I was thinking about the GelfandNaimark theorem asserting the isometric * isomorphism between a commutative C* algebra (with unit) A and the C* algebra of continuous complexvalued functions on its spectrum (via the Gelfand transform). Explicitly: let spec(A) denote the spectrum of A and C(X) the algebra of complex continuous functions on X. Then spec and C define contravariant functors from commC* alg to CompHausTop, which (correct me if i'm wrong) establish an equivalence between the two categories. GelfandNaimark theorem has a noncommutative analogue, which is based on the socalled GNS construction and which shows that every non commutative C* algebras has a faithful isometric *representation on a Hilbert space H. In this case I can't see an analogue of the preceding equivalence of categories, which is equally meaningful. Does it exist?

The proper analogue is rather based on the characterization of the state space of a unital C*algebra found in (sorry about the selfadvertisement) E. Alfsen, H. HancheOlsen and F.W. Shultz: State Spaces of C∗Algebras, Acta Math. 144 (1980) 267–305. So the category to replace CompHausTop would be the category of state spaces equipped with orientations on their facial 3balls, and whose morphisms are certain affine maps between these compact convex sets. In this context, a compact Hausdorff space X is represented by the set of probability Baire measures on X, which is in particular a Choquet simplex. 

