Simple nuclear C *-algebras not isomorphic to their opposites

Abstract

We show that it is consistent with Zermelo-Fraenkel set theory with the axiom of choice (ZFC) that there is a simple nuclear nonseparable C * -algebra, which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O 2 or of the canonical anticommutation relations (CAR) algebra.C*-algebras | Jensen's diamond | opposite algebra | Naimark's problem | Glimm dichotomy

Trace spaces of counterexamples to Naimark's problem

“…For the reader's convenience we quickly recall the construction here. All omitted details can be found in [FH17], where a continuous model-theoretic equivalent version of ♦, more suitable for working with C * -algebras, is introduced.…”

“…Theorem 2.1 is applied in the proof of the following lemma. The algebra B is A ⋊ α Z, where α ∈ Aut(A) is provided by theorem 2.1 for two sequences of inequivalent pure states which depend on X , Y and E. Back to the construction in [FH17], given A β , A β+1 = A β ⋊ α Z is obtained by an application of lemma 2.2, where X , Y and E are chosen accordingly to ♦.…”

“…The starting point for the proof of theorem 2 is the techniques developed in [AW04] and [FH17], which both rely on an application of the Kishimoto-Ozawa-Sakai theorem in [KOS03]. As we shall see in the next section, the main effort to prove theorem 2 will be to refine the results in [KOS03] in order to have a better control on the trace space of the crossed product obtained from the automorphism provided by the Kishimoto-Ozawa-Sakai theorem (see theorem A in section 2).…”

A Rigid Hyperfinite Type II1 Factor

“…As in [FH17] and [Far19, §11.2], our construction will utilize codes for metric structures 1 but the coding used here is somewhat simplified. Suppose d is a metric on an ordinal θ of diameter 2 and A is its metric completion.…”

“…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]).…”