I was thinking about the Gelfand-Naimark theorem asserting the isometric \* isomorphism between a commutative C\* algebra (with unit) A and the C\* algebra of continuous complex-valued 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. 
Gelfand-Naimark theorem has a non-commutative analogue, which is based on the so-called 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?



Gelfand-Naimark from the category-theoretic point of view
The proper analogue is rather based on the characterization of the state space of a unital C\*-algebra found in (sorry about the self-advertisement) E. Alfsen, H. Hanche-Olsen 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 3-balls, 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.


