 Commutative Gelfand--Naimark theorem

Every commutative unital C-star algebra $A$ is isometrically star-isomorphic to $C({\Phi }_A)$ through its Gelfand transform, where ${\Phi }_A$ is its character space of an algebra. Thus abstract algebra elements can be treated as continuous functions on a compact Hausdorff space.

This is the isometric commutative C-star specialization of the Gelfand representation.