scholarly journals Compressions of free products of von Neumann algebras

2000 ◽  
Vol 316 (1) ◽  
pp. 61-82 ◽  
Author(s):  
Kenneth J. Dykema ◽  
Florin Radulescu
Keyword(s):  
Von Neumann Algebras ◽  
Free Products ◽  
Von Neumann

W*-superrigidity for wreath products with groups having positive first ℓ2-Betti number

2015 ◽  
Vol 26 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Mihaita Berbec
Keyword(s):  
Wreath Product ◽  
Betti Number ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Wreath Products ◽  
Hyperbolic Groups ◽  
Free Products ◽  
Von Neumann ◽  
Product Group ◽  
Group Von Neumann Algebra

In [M. Berbec and S. Vaes, W*-superrigidity for group von Neumann algebras of left–right wreath products, Proc. London Math. Soc.108 (2014) 1116–1152] we have proven that, for all hyperbolic groups and for all nontrivial free products Γ, the left–right wreath product group 𝒢 ≔ (ℤ/2ℤ)(Γ) ⋊ (Γ × Γ) is W*-superrigid, in the sense that its group von Neumann algebra L𝒢 completely remembers the group 𝒢. In this paper, we extend this result to other classes of countable groups. More precisely, we prove that for weakly amenable groups Γ having positive first ℓ2-Betti number, the same wreath product group 𝒢 is W*-superrigid.

scholarly journals Complete Gradient Estimates of Quantum Markov Semigroups

2021 ◽  
Vol 387 (2) ◽  
pp. 761-791
Author(s):  
Melchior Wirth ◽  
Haonan Zhang
Keyword(s):  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Wasserstein Distance ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
Markov Semigroups ◽  
Free Products ◽  
Von Neumann ◽  
Symmetric Quantum ◽  
Poisson Type

AbstractIn this article we introduce a complete gradient estimate for symmetric quantum Markov semigroups on von Neumann algebras equipped with a normal faithful tracial state, which implies semi-convexity of the entropy with respect to the recently introduced noncommutative 2-Wasserstein distance. We show that this complete gradient estimate is stable under tensor products and free products and establish its validity for a number of examples. As an application we prove a complete modified logarithmic Sobolev inequality with optimal constant for Poisson-type semigroups on free group factors.

scholarly journals Radial multipliers on amalgamated free products of II1-factors

2014 ◽  
Vol 25 (03) ◽  
pp. 1450026
Author(s):  
Sören Möller
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Hankel Matrix ◽  
Trace Class ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Radial Multipliers ◽  
Completely Bounded Map

Let ℳi be a family of II1-factors, containing a common II1-subfactor 𝒩, such that [ℳi : 𝒩] ∈ ℕ0 for all i. Furthermore, let ϕ: ℕ0 → ℂ. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the amalgamated free product of the ℳi with amalgamation over 𝒩, which acts as a radial multiplier. Hereby, we extend a result of Haagerup and the author for radial multipliers on reduced free products of unital C*- and von Neumann algebras.

scholarly journals A Random Matrix Approach to Absorption in Free Products

2020 ◽  
Author(s):  
Ben Hayes ◽  
David Jekel ◽  
Brent Nelson ◽  
Thomas Sinclair
Keyword(s):  
Random Matrix ◽  
Von Neumann Algebras ◽  
Concentration Of Measure ◽  
Unified Approach ◽  
Free Products ◽  
Von Neumann ◽  
Free Entropy ◽  
Famous Result ◽  
Random Matrix Models ◽  
Entropy Zero

Abstract This paper gives a free entropy theoretic perspective on amenable absorption results for free products of tracial von Neumann algebras. In particular, we give the 1st free entropy proof of Popa’s famous result that the generator MASA in a free group factor is maximal amenable, and we partially recover Houdayer’s results on amenable absorption and Gamma stability. Moreover, we give a unified approach to all these results using $1$-bounded entropy. We show that if ${\mathcal{M}} = {\mathcal{P}} * {\mathcal{Q}}$, then ${\mathcal{P}}$ absorbs any subalgebra of ${\mathcal{M}}$ that intersects it diffusely and that has $1$-bounded entropy zero (which includes amenable and property Gamma algebras as well as many others). In fact, for a subalgebra ${\mathcal{P}} \leq{\mathcal{M}}$ to have this absorption property, it suffices for ${\mathcal{M}}$ to admit random matrix models that have exponential concentration of measure and that “simulate” the conditional expectation onto ${\mathcal{P}}$.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

2021 ◽  
pp. 1-54 ◽  
Author(s):  
Michael Brannan ◽  
Li Gao ◽  
Marius Junge
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Beltrami Operator ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Quantum Tori ◽  
Bounded Below

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

E0-semigroups of free Araki-Woods factors

2017 ◽  
Vol 28 (10) ◽  
pp. 1750075
Author(s):  
Panchugopal Bikram ◽  
Kunal Mukherjee
Keyword(s):  
Large Class ◽  
Von Neumann Algebras ◽  
Type I ◽  
Free Products ◽  
Von Neumann

Analyzing endomorphisms and shift E0-semigroups of free products of von Neumann algebras, we construct uncountably many pairwise cocycle inequivalent E0-semigroups on a large class of free Araki-Woods factors. These E0-semigroups are equimodular but not modularly extendable. We also produce conjugate but not cocycle equivalent E0-semigroups on non-type I factors.

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