Ken Dykema, Uffe Haagerup
We show that Voiculescu's circular operator and, more generally, each circular free Poisson operator has a continuous family of invariant subspaces relative to the von Neumann algebra it generates. The proof relies on upper triangular random matrix models and consequent realizations of these operators as upper triangular matrices of free random variables.
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The DT-operators are introduced, one for every pair (μ,c) consisting of a compactly supported Borel probability measure μon the complex plane and a constant c>0. These are operators on Hilbert space that are defined as limits in *-moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II_1-factor, whose isomorphism class depends only on the number and sizes of a…
Random matrices have their roots in multivariate analysis in statistics, and since Wigner's pioneering work in 1955, they have been a very important tool in mathematical physics. In functional analysis, random matrices and random structures have in the last two decades been used to construct Banach spaces with surprising properties. After Voiculescu in 1990--1991 used random matrices to classification problems for von Neumann algebras, they have played a key role in von Neumann algebra theory. In this lecture we will discuss some new applications of random matrices to operator algebra theory, …
We show that every R-diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to find the Brown measure of x and to find a new conceptual proof that Voiculescu's S-transform is multiplicative. Our considerations base on a new concept of R-diagonality with amalgamation, for which we give several equivalent characterizations.
$B$-valued semi-circular system and the free Poincaré inequality
本文通过$B$-值自由概率的庞加莱不等式刻画了$B$-值半圆系统，推广了Biane定理，并证明Voiculescu关于$B$-值自由庞加莱不等式的猜想不完全成立。
Let T be the quasi-nilpotent DT-operator. By use of Voiculescu's amalgamated R-transform we compute the moments of $(T-λ1)^*(T-λ1)$, where $λ\in \mathbb C$, and the Brown-measure of $T+\sqrtε Y$, where Y is a circular element *-free from T for $ε>0$. Moreover we give a new proof of Śniady's formula for the moments $τ(((T^*)^k T^k)^n)$ for $k,n\in \mathbb N$.
We show that if A is a Hilbert-space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra vN(A) that is generated by A, is independent of the representation of vN(A), thought of as an abstract W^*-algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B-circular operators as a special case of Speicher's B-Gaussian operators in free probability theory, and we prove several results about …
We present a Hilbert space approach to the limit joint *-distributions of complex independent Gaussian random matrices. For that purpose, we use a suitably defined family of creation and annihilation operators living in some direct integral of Hilbert spaces. These operators are decomposed in terms of continuous circular systems of operators acting between the fibers of the considered Hilbert space direct integral. In the case of square matrices with i.i.d. entries, we obtain the circular operators of Voiculescu, whereas in the case of upper-triangular matrices with i.i.d. entries, we obtain t…
In the process of developing the theory of free probability and free entropy, Voiculescu introduced in 1991 a random matrix model for a free semicircular system. Since then, random matrices have played a key role in von Neumann algebra theory (cf. [V8], [V9]). The main result of this paper is the following extension of Voiculescu's random matrix result: Let X_1^(n),...,X_r^(n) be a system of r stochastically independent n by n Gaussian self-adjoint random matrices as in Voiculescu's random matrix paper [V4], and let (x_1,...,x_r) be a semi-circular system in a C*-probability space. Then for ev…
In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement that their two-dimensional $R$-transform (or free cumulants) have a special diagonal form. We show that the class of such $R$-diagonal pairs has an absorption property under nested multiplication of free pairs. This implies that in the polar decomposition of such an element the polar part and the absolute value are free. Our calculations are based on combin…
The Brown measure of a sum of two free random variables, one of which is triangular elliptic
本文研究了三角椭圆算子与自由随机变量之和的 Brown 测度，通过前推测度公式证明其密度绝对连续且有界，并将算子值从属函数推广至无界算子。
We prove the operator space Grothendieck inequality for bilinear forms on subspaces of noncommutative $L_p$-spaces with $2<p<\infty$. One of our results states that given a map $u: E\to F^*$, where $E, F\subset L_p(M)$ ($2<p<\infty$, $M$ being a von Neumann algebra), $u$ is completely bounded iff $u$ factors through a direct sum of a $p$-column space and a $p$-row space. We also obtain several operator space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative $L_p$-space ($2<p<\infty$) with values in a $q$-column space for every $q\in [p'…
Brown measure of the sum of an elliptic operator and a free random variable in a finite von Neumann algebra
本文研究有限冯·诺依曼代数中任意算子与自由椭圆算子之和的布朗测度，通过 Hermitian 约化和从属函数给出极限分布公式，并将结果推广到扭曲椭圆算子及 R-对角算子情形。
We prove a Strong Haagerup inequality with operator coefficients. If for an integer d, H_d denotes the subspace of the von Neumann algebra of a free group F_I spanned by the words of length d in the generators (but not their inverses), then we provide in this paper an explicit upper bound on the norm on M_n(H_d), which improves and generalizes previous results by Kemp-Speicher (in the scalar case) and Buchholz and Parcet-Pisier (in the non-holomorphic setting). Namely the norm of an element of the form $\sum_{i=(i_1,..., i_d)} a_i \otimes λ(g_{i_1} ... g_{i_d})$ is less than $4^5 \sqrt e (\|M_…
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