Every classifiable simple C*-algebra has a Cartan subalgebra

Abstract

A. We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra. IClassification of C*-algebras has seen tremendous advances recently. In the unital case, the classification of unital separable simple nuclear Z-stable C*-algebras satisfying the UCT is by now complete. This is the culmination of work by many mathematicians. The reader may consult [26,36,22,14, 46] and the references therein. In the stably projectionless case, classification results are being developed (see [17,15,16,20,21]). It is expected that -once the stably projectionless case is settled -the final result will classify all separable simple nuclear Z-stable C*-algebras satisfying the UCT by their Elliott invariants. This class of C*-algebras is what we refer to as "classifiable C*-algebras".To complete these classification results, it is important to construct concrete models realizing all possible Elliott invariants by classifiable C*-algebras. Such models have been constructed -in the greatest possible generality -in [13] (see also [45] which covers special cases). In the stably finite unital case, the reader may also find such range results in [22], where the construction follows the ideas in [13] (with slight modifications, so that the models belong to the special class considered in [22]). In the stably projectionless case, models have been constructed in a slightly different way in [21] (again to belong to the special class of algebras considered) under the additional assumption of a trivial pairing between K-theory and traces.Recently, the notion of Cartan subalgebras in C*-algebras [27,38] has attracted attention, due to connections to topological dynamics [28,29,30] and the UCT question [3,4]. In particular the reformulation of the UCT question in [3,4] raises the following natural question (see [31, Question 5.9], [44, Question 16] and [5, Problems 1 and 2]): Question 1.1. Which classifiable C*-algebras have Cartan subalgebras?By [27,38], we can equally well ask for groupoid models for classifiable C*-algebras. In the purely infinite case, groupoid models and hence Cartan subalgebras have been constructed in [43] (see also [31, § 5]). For special classes of stably finite unital C*-algebras, groupoid models have been constructed in [10, 37] using topological dynamical systems. Using a new approach, the goal of this paper is to answer Question 1.1 by constructing Cartan subalgebras in all the C*-algebra models constructed in [13,22,21], covering all classifiable stably finite C*-algebras, in particular in all classifiable unital C*-algebras. Generally speaking, Cartan subalgebras allow us to introduce ideas from geometry and dynamical systems to the study of C*-algebras. More concretely, in view of [3,4], we expect that our answer to Question 1.1 will lead to progress on the UCT question.The following two theorems are the main results of this paper. The r...

Cartan subalgebras and the UCT problem, II

“…This is the first general result on Cartan subalgebras in inductive limit C*-algebras, and turns out to have applications going far beyond the scope of the present paper. For instance, among other things, the second author has used this new tool to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras in [30].…”

Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

“…Until now, it was not clear whether any non-trivial functoriality also applies to the Weyl groupoid representation. In fact, it was not uncommon to hear C*-algebraists claim that there are no natural morphisms for étale groupoids, a myth that was finally dispelled in [AM18], [AG19] and [Li20]. Also, the original Weyl groupoid construction is perhaps not as general as one might like, dealing only with effective groupoids and their line bundles.…”

Simplicity of twisted C*-algebras of Deaconu--Renault groupoids

“…The study of twisted C*-algebras associated to continuous groupoid 2-cocycles dates back to Renault's seminal work [31]. They serve both as a very flexible C*-algebraic framework for modelling dynamical systems, and as a source of tractable models for classifiable C*-algebras [29,16,8,27]. So it is important to be able to determine when a given twisted groupoid C*-algebra is simple; but this is in general a complicated question.…”