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1. 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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2. 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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3. Unitary groups and augmented Cuntz semigroups of separable simple Z-Stable C∗-algebras

   https://doi.org/10.1142/S0129167X22500185  

   Lin, Huaxin (February 2022, International journal of mathematics)

   Let A be a separable simple exact Z-stable C∗-algebra. We show that the unitary group of \tilde{A} has the cancellation property. If A has continuous scale then the Cuntz semigroup of A has strict comparison property and a weak cancellation property. Let C be a 1-dimensional noncommutative CW complex with K1(C) = {0}. Suppose that λ : Cu∼(C) → Cu∼(A) is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms φn : C → A such that limn→∞ Cu∼(φn) = λ. This result leads to the proof that every separable amenable simple C∗-algebra in the UCT class has rationally generalized tracial rank at most one. 

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4. 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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5. 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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