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 ${\mathcal{P}}$.

FREE ENTROPY

2002 ◽  
Vol 34 (3) ◽  
pp. 257-278 ◽  
Author(s):  
DAN VOICULESCU
Keyword(s):  
Probability Theory ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Von Neumann Algebras ◽  
Matrix Theory ◽  
Free Probability ◽  
Open Problems ◽  
Von Neumann ◽  
Free Probability Theory ◽  
Free Entropy

Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

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 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

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.

A UNIFIED APPROACH TO GENERALIZED CONDITIONAL EXPECTATIONS OPERATOR VALUED WEIGHTS AND RADON–NIKODYM DERIVATIVES ON VON NEUMANN ALGEBRAS

2002 ◽  
Vol 05 (03) ◽  
pp. 409-419
Author(s):  
STEFANO CAVALLARO ◽  
CARLO CECCHINI
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Unified Approach ◽  
Von Neumann ◽  
Canonical Map ◽  
Conditional Expectations

Given two von Neumann algebras, ℳ and [Formula: see text] with [Formula: see text], and two normal semifinite faithful weights, φ and ψ on ℳ and [Formula: see text] respectively, we define a canonical map from {b ∈ ℳ+ | φ(b)< ∞} to the set of positive forms on the Hilbert space of the GNS representation of [Formula: see text] associated to ψ. We show that generalized conditional expectations, operator valued weights and Radon–Nikodym derivatives on von Neumann algebras can be obtained from particular cases of this canonical map.

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.

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