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1. Crossed products and $$\mathrm{C}^{*}$$-covers of semi-Dirichlet operator algebras

   https://doi.org/10.4171/DM/1007  

   Humeniuk, Adam; Katsoulis, Elias G; Ramsey, Christopher (June 2025, Documenta Mathematica)

   Winter, W (Ed.)

   In this paper, we show that the semi-Dirichlet\mathrm{C}^{*}-covers of a semi-Dirichlet operator algebra form a complete lattice, establishing that there is a maximal semi-Dirichlet\mathrm{C}^{*}-cover. Given an operator algebra dynamical system we prove a dilation theory that shows that the full crossed product is isomorphic to the relative full crossed product with respect to this maximal semi-Dirichlet cover. In this way, we can show that every semi-Dirichlet dynamical system has a semi-Dirichlet full crossed product. 

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2. Wreath-like products of groups and their von Neumann algebras I: W^∗-superrigidity

   https://doi.org/10.4007/annals.2023.198.3.6  

   Chifan, Ionuţ; Ioana, Adrian; Osin, Denis; Sun, Bin (November 2023, Annals of Mathematics)

   We introduce a new class of groups called {\it wreath-like products}. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many wreath-like products have Kazhdan's property (T). In this paper, we prove that any group $$G$$ in a natural family of wreath-like products with property (T) is W$^*$-superrigid: the group von Neumann algebra $$\text{L}(G)$$ remembers the isomorphism class of $$G$$. This allows us to provide the first examples (in fact, $$2^{\aleph_0}$$ pairwise non-isomorphic examples) of W$^*$-superrigid groups with property (T). 

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3. Globally irreducible Weyl modules for quantum groups

   Garibaldi, Skip; Nakano, Daniel K.; Guralnick, Robert M. (January 2018, Springer Proceedings in Mathematics and Statistics (Geometric and Topological Aspects of the Representation Theory of Finite Groups), Springer-Verlag)

   The authors proved that a Weyl module for a simple algebraic group is irreducible over every field if and only if the module is isomorphic to the adjoint representation for $$E_{8}$$ or its highest weight is minuscule. In this paper, we prove an analogous criteria for irreducibility of Weyl modules over the quantum group $$U_{\zeta}({{\mathfrak g}})$$ where $${\mathfrak g}$$ is a complex simple Lie algebra and $$\zeta$$ ranges over roots of unity. 

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4. Crossed Product Equivalence of Quantum Automorphism Groups of Finite Dimensional C*-Algebras

   https://doi.org/10.1093/imrn/rnad060  

   Brannan, Michael; Elzinga, Floris; Harris, Samuel J; Yamashita, Makoto (March 2023, International Mathematics Research Notices)

   Abstract We compare the algebras of the quantum automorphism group of finite-dimensional C$$^\ast $$-algebra $$B$$, which includes the quantum permutation group $$S_N^+$$, where $$N = \dim B$$. We show that matrix amplification and crossed products by trace-preserving actions by a finite Abelian group $$\Gamma $$ lead to isomorphic $$\ast $$-algebras. This allows us to transfer various properties such as inner unitarity, Connes embeddability, and strong $$1$$-boundedness between the various algebras associated with these quantum groups. 

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5. 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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