OAI.NaimarkZFC.main

The theorem states that, for every universe level choice, there exists a type A carrying a C*-algebra structure (over ℂ) with the following five properties. First, A is simple in the sense that it is nontrivial and its only closed two-sided ideals are {0} and A itself. Second, A is not finite-dimensional as a complex vector space. Third, there is a continuous linear functional τ : A → ℂ that is a faithful tracial state: τ has norm 1, τ(aa) is a nonnegative real number for every a, τ(ab) = τ(ba) for all a and b, and τ(aa) > 0 whenever a ≠ 0. Fourth, A has a unique irreducible representation up to unitary equivalence: whenever π and ρ are nonzero star-homomorphisms (non-unital algebra homomorphisms preserving star) from A into the bounded operators on complete complex inner product spaces H and K, taken in the stated universe levels, and each has no closed invariant subspace other than 0 and the whole space, there is a linear isometric isomorphism U : H → K with U(π(a)x) = ρ(a)(Ux) for all a and x. Fifth, for every complete complex inner product space H in the stated universe, A is not isomorphic to the algebra of compact operators on H, meaning there is no injective star-homomorphism φ : A → B(H) whose range is exactly the set of compact operators on H.