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1. Quasinormalizers in crossed products of von Neumann algebras

   https://doi.org/10.1016/j.aim.2024.109535  

   Bannon, Jon P; Cameron, Jan; Chifan, Ionuţ; Mukherjee, Kunal; Smith, Roger; Wiggins, Alan (May 2024, Advances in Mathematics)

   We study the relationship between the dynamics of the action $$\alpha$$ of a discrete group $$G$$ on a von Neumann algebra $$M$$, and structural properties of the associated crossed product inclusion $$L(G) \subseteq M \rtimes_\alpha G$$, and its intermediate subalgebras. This continues a thread of research originating in classical structural results for ergodic actions of discrete, abelian groups on probability spaces. A key tool in the setting of a noncommutative dynamical system is the set of quasinormalizers for an inclusion of von Neumann algebras. We show that the von Neumann algebra generated by the quasinormalizers captures analytical properties of the inclusion $$L(G) \subseteq M \rtimes_\alpha G$$ such as the Haagerup Approximation Property, and is essential to capturing ``almost periodic" behavior in the underlying dynamical system. Our von Neumann algebraic point of view yields a new description of the Furstenberg-Zimmer distal tower for an ergodic action on a probability space, and we establish new versions of the Furstenberg-Zimmer structure theorem for general, tracial $W^*$-dynamical systems. We present a number of examples contrasting the noncommutative and classical settings which also build on previous work concerning singular inclusions of finite von Neumann algebras. 

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2. Cartan subalgebras in von Neumann algebras associated with graph product groups

   https://doi.org/10.4171/GGD/756  

   Chifan, Ionuţ; Kunnawalkam_Elayavalli, Srivatsav (April 2024, Groups, Geometry, and Dynamics)

   We completely classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces. 

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3. 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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4. A quantization of coarse spaces and uniform Roe algebras

   https://doi.org/10.2140/pjm.2025.338.163  

   Braga, Bruno M; Eisner, Joseph; Sherman, David (January 2025, Pacific Journal of Mathematics)

   We propose a quantization of coarse spaces and uniform Roe algebras. The objects are based on the quantum relations introduced by N. Weaver and require the choice of a represented von Neumann algebra. In the case of the diagonal inclusion ell_infty(X) subset B(ell_2(X)), they reduce to the usual constructions. Quantum metric spaces furnish natural examples parallel to the classical setting, but we provide other examples that are not inspired by metric considerations, including the new class of support expansion C*-algebras. We also work out the basic theory for maps between quantum coarse spaces and their consequences for quantum uniform Roe algebras. 

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5. Boundary algebras of the Kitaev quantum double model

   https://doi.org/10.1063/5.0212164  

   Chuah, Chian Yeong; Hungar, Brett; Kawagoe, Kyle; Penneys, David; Tomba, Mario; Wallick, Daniel; Wei, Shuqi (October 2024, Journal of Mathematical Physics)

   The recent article by Jones et al. [arXiv:2307.12552 (2023)] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev’s Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev’s Quantum Double model for a finite group G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial G-symmetry protected topological phase (G-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3 + 1)D Quantum Double model associated to an Abelian group. 

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