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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?

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up vote 14 down vote accepted

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.

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