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1. Multivariate trace inequalities, p-fidelity, and universal recovery beyond tracial settings

   https://doi.org/10.1063/5.0066653  

   Junge, Marius ; LaRacuente, Nicholas ( December 2022 , Journal of Mathematical Physics)

   Trace inequalities are general techniques with many applications in quantum information theory, often replacing the classical functional calculus in noncommutative settings. The physics of quantum field theory and holography, however, motivates entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki Lp spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras. In particular, we show this for the generalized Araki–Lieb–Thirring and Golden–Thompson inequalities from the work of Sutter et al. [Commun. Math. Phys. 352(1), 37 (2017)]. Then, using the Haagerup approximation method, we prove a general von Neumann algebra version of universal recovery map corrections to the data processing inequality for relative entropy. We also show subharmonicity of a logarithmic p-fidelity of recovery. Furthermore, we prove that the non-decrease of relative entropy is equivalent to the existence of an L1-isometry implementing the channel on both input states. 

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2. Brauer groups and Galois cohomology of commutative ring spectra

   https://doi.org/10.1112/S0010437X21007065  

   Gepner, David ; Lawson, Tyler ( June 2021 , Compositio Mathematica)

   null (Ed.)

   In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$ -categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$ , we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $\pi _0(E)$ , recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\mathbb {Z}/8\times \mathbb {Z}/2$ . Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$ -algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$ -algebra with the same coefficient ring as $\mathrm {End}_{KO}(KU)$ . This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$ , previously studied by Mathew and Stojanoska. 

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3. 3D Mirror Symmetry for Instanton Moduli Spaces

   https://doi.org/10.1007/s00220-023-04831-5  

   Koroteev, Peter ; Zeitlin, Anton M. ( September 2023 , Communications in Mathematical Physics)

   We prove that the Hilbert scheme ofkpoints on$${\mathbb {C}}^2$$$C^2$($$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$) is self-dual under three-dimensional mirror symmetry using methods of geometry and integrability. Namely, we demonstrate that the corresponding quantum equivariant K-theory is invariant upon interchanging its Kähler and equivariant parameters as well as inverting the weight of the$${\mathbb {C}}^\times _\hbar $$$C_{ħ}^{\times }$-action. First, we find a two-parameter family$$X_{k,l}$$$X_{k,l}$of self-mirror quiver varieties of type A and study their quantum K-theory algebras. The desired quantum K-theory of$$\hbox {Hilb}^k[{\mathbb {C}}^2]$$${\text{Hilb}}^k\left[C^2\right]$is obtained via direct limit$$l\longrightarrow \infty $$$l⟶\infty$and by imposing certain periodic boundary conditions on the quiver data. Throughout the proof, we employ the quantum/classical (q-Langlands) correspondence between XXZ Bethe Ansatz equations and spaces of twisted$$\hbar $$$ħ$-opers. In the end, we propose the 3d mirror dual for the moduli spaces of torsion-free rank-Nsheaves on$${\mathbb {P}}^2$$$P^2$with the help of a different (three-parametric) family of type A quiver varieties with known mirror dual.

    

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4. Duality for Optimal Couplings in Free Probability

   https://doi.org/10.1007/s00220-022-04480-0  

   Gangbo, Wilfrid ; Jekel, David ; Nam, Kyeongsik ; Shlyakhtenko, Dimitri ( December 2022 , Communications in Mathematical Physics)

   Abstract We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on $${\mathbb {R}}^m$$ R m are replaced by non-commutative laws of m -tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative $$L^2$$ L 2 -Wasserstein distance using a new type of convex functions. As a consequence, we show that if ( X ,  Y ) is a pair of optimally coupled m -tuples of non-commutative random variables in a tracial $$\mathrm {W}^*$$ W ∗ -algebra $$\mathcal {A}$$ A , then $$\mathrm {W}^*((1 - t)X + tY) = \mathrm {W}^*(X,Y)$$ W ∗ ( ( 1 - t ) X + t Y ) = W ∗ ( X , Y ) for all $$t \in (0,1)$$ t ∈ ( 0 , 1 ) . Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m -tuples is not separable with respect to the Wasserstein distance for $$m > 1$$ m > 1 . 

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5. Banach-Valued Multilinear Singular Integrals with Modulation Invariance

   https://doi.org/10.1093/imrn/rnaa234  

   Di Plinio, Francesco ; Li, Kangwei ; Martikainen, Henri ; Vuorinen, Emil ( September 2020 , International Mathematics Research Notices)

   Abstract We prove that the class of trilinear multiplier forms with singularity over a one-dimensional subspace, including the bilinear Hilbert transform, admits bounded $L^p$-extension to triples of intermediate $\operatorname{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\operatorname{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\textrm{UMD}$-valued setting. This is then employed to obtain appropriate single-tree estimates by appealing to the $\textrm{UMD}$-valued bound for bilinear Calderón–Zygmund operators recently obtained by the same authors. 

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