Trace spaces of counterexamples to Naimark's problem

Abstract

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. For any such counterexample 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 here assuming diamond. In particular, we adapt the Akemann-Weaver construction to 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.

Nonseparable CCR algebras

“…A programme analogous to one given by Question 5.1 was pursued in [22] and [21]. In these papers the authors studied the variety of the invariants of counterexamples to Naimark's Problem (using Jensen's diamond).…”

A Rigid Hyperfinite Type II1 Factor

“…Notably, these obstructions are 'irreversible' in the sense that β cannot be extended to any further hyperfinite extension. This should be contrasted to the 'fleeting' obstructions used in [AW04], [FH17], and [Vac18] where at each step of the construction one had to take care of all objects captured in the earlier stages of the construction.…”

“…While the set theoretic machinery we use here is similar to the one used in those papers (albeit somewhat simplified), the operator algebraic techniques turn out to be very different in nature. The results in [AW04,FH17,Vac18] rely on studying the action of outer automorphisms and antiautomorphisms on the pure states of a separable C * -algebra, and use in an essential way results due to Kishimoto ([Kis81]) and work of Kishimoto, Ozawa, and Sakai ( [KOS03]) about the homogeneity of the pure state space of separable C * -algebras. Beyond the fact that pure states of von Neumann algebras are generally not normal, the homogeneity result of Kishimoto-Ozawa-Sakai breaks down for non-separable C * -algebras, and in particular for type II 1 -factors ([KOS03, Remark 2.3]).…”

“…The Akemann-Weaver construction was subsequently adapted in [FH17] to construct a simple nuclear C * -algebra which is not isomorphic to its opposite. Those techniques were further refined in [Vac18]. While the set theoretic machinery we use here is similar to the one used in those papers (albeit somewhat simplified), the operator algebraic techniques turn out to be very different in nature.…”

C*-algebras and the Uncountable: A Systematic Study of the Combinatorics of the Uncountable in the Noncommutative Framework

“…Some elementary observations on the structural properties of these C * -algebras seem to suggest that the trace space (a classical C * -algebraic invariant) of a counterexample is trivial. We prove that this is not the case and we show that, assuming ♦, almost every trace space of a separable C * -algebra also occurs as the trace space of a counterexample (see also [4]).…”