Gelfand–Naimark Theorem

The theorem that every commutative unital C*-algebra A is *-isomorphic to C(X), the algebra of continuous complex-valued functions on a compact Hausdorff space X (the space of characters or maximal ideals of A), via the Gelfand transform; it establishes a duality between the category of commutative unital C*-algebras and the category of compact Hausdorff spaces.

The theorem that every commutative C*-algebra is isometrically *-isomorphic to C0(X), the algebra of continuous complex-valued functions vanishing at infinity on a locally compact Hausdorff space X (its spectrum), via the Gelfand transform, realizing a duality between spaces and commutative operator algebras.