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1. 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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2. 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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3. Quantum Cuntz-Krieger algebras

   https://doi.org/10.1090/btran/88  

   Brannan, Michael ; Eifler, Kari ; Voigt, Christian ; Weber, Moritz ( October 2022 , Transactions of the American Mathematical Society, Series B)

   Motivated by the theory of Cuntz-Krieger algebras we define and study C ∗ C^\ast -algebras associated to directed quantum graphs. For classical graphs the C ∗ C^\ast -algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to K K KK -equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these C ∗ C^\ast -algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of K K KK -theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general. 

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4. 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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5. 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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