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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. K-Theory and the Universal Coefficient Theorem for Simple Separable Exact C*-Algebras Not Isomorphic to Their Opposites

   https://doi.org/10.1093/imrn/rnac358  

   Phillips, N Christopher; Viola, Maria Grazia (March 2023, International Mathematics Research Notices)

   Abstract We construct uncountably many mutually nonisomorphic simple separable stably finite unital exact C*-algebras that are not isomorphic to their opposite algebras. In particular, we prove that there are uncountably many possibilities for the $$K_0$$-group, the $$K_1$$-group, and the tracial state space of such an algebra. We show that these C*-algebras satisfy the Universal Coefficient Theorem, which is new even for the already known example of an exact C*-algebra nonisomorphic to its opposite algebra produced in an earlier work. 

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3. 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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4. Coupling capacity in C*-algebras

   https://doi.org/10.1017/prm.2023.81  

   Skalski, Adam; Todorov, Ivan G; Turowska, Lyudmila (February 2025, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further, we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras, our parameters are related to quantitative versions of Arveson's null set theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalize quantum versions of Strassen's theorem. We show that in the latter case our parameters can detect maximal entanglement and separability. 

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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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