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1. 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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2. Simple stably projectionless C*-algebras with generalized tracial rank one

   https://doi.org/Volume 14, Issue 1, 2020, pp. 251–347 DOI: 10.4171/JNCG/367  

   Elliott, G. A. ; Gong, G ; Lin, Huaxin ; Niu, Z. ( May 2020 , Journal of noncommutative geometry)

   We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some struc- tural questions concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras. 

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3. Nonamenable simple 𝐶*-algebras with tracial approximation

   https://doi.org/https://doi.org/10.1017/fms.2021.79  

   Fu, X. ; Lin, Huaxin ( January 2022 , Forum mathematicum)

   We construct two types of unital separable simple 𝐶∗-algebras: 𝐴𝐶1 𝑧 and 𝐴𝐶2 𝑧 , one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely, 𝐴𝐶𝑖 𝑧 has a unique tracial state,  𝐾0  𝐴𝐶𝑖 𝑧  , 𝐾0  𝐴𝐶𝑖 𝑧  + ,  1 𝐴𝐶𝑖 𝑧  = (Z, Z+, 1), and 𝐾1  𝐴𝐶𝑖 𝑧  = {0} (𝑖 = 1, 2). We show that 𝐴𝐶𝑖 𝑧 (𝑖 = 1, 2) is essentially tracially in the class of separable 𝒵-stable 𝐶∗-algebras of nuclear dimension 1. 𝐴𝐶𝑖 𝑧 has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) 𝐶∗-algebras which are essentially tracially in the class of simple separable nuclear𝒵-stable 𝐶∗-algebras, and the models exhaust all possible weakly unperforated Elliott invariants.We also discuss some basic properties of essential tracial approximation. 1. 

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4. On classification of non-unital amenable simple C*-algebras, II

   https://doi.org/doi.org/10.1016  

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

   null (Ed.)

   We present a classification theorem for separable amenable simple stably projectionless C -algebras with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C -algebras in the class is denoted by Z0 which has a unique tracial state, K_0(Z_0) = Z and K1(Z_0) = {0}. Let A and B be two separable amenable simple C -algebras satisfying the UCT. We show that A ⊗ Z_0 = B ⊗ Z_0 if and only if Ell(A ⊗ Z_0 ) = Ell(B ⊗ Z_0 ). A class of simple separable C -algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C -algebras of the form A ⊗ Z_0 , where A is any finite separable simple amenable C -algebras. 

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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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