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1. A classification of finite simple amenable Z-stable C∗-algebras, I: C*-algebras with generalized tracial rank one

   Gong, G. ; Lin, Huaxin ; Niu, Z. ( December 2020 , C. R. Math. Rep. Acad. Sci. Canada)

   Elliott, G.A. (Ed.)

   A class of C*-algebras, to be called those of generalized tracial rank one, is introduced. A second class of unital simple separable amenable C*-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable Z-stable C*- algebras. A number of results toward the classification of the second class are presented including an isomorphism theorem for a special sub-class of the first class, leading to the general classification of all unital simple C*- algebras with rational generalized tracial rank one in Part II. 

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2. Extensions of C *-Algebras by a Small Ideal

   https://doi.org/10.1093/imrn/rnac115  

   Lin, Huaxin ; Ng, PingWong ( May 2022 , International Mathematics Research Notices)

   Abstract We classify all essential extensions of the form $$ \begin{align*} &0 \rightarrow {\mathcal{W}} \rightarrow {D} \rightarrow A \rightarrow 0,\end{align*}$$where ${\mathcal {W}}$ is the unique separable simple C*-algebra with a unique tracial state, which is $KK$-contractible and has finite nuclear dimension, and $A$ is a separable amenable ${\mathcal {W}}$-embeddable C*-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable $C^*$-algebras, which have only one proper closed ideal ${\mathcal {W}}.$ 

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3. A classification of finite simple amenable Z-stable C*-algebras, II: C*-algebras with rational generalized tracial rank one

   Gong, G. ; Lin, Huaxin ; Niu, Z. ( December 2020 , C. R. Math. Rep. Acad. Sci. Canada)

   Elliott, G.A. (Ed.)

   A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possi- ble values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C∗-algebras which satisfy the UCT and have finite rational tracial rank. 

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4. The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

   https://doi.org/10.1016/j.geomphys.2020.103861  

   Elliott, G.A ; Gong, G. ; Lin, Huaxin ; Niu, Z. ( December 2020 , Journal of geometry and physics)

   null (Ed.)

   The class of simple separable KK-contractible (KK-equivalent to \{0\} ) C*-algebra s which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras A\otimes K is classifiable, where A is a simple separable C*-algebra with finite nuclear dimension and is the simple inductive limit of Razak algebras with unique trace, which is bounded 

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5. Complexity rank for C*-algebras

   Jaime, Arturo ; Willett, Rufus ( January 2023 , Münster journal of mathematics)

   Cuntz, Joachim (Ed.)

   Complexity rank for C*-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most n if you can repeatedly cut the C∗-algebra in half at most n times, and end up with something finite-dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as “two-colored local finite-dimensionality”. We first show that, for separable, unital, and simple C*-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear C*-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero K-theory groups. However, we also show using a K-theoretic obstruction (torsion in K1) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg–Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it equals one when the K1-group is torsion-free, and equals two otherwise. 

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