Localizing the Elliott conjecture at strongly self-absorbing C*-algebras

By W. Winter. 2007

132 citations recorded by the source corpus.

We formally introduce the concept of localizing the Elliott conjecture at a given strongly self-absorbing C*-algebra $D$; we also explain how the known classification theorems for nuclear C*-algebras fit into this concept. As a new result in this direction, we employ recent results of Lin to show that (under a mild K-theoretic condition) the class of separable, unital, simple C*-algebras with locally finite decomposition rank and UCT, and for which projections separate traces, satisfies the Elliott conjecture localized at the Jiang-Su algebra Z. Our main result is formulated in a more general way; this allows us to outline a strategy to possibly remove the trace space condition as well as the K-theory restriction entirely. When regarding both our result and the recent classification theorem of Elliott, Gong and Li as generalizations of the real rank zero case, the two approaches are perpendicular in a certain sense. The strategy to attack the general case aims at combining these two approaches. Our classification theorem covers simple ASH algebras for which projections separate traces (and the K-groups of which have finitely generated torsion part); it does, however, not at all depend on an inductive limit structure. Also, in the monotracial case it does not rely on the existence or absence of projections in any way. In fact, it is the first such result which, in a natural way, covers all known unital, separable, simple, nuclear and stably finite C*-algebras of real rank zero (the K-groups of which have finitely generated torsion part) as well as the (projectionless) Jiang-Su algebra itself.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ba6ef41e3ca86a4b2d6e008ddd50fe57a9d158db

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Citations in the corpus

The C∗-algebra of a minimal homeomorphism of zero mean dimension (2014, methodology): “Therefore it is covered by the classification theorem of [19], [5], [6].”
On the classification of simple amenable C*-algebras with finite decomposition rank (2015, methodology): “For TAI algebras, based on Winter’s deformation technique ([25] and [18]) and on [19], one has the following classification theorem.”
A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, methodology): “Once we overcame these new hurdles arising from the Winter approach, we were able to classify the class of all unital separable simple Z-stable C∗-algebras A which satisfy the UCT and whose tensor products with all UHF-algebras of infinite type are of tracial rank zero, by mean of the Elliott invari”
Completely positive maps of order zero (2009, methodology): “Order zero maps with finite-dimensional domains have been used in [11], [10], [6] and [5] as building blocks of noncommutative partitions of unity to define noncommutative versions of topological covering dimension; see [9] and [12] for related applications.”
Quasidiagonality of nuclear C*-algebras (2015, methodology): “iv preprints still under review. QUASIDIAGONALITY OF NUCLEAR C∗-ALGEBRAS 41 [62, Theorem 8.4.1]) and doesn’t use Theorem A. The monotracial situation follows from [50, Theorem 5.4] (which generalises [81]); since Theorem A makes quasidiagonality automatic, the tracial rank hypothesis required in [50”
Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “1]), allowing the application of the classification technique from [66], refined by Lin and Lin-Niu in [33, 34].”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, result): “This strategy of proving results for Z-stable algebras via the UHFstable case has its origins in [46].”
Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, background): “A class of unital simple amenable Z-stable C∗-algebras A whose tensor products with all UHF-algebras of infinite type are of tracial rank zero were shown to be classified by the Elliott invariant in [96] and [66].”