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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. Tracial approximation in simple $C*$-algebras

   https://doi.org/10.4153/S0008414X21000158  

   Fu, X.; Lin, H. (February 2021, Canadian journal of mathematics)

   We revisit the notion of tracial approximation for unital simple C*-algebras. We show that a unital simple separable in nite dimensional C*-algebra A is asymptotically tracially in the class of C-algebras with nite nuclear dimension if and only if A is asymptotically tracially in the class of nuclear Z-stable C-algebras. 1 

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