Gelfand-Naimark theorem

(See @strocchi2008introduction page 28)
An (abstract) commutative c-star algebra $\mathcal{A}$ (with identity) is isometrically isomorphic to the algebra of complex continuous functions $C(X)$ on a compact Hausdorff topological space $X$, where $X$ is intrinsically defined as the Gelfand spectrum of $\mathcal{A}$.
If the algebra does not have an identity one has a locally compact Hausdorff space.

Sketch of the construction: there is an sketch of the construction in the case of probabilistic spaces here.