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1. The closure ordering conjecture on local Arthur packets of classical groups

   https://doi.org/10.1515/crelle-2025-0012  

   Hazeltine, Alexander; Liu, Baiying; Lo, Chi-Heng; Zhang, Qing (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove the closure ordering conjecture on the local 𝐿-parameters of representations in local Arthur packets of ${\mathrm{G}}_n={\mathrm{Sp}}_{2n},{\mathrm{SO}}_{2n+1}$\mathrm{G}_{n}=\mathrm{Sp}_{2n},\mathrm{SO}_{2n+1}over a non-Archimedean local field of characteristic zero.Precisely, given any representation 𝜋 in a local Arthur packet ${\mathrm{\Pi }}_{\psi }$\Pi_{\psi}, the closure of the local 𝐿-parameter of 𝜋 in the Vogan variety must contain the local 𝐿-parameter corresponding to 𝜓.This conjecture reveals a geometric nature of local Arthur packets and is inspired by the work of Adams, Barbasch and Vogan, and the work of Cunningham, Fiori, Moussaoui, Mracek and Xu, on ABV-packets.As an application, for general quasi-split connected reductive groups, we show that the closure ordering conjecture implies the enhanced Shahidi conjecture, under certain reasonable assumptions.This provides a framework towards the enhanced Shahidi conjecture in general.We verify these assumptions for ${\mathrm{G}}_n$\mathrm{G}_{n}, hence give a new proof of the enhanced Shahidi conjecture.Finally, we show that local Arthur packets cannot be fully contained in other ones, which is in contrast to the situation over Archimedean local fields and is of independent interest. 

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2. Block perturbation of symplectic matrices in Williamson’s theorem

   Babu, Gajendra; Mishra, Hemant K (August 2023, Canadian Mathematical Bulletin)

   Williamson's theorem states that for any 2n×2n real positive definite matrix A, there exists a 2n×2n real symplectic matrix S such that STAS=D⊕D, where D is an n×n diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of A. Let H be any 2n×2n real symmetric matrix such that the perturbed matrix A+H is also positive definite. In this paper, we show that any symplectic matrix S̃ diagonalizing A+H in Williamson's theorem is of the form S̃ =SQ+(‖H‖), where Q is a 2n×2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S̃ and S can be chosen so that ‖S̃ −S‖=(‖H‖). Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017]. 

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3. 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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4. Fell’s absorption principle for semigroup operator algebras

   https://doi.org/10.4171/JNCG/566  

   Katsoulis, Elias G (April 2025, Journal of Noncommutative Geometry)

   Cuntz, J (Ed.)

   Fell’s absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica-covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper, we explain how to extend Fell’s absorption principle to an arbitrary submonoidPof a groupGby using an enhanced version of the left regular representation. Li’s semigroup\mathrm{C}^{*}-algebra\mathrm{C}^{*}_{s}(P)and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for\mathrm{C}^{*}_{s}(P)but we also resolve open problems and obtain very transparent proofs of earlier results. In particular, we address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell’s absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouâtre and Dor-On on the residual finite dimensionality of certain\mathrm{C}^{*}-algebras associated with such tensor algebras. As another application, we give yet another proof for the existence of a\mathrm{C}^{*}-algebra which is co-universal for equivariant, Li-covariant representations of a submonoidPof a groupG. 

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5. Covering entropy for types in tracial W^*-algebras

   https://doi.org/10.4115/jla.2023.15.2  

   Jekel, David (June 2023, Journal of Logic and Analysis)

   We study embeddings of tracial $$\mathrm{W}^*$$-algebras into a ultraproduct of matrix algebras through an amalgamation of free probabilistic and model-theoretic techniques.  Jung implicitly and Hayes explicitly defined \emph{$$1$$-bounded entropy} through the asymptotic covering numbers of Voiculescu's microstate spaces, that is, spaces of matrix tuples $$(X_1^{(N)},X_2^{(N)},\dots)$$ having approximately the same $$*$$-moments as the generators $$(X_1,X_2,\dots)$$ of a given tracial $$\mathrm{W}^*$$-algebra.  We study the analogous covering entropy for microstate spaces defined through formulas that use suprema and infima, not only $$*$$-algebra operations and the trace | formulas such as arise in the model theory of tracial $$\mathrm{W}^*$$-algebras initiated by Farah, Hart, and Sherman.  By relating the new theory with the original $$1$$-bounded entropy, we show that if $$\mathcal{M}$$ is a separable tracial $$\mathrm{W}^*$$-algebra with $$h(\cN:\cM) \geq 0$$, then there exists an embedding of $$\cM$$ into a matrix ultraproduct $$\cQ = \prod_{n \to \cU} M_n(\C)$$ such that $$h(\cN:\cQ)$$ is arbitrarily close to $$h(\cN:\cM)$$.  We deduce that if all embeddings of $$\cM$$ into $$\cQ$$ are automorphically equivalent, then $$\cM$$ is strongly $$1$$-bounded and in fact has $$h(\cM) \leq 0$$. 

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