We study Voiculescu's microstate free entropy for a single non-selfadjoint random variable. The main result is that certain additional constraints on eigenvalues of microstates do not change the free entropy. Our tool is the method of random regularization of Brown measure which was studied recently by Haagerup and the author. As a simple application we present an upper bound for the free entropy of a single non--selfadjoint operator in terms of its Brown measure and the second moment. We furthermore show that this inequality becomes an equality for a class of DT-operators which was introduced recently by Dykema and Haagerup.
We show that the Voiculescu-Brown entropy of a noncommutative toral automorphism arising from a matrix S in GL(d,Z) is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu-Brown entropy with respect to single unitaries. In particular we show that if S has no roots of unity as eigenvalues then the local Voiculescu-Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital ve…
A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are "chaotic." While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of C*-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author's talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottaw…
Certain classes of automorphisms of recued amalgamated free products of C*-algebras are shown to have Brown-Voiculescu topological entropy zero. Also, for automorphisms of exact C*-algebras, the Connes-Narnhofer-Thirring entropy is shown to be bounded above by the Brown-Voiculescu entropy. These facts are applied to generalize Stormer's result about entropy of automorphisms of the II_1-factor of a free group.
In this paper we generalize Brown's spectral distribution measure to a large class of unbounded operators affiliated with a finite von Neumann algebra. Moreover, we compute the Brown measure of all unbounded R-diagonal operators in this class. As a particular case, we determine the Brown measure of z=xy^{-1}, where (x,y) is a circular system in the sense of Voiculescu, and we prove that for all positive integers n, z^n is in L^p(M) iff 0<p< 2/(n+1).
We consider the topological entropy of state space and quasi-state space homeomorphisms induced from C*-algebra automorphisms. Our main result asserts that, for automorphisms of separable exact C*-algebras, zero Voiculescu-Brown entropy implies zero topological entropy on the quasi-state space (and also more generally on the entire unit ball of the dual). As an application we obtain a simple description of the topological Pinsker algebra in terms of local Voiculescu-Brown entropy.
We define a notion of dynamical pressure at a self-adjoint element for a contractive completely positive self-map of an exact C*-algebra which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev-Stormer unital-nuclear pressure. A variational inequality bounding the pressure below by the free energies with respect to the Sauvageot-Thouvenot entropy is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C*-…
We use Voiculescu's concept of free probability to construct a completely isomorphic embedding of the operator space OH in the predual of a von Neumann algebra. We analyze the properties of this embedding and determine the operator space projection constant of OH_n: It is of the order (n/1+ln n)^{1/2} The lower estimate is a recent result of Pisier and Shlyakhtenko that improves an estimate of order 1/(1+ln n) of the author. The additional factor 1/(1+ln n)^{1/2} indicates that the operator space OH_n behaves differently than its classical counterpart l_2^n. We give an application of this form…
Biane proved the free analog of the logarithmic Sobolev inequality for probability measures on the real line by means of random matrix approximation procedure. We show that the same method can be applied to reprove Biane and Voiculescu's free analog of Talagrand's transportation cost inequality for measures on the real line. Furthermore, we prove the free analogs of the logarithmic Sobolev inequality and the transportation cost inequality for measures on 1-dimensional torus as well by extending the method to special unitary random matrices.
We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our me…
We give a method to obtain, from Voiculescu's inequality, norm estimates for sums of free variables with amalgamation in general fully symmetric spaces. We use these estimates to interpolate the Burkholder inequalities for non commutative martingales. The method is also applicable to other similar settings. In that spirit, we improve known results on the non commutative Johnson-Schechtman inequalities and recover Khintchine inequalities associated to free groups.
This paper is part of a series aiming at proving that the $\limsup$ and $\liminf$ variants of Voiculescu's free entropy coincide. This is based on a Laplace principle (implying a large deviation principle) for hermitian brownian motion on $[0,1]$. In the current paper, we show that microstates free entropy $χ(X_1,...,X_m)$ and non-microstate free entropy $χ^*(X_1,...,X_m)$ coincide for self-adjoint variables $(X_1,...,X_m)$ satisfying a Schwinger-Dyson equation for subquadratic, bounded below, strictly convex potentials with Lipschitz derivative sufficiently approximable by non-commutative pol…
We define a classical probability analogue of Voiculescu's free entropy dimension that we shall call the classical probability entropy dimension of a probability measure on $\mathbb{R}^n$. We show that the classical probability entropy dimension of a measure is related with diverse other notions of dimension. First, it can be viewed as a kind of fractal dimension. Second, if one extends Bochner's inequalities to a measure by requiring that microstates around this measure asymptotically satisfy the classical Bochner's inequalities, then we show that the classical probability entropy dimension c…
Well-known subadditivity results for positive operators (of Brown-Kosaki and Rotfeld/Ando-Zhan types) are extended to Hermitian and normal ones. Applications to Cartesian decomposition and block-matrices are given.
Let $\mathcal{A}$ denote the reduced amalgamated free product of a family $\mathsf{A}_1, \mathsf{A}_2, ..., \mathsf{A}_n$ of von Neumann algebras over a von Neumann subalgebra $\Be$ with respect to normal faithful conditional expectations $\Es_k: \mathsf{A}_k \to \Be$. We investigate the norm in $L_p(\Al)$ of homogeneous polynomials of a given degree $d$. We first generalize Voiculescu's inequality to arbitrary degree $d \ge 1$ and indices $1 \le p \le \infty$. This can be regarded as a free analogue of the classical Rosenthal inequality. Our second result is a length-reduction formula from wh…
By proving that certain free stochastic differential equations have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain $n$-tuples $X_{1},...,X_{n}$: we show that Abstract. By proving that certain free stochastic differential equations with analytic coefficients have stationary solutions, we give a lower estimate on the microstates free entropy dimension of certain n-tuples X_{1},...,X_{n}. In particular, we show that δ_{0}(X_{1},...,X_{n})\geq\dim_{M\bar{\otimes}M^{o}}V where M=W^{*}(X_{1},...,X_{n}) and V=\{(\partial(X_{1}),...,\partial(X_{n}))…
