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?