Random matrices, amalgamated free products and subfactors of the von Neumann algebra of a free group, of noninteger index

1994 ◽  
Vol 115 (1) ◽  
pp. 347-389 ◽  
Author(s):  
Florin R�dulescu
Keyword(s):  
Random Matrices ◽  
Free Group ◽  
Von Neumann Algebra ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products

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 On spectra and Brown's spectral measures of elements in free products of matrix algebras

2008 ◽  
Vol 103 (1) ◽  
pp. 77
Author(s):  
Junsheng Fang ◽  
Don Hadwin ◽  
Xiujuan Ma
Keyword(s):  
Free Product ◽  
Von Neumann Algebra ◽  
Single Point ◽  
Diagonal Operator ◽  
Spectral Measures ◽  
Free Products ◽  
Hyperinvariant Subspace ◽  
Von Neumann ◽  
Matrix Algebras ◽  
Adjoint Operators

We compute spectra and Brown measures of some non self-adjoint operators in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$, the reduced free product von Neumann algebra of $M_2(\mathsf {C})$ with $M_2(\mathsf {C})$. Examples include $AB$ and $A+B$, where $A$ and $B$ are matrices in $(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})*1$ and $1*(M_2(\mathsf {C}), \frac{1}{2}\mathrm{Tr})$, respectively. We prove that $AB$ is an R-diagonal operator (in the sense of Nica and Speicher [12]) if and only if $\mathrm{Tr}(A)=\mathrm{Tr}(B)=0$. We show that if $X=AB$ or $X=A+B$ and $A,B$ are not scalar matrices, then the Brown measure of $X$ is not concentrated on a single point. By a theorem of Haagerup and Schultz [9], we obtain that if $X=AB$ or $X=A+B$ and $X\neq \lambda 1$, then $X$ has a nontrivial hyperinvariant subspace affiliated with $(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})*(M_2(\mathsf{C}), \frac{1}{2}\mathrm{Tr})$.

scholarly journals Anisotropic random walks on free products of cyclic groups, irreducible representations and idempotents of C*reg(G)

1992 ◽  
Vol 128 ◽  
pp. 95-120 ◽  
Author(s):  
Gabriella Kuhn
Keyword(s):  
Harmonic Analysis ◽  
Homogeneous Space ◽  
Free Product ◽  
Dual Space ◽  
Von Neumann Algebra ◽  
Point Of View ◽  
Irreducible Representations ◽  
Free Products ◽  
Von Neumann ◽  
Cyclic Groups

Let be the free product of q + 1 copies of Zn+1 and let denote its Cayley graph (with respect to aj, 1 ≤ j ≤ q + 1). We may think of G as a group acting on the “homogeneous space” , This point of view is inspired by the case of SL2(R) acting on the hyperbolic disk and is developed in [FT-P] [I-P] [FT-S] [S] (but see also [C]).Since G is a group we may investigate some classical topics: the full (reductive) C* algebra, its dual space, the regular Von Neumann algebra and so on. See [B] [P] [L] [V] and also [H]. These approaches give results pointing up the analogy between harmonic analysis on these groups and harmonic analysis on more classical objects.

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.

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.

scholarly journals On the functionTreH+itK

2002 ◽  
Vol 29 (7) ◽  
pp. 389-393 ◽  
Author(s):  
Mark Fannes ◽  
Dénes Petz
Keyword(s):  
Random Matrices ◽  
Von Neumann Algebra ◽  
Positive Definite ◽  
Von Neumann

We prove, for two free semicircularly distributed selfadjoint elementsaandbin a typeII1von Neumann algebra with faithful traceτ, that the functiont∈ℝ↦τ(exp(a+itb))is positive definite. This shows that the Bessis-Moussa-Vilani conjecture holds for large random matrices in an asymptotic sense.

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