Andrea Vaccaro, University of Pisa/York University
"Trace spaces of Counterexamples to Naimark's Problem"
A counterexample to Naimark's problem is a C*-algebra that is not isomorphic to the algebra of compact operators on some Hilbert space, yet still has only one irreducible representation up to unitary equivalence. It is well-known that such algebras must be nonseparable, and in 2004 Akemann and Weaver used the diamond principle (a set theoretic axiom independent from ZFC) to give the first counterexamples. After a quick overview on the current progress on Naimark's problem, we undertake a general study of the trace spaces of unital counterexamples to Naimark's problem. For any such algebra A, the unitary group U(A) acts transitively on the pure states, which are the extreme points of the state space S(A). It is conceivable that this implies (as happens for finite-dimensional simplexes) that the action of U(A) on S(A) has at most one fixed point, i.e. A has at most one trace. We give a strong negative answer assuming diamond. Using a refinement of the Kishimoto-Ozawa-Sakai theorem on the transitivity of the action of automorphisms on the pure state space of a separable simple unital C*-algebra, we show that the trace space of a counterexample to Naimark's problem can be affinely homeomorphic to any metrizable Choquet simplex, and can also be nonseparable.
MC 5417