  A counterexample to Naimark's problem is a $C^\ast$-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.
