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1. Embeddings of von Neumann algebras into uniform Roe algebras and quasi-local algebras

   https://doi.org/10.1016/j.jfa.2023.110186  

   Baudier, Florent P; Braga, Bruno M; Farah, Ilijas; Vignati, Alessandro; Willett, Rufus (January 2024, Journal of Functional Analysis)

   We study which von Neumann algebras can be embedded into uniform Roe algebras and quasi-local algebras associated with a uniformly locally finite metric space X. Under weak assumptions, these C*-algebras contain embedded copies of certain matrix algebras. We aim to show they cannot contain any other von Neumann algebras. One of our main results shows that the only embedded von Neumann algebras are the “obvious” ones. 

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2. On Gorenstein algebras of finite Cohen-Macaulay type: Dimer tree algebras and their skew group algebras

   https://doi.org/10.1016/j.jalgebra.2024.07.027  

   Schiffler, Ralf; Serhiyenko, Khrystyna (December 2024, Journal of Algebra)
3. Root of unity quantum cluster algebras and Cayley–Hamilton algebras

   https://doi.org/10.1090/tran/8904  

   Huang, Shengnan; Lê, Thang; Yakimov, Milen (October 2023, Transactions of the American Mathematical Society)

   We prove that large classes of algebras in the framework of root of unity quantum cluster algebras have the structures of maximal orders in central simple algebras and Cayley–Hamilton algebras in the sense of Procesi. We show that every root of unity upper quantum cluster algebra is a maximal order and obtain an explicit formula for its reduced trace. Under mild assumptions, inside each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra, such that the pair is a Cayley–Hamilton algebra; its fully Azumaya locus is shown to contain a copy of the underlying cluster A \mathcal {A} -variety. Both results are proved in the wider generality of intersections of mixed quantum tori over subcollections of seeds. Furthermore, we prove that all monomial subalgebras of root of unity quantum tori are Cayley–Hamilton algebras and classify those ones that are maximal orders. Arbitrary intersections of those over subsets of seeds are also proved to be Cayley–Hamilton algebras. Previous approaches to constructing maximal orders relied on filtration and homological methods. We use new methods based on cluster algebras. 

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4. q-Schur algebras corresponding to Hecke algebras of type B

   Lai, Chun-Ju; Nakano, Daniel K.; Xiang, Ziqing (January 2022, Transformation groups)
5. q-Schur algebras corresponding to Hecke algebras of type B

   https://doi.org/10.1007/s00031-020-09628-7  

   Lai, C.J; Nakano, Daniel K.; Xiang, Ziqing (October 2020, Transformation groups)

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