I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued functions on its spectrum (via the Gelfand transform). Explicitly: let $\operatorname{spec}(\mathcal A)$ 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?

## 1 Answer

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.