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- W2963165563 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." @default.
- W2963165563 created "2019-07-30" @default.
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- W2963165563 date "2018-11-01" @default.
- W2963165563 modified "2023-09-29" @default.
- W2963165563 title "Trace spaces of counterexamples to Naimark's problem" @default.
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- W2963165563 doi "https://doi.org/10.1016/j.jfa.2018.06.018" @default.
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