparent coordinate constraints. They also explain why general states correspond to regular Borel probability measures rather than spatial maps to a point: integration is positive and normalized, but it is usually not a multiplicative algebra homomorphism.