Classification of inductive limits of 1-dimensional NCCW complexes

By Leonel Robert. 2010

128 citations recorded by the source corpus.

A classification result is obtained for the C*-algebras that are (stably isomorphic to) inductive limits of 1-dimensional noncommutative CW complexes with trivial $K_1$-group. The classifying functor Cu is defined in terms of the Cuntz semigroup of the unitization of the algebra. For the simple C*-algebras covered by the classification, Cu reduces to the ordered $K_0$-group, the cone of traces, and the pairing between them. As an application of the classification, it is shown that the crossed products by a quasi-free action $O_2\rtimes_\lambda \mathbb{R}$ are all isomorphic for a dense set of positive irrational numbers $\lambda$.

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

View Paper: https://www.semanticscholar.org/paper/2b163a6aede3307f0032b39bce73600a502e68e6

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In Algebra and Number Theory

1. W. Winter
2. Z. Niu
3. Huaxin Lin
4. G. Elliott
5. G. Gong
6. Andrew S. Toms
7. Stuart White
8. Aaron Tikuisis

In Mathematical Physics

1. W. Winter
2. Z. Niu
3. G. Elliott
4. Huaxin Lin
5. Andrew S. Toms
6. Hannes Thiel
7. G. Gong
8. Stuart White

In Statistical and Nonlinear Physics

1. Z. Niu
2. G. Gong
3. Stuart White
4. Aaron Tikuisis
5. W. Winter
6. G. Elliott
7. Hannes Thiel
8. Huaxin Lin

Citations in the corpus

• On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “Applied to the canonical unital map Cu(C1) → Cu(C) ∼= Cu(A), Theorem 1 of [22] implies that there is a unital homomorphism ψ : C1 → A giving rise to this map, and in particular such that (2.”
• Tensor products and regularity properties of Cuntz semigroups (2014, background): “[76] Robert, Leonel, Classification of inductive limits of 1-dimensional NCCW complexes, Adv.”
• The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension (2016, background): “1 of [42], as A has stable rank one (by 11.”
• Quasidiagonality of nuclear C*-algebras (2015, background): “The main result of [29] (which in turn relies on [30, 55, 73] to reduce to the classification theorem of [32], 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”
• Pullbacks, $C(X)$-algebras, and their Cuntz semigroup (2011, methodology): “The classification of inductive limits of one-dimensional NCCW-complexes with trivial K1-group was carried out in [24] using the functor Cu∼ which is related to the functor Cu.”
• Classifying crossed product C*-algebras (2013, background): “Let be the induced map (cf. [22, Section 3]), i.e., then κ ∼ i ∈ Mor ( Cu ) .”