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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. 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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3. 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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4. Lebesgue Decomposition of Non-Commutative Measures

   https://doi.org/10.1093/imrn/rnaa231  

   Jury, Michael T; Martin, Robert T (September 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $$C^{\ast }-$$algebra, the $$C^{\ast }-$$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $$C^{\ast }-$$algebras; any *−representation of the Cuntz–Toeplitz $$C^{\ast }-$$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory. 

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5. Explicit isomorphisms of quaternion algebras over quadratic global fields

   https://doi.org/10.1007/s40993-022-00380-3  

   Csahók, Tímea; Kutas, Péter; Montessinos, Mickaël; Zábrádi, Gergely (December 2022, Research in Number Theory)

   Abstract Let L be a separable quadratic extension of either $${\mathbb {Q}}$$ Q or $${\mathbb {F}}_q(t)$$ F q ( t ) . We exhibit efficient algorithms for finding isomorphisms between quaternion algebras over L . Our techniques are based on computing maximal one-sided ideals of the corestriction of a central simple L -algebra. 

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