Classifying crossed product C*-algebras

By W. Winter. 2013

48 citations recorded by the source corpus.

I combine recent results in the structure theory of nuclear C*-algebras and in topological dynamics to classify certain types of crossed products in terms of their Elliott invariants. In particular, transformation group C*-algebras associated to free minimal Z^d-actions on the Cantor set with compact space of ergodic measures are classified by their ordered K-theory. In fact, the respective statement holds for finite dimensional compact metrizable spaces, provided that projections of the crossed products separate tracial states. Moreover, C*-algebras associated to certain minimal homeomorphisms of odd dimensional spheres are only determined by their spaces of invariant Borel probability measures (without a condition on the space of ergodic measures). Finally, I show that for a large collection of classifiable C*-algebras, crossed products by Z^d-actions are generically again classifiable.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/275f8fb5d3741b45af6e96bcdd00883ee0b4f322

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. W. Winter
2. Z. Niu
3. G'abor Szab'o
4. Huaxin Lin
5. G. Elliott
6. G. Gong
7. Stuart White
8. Aaron Tikuisis

In Mathematical Physics

1. W. Winter
2. G'abor Szab'o
3. Z. Niu
4. G. Elliott
5. Huaxin Lin
6. David Kerr
7. Stuart White
8. G. Gong

In Statistical and Nonlinear Physics

1. G'abor Szab'o
2. W. Winter
3. Z. Niu
4. Stuart White
5. G. Gong
6. Aaron Tikuisis
7. J. Zacharias
8. G. Elliott

Citations in the corpus

• Rokhlin dimension for actions of residually finite groups (2014, background): “In view of the astonishing progess of C∗-algebra classification over the last few years [18, 13, 12, 73], which is largely driven by the discovery of various regularity properties (see for example [83, 81, 82, 59, 48, 36, 75, 65, 50, 84, 66, 5]), a natural and important question appears, which can b”
• The Rokhlin dimension of topological ℤm‐actions (2013, background): “tion of simple transformation group C∗-algebras of this type. This works under the prevalence of finite nuclear dimension and a technical condition involving the set of invariant ergodic measures (see [20]). An alternative immediate application is to combine the main results of [12] and [14] to obtai”
• On the classification of simple amenable C*-algebras with finite decomposition rank (2015, unknown): “ive maps Hn : A→ Csuch that lim n→∞ kHn(ab) −Hn(a)Hn(b)k = 0, a,b∈ A, and lim n→∞ sup{|τ◦ Hn(a)−ΓAff(a)(τ)| : τ∈ T(C)} = 0, a∈ A. It follows fromLemma 3.4 of [22](which used resultsobtained in [25]and [32]) that gTR(A⊗Q) ≤ 1. Since A∼= A⊗Q, Ais Z-stable and therefore it is in N1. 20 Theorem 1.1 follo”
• Nuclear dimension and $$\mathcal Z$$Z-stability (2014, unknown): “This leads to classification by K -theory ([44]; see [36,50] for the case of Zd actions).”
• The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “1 of [53] (the argument does not require the C*-algebra to be unital; the hypothesis of strict comparison is sufficient for the argument to proceed).”
• Quasidiagonality of nuclear C*-algebras (2015, background): “The main result of [31] (which in turn relies on [33, 59, 77] to reduce to the classification theorem of [36], a result which builds on a large body of work over decades) classifies separable, unital, simple, nuclear C∗-algebras with finite nuclear dimension (the noncommutative notion of covering di”