Dan Voiculescu

Limit laws for Random matrices and free products

In earlier articles we studied a kind of probability theory in the framework of operator algebras, with the tensor product replaced by the free product. We prove here that free random variables naturally arise as limits of random matrices and that Wigners semicircle law is a consequence of the central limit theorem for free random variables. In this way we obtain a non-commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra, Wigners law being given by the trace of the spectral measure of the selfadjoint component of this operator

The analogues of entropy and of Fisher's information measure in free probability theory. I

Analogues of the entropy and Fisher information measure for random variables in the context of free probability theory are introduced. Monotonicity properties and an analogue of the Cramer-Rao inequality are proved.

Addition of certain non-commuting random variables

Soit G le groupe libre non commutatif sur deux generateurs et soit U j (j=1,2) les unites dans l 2 (G) correspondant a la translation a gauche par les generateurs et on considere ξ∈l 2 (G) la fonction ξ(g)=δ g,e . Soient X j des operateurs de la forme X j =φ j (u j ). Les operateurs X j peuvent etre consideres comme des variables aleatoires de moments , ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques μ j , ou μ j (f)= . Avec ces conventions la distribution de X 1 +X 2 depend seulement des distributions de X 1 et X 2 . On explique cette relation

Dynamical approximation entropies and topological entropy in operator algebras

Dynamical entropy invariants, based on a general approximation approach are introduced for C*-and W*-algebra automorphisms. This includes a noncommutative extension of topological entropy.

Free analysis questions I: duality transform for the coalgebra of ∂X:B

We construct a duality transform for the coalgebra of the free difference quotient derivation-comultiplication of an operator with respect to a free algebra of scalars. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.

Free Probability Theory: Random Matrices and von Neumann Algebras

Independence in usual noncommutative probability theory (or in quantum physics) is based on tensor products. This lecture is about what happens if tensor products are replaced by free products. The theory one obtains is highly noncommutative: freely independent random variables do not commute in general. Also, at the level of groups, this means instead of ℤ n we will consider the noncommutative free group F(n) = ℤ* ⋯ *ℤ or, looking at the Cayler graphs, a lattice is replaced by a homogeneous tree.

Regularity Questions for Free Convolution

Free additive convolution is a binary operation on the set M of all probability measures on the real line R. This operation was first defined in [7] for measures with finite moments of all orders (in particular for compactly supported measures). Maassen [5] extended this operation to measures with finite variance, and the extension to arbitrary measures was done in [1]. The free convolution of µ,v ∈ M is denoted µ ⊞ v. Unlike classical convolution, free convolution is a highly nonlinear operation, and therefore it is not obvious that various regularity properties of µ (like absolute continuity, differentiability, etc.) should be passed on to µ ⊞ v. In some respects however free convolution has a stronger regularizing effect than classical convolution. It is our purpose in this paper to examine a few instances in which regularity properties of µ ⊞ v can be inferred. Some of the earlier results in this direction were only proved for measures with compact support. We will extend these results to general probability measures, giving as much of the technical detail as necessary. Among the new results, we show that there may be a loss of smoothness under free convolution. We also give a complete description of the atoms of a free convolution of probability measures.

Entropy of Bogoliubov automorphisms of the canonical anticommutation relations

We compute the entropyhωA(αU) in the sense of Connes, Narnhofer and Thirring of Bogoliubov automorphismsαU of the CAR-algebra with respect to invariant quasifree statesωA with 0≦A≦1 having pure point spectrum.

Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability theory (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannons entropy power inequality ([6, 1]) has an analogue for the free entropy χ(X) (Theorem 2.1).The free entropy, consistent with Boltzmanns formulaS=klogW, was defined via volumes of matricial microstates. Proving the free entropy power inequality naturally becomes a geometric question.Restricting the Minkowski sum of two sets means to specify the set of pairs of points which will be added. The relevant inequality, which holds when the set of addable points is sufficiently large, differs from the Brunn-Minkowski inequality by having the exponent 1/n replaced by 2/n. Its proof uses the rearrangement inequality of Brascamp-Lieb-Lüttinger ([2]). Besides the free entropy power inequality, note that the inequality for restricted Minkowski sums may also underlie the classical Shannon entropy power inequality (see 3.2 below).

s-Numbers of singular integrals for the invariance of absolutely continuous spectra in fractional dimensions

An important consequence of the Kato-Rosenblum theorem in abstract scattering theory [5, 81 is the invariance, up to unitary equivalence, of absolutely continuous spectra under trace-class perturbations. The extension of this fact to higher (integral) dimensions was given in [lo]. A careful inspection of trace-class scattering theory, reveals the crucial role of L2-boundedness of the Hilbert-transform in the one-dimensional case. In the n-dimensional generalization this is replaced by the asymptotics of Fourier-coefficients of (2, 1 )(x , ~ k c n 1~~ 11 2, ~~ ’ on the torus, or equivalently by the asymptotics of s-numbers of corresponding singular integrals. The general functional analysis machinery developed in [9, lo] for studying normed ideal perturbations of systems of Hilbert space operators reduces this kind of operator theory questions to questions about the existence or non-existence of quasicentral approximate units relative to normed ideals. The existence of quasicentral approximate units, in turn is related to commutator questions ([ 11-J; for related considerations in the quasidiagonality context, see [7] ). The natural operators to examine are then singular integral operators. In this paper we consider the fractional-dimensional case. The natural singular integral problem, in this case, is to estimate the s-numbers of singular integrals with kernels ,yjl;/lx~1’ in L’(p), where ~1 is a compactly supported probability measure on [w” of “fractional dimension” (like Hausdorff measure restricted to suitable compacta). The results we