On a class of free Lévy laws related to a regression problem

The free Meixner laws arise as the distributions of orthogonal polynomials with constant-coefficient recursions. We show that these are the laws of the free pairs of random variables which have linear regressions and quadratic conditional variances when conditioned with respect to their sum. We apply this result to describe free Lévy processes with quadratic conditional variances, and to prove a converse implication related to asymptotic freeness of random Wishart matrices.     

Two-state free Brownian motions

In a two-state free probability space (A,φ,ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function Rφ,ψ(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states φ, ψ are normal, and φ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to φ=ψ), these processes only exist for finite time.     

The non-microstates free entropy dimension of DT-operators

Dykema and Haagerup introduced the class of DT-operators (Amer. J. Math. 126 (2004) 121-189) and also showed that every DT-operator generate (J. Funct. Anal. 209 (2004) 332-366), the von Neumann algebra generated by the free group on two generators. In this paper, we prove that Voiculescu's non-microstates free entropy dimension is 2 for all DT-operators.     

r-Qsym is free over Sym

Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x₁,x₂,…,xn are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-Quasi-Symmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2) (2003) 217-263].     

Free Jacobi Process

In this paper, we define and study two parameters dependent free processes (λ,θ) called free Jacobi, obtained as the limit of its matrix counterpart when the size of the matrix goes to infinity. The main result we derive is a free SDE analogous to that satisfied in the matrix setting, derived under injectivity assumptions. Once we did, we examine a particular case for which the spectral measure is explicit and does not depend on time (stationary). This allows us to determine easily the parameters range ensuring our injectivity requirements so that our result applies. Then, we show that under an additional condition of invertibility at time t=0, this range extends to the general setting. To proceed, we set a recurrence formula for the moments of the process via free stochastic calculus.     

An Alternative to Free Entropy for Free Group Factors

In this paper we define a very simple invariant η(V^-) for a k-tuple V^-of unitaries in a finite factor von Neumann algebra, and we show how this invariant can replace free entropy in many of the important applications. We also introduce a notion of metric free entropy and some related concepts.We include proofs, using η, of the theorems of Liming Ge and of D. Voiculescu, respectively, on the primeness of and on the absence of Cartan snbalgebras in the noncommutative free group factors.     

On the S-transform over a Banach algebra

The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued Banach noncommutative probability space, for an arbitrary unital complex Banach algebra B. Also, a new proof of the additivity of the R-transform in this setting is given.     

On Some Ideals in the Free Nonassociative Algebra

The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals.     

Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

Heat Conduction with Flux Condition on a Free Patch

A new free boundary or free patch problem for the heat equation is presented. In the problem a nonlinear heat flux condition is prescribed on a free portion of the boundary, the patch, the position of which depends on the solution. The existence of a weak solution is established using the theory of set-valued pseudomonotone operators.     

On classical analogues of free entropy dimension

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 Rn. 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 controls the rate of increase of optimal constants in Bochner's inequality for a measure regularized by convolution with the Gaussian law as the regularization is removed. We introduce a free analogue of the Bochner inequality and study the related free entropy dimension quantity. We show that it is greater or equal to the non-microstates free entropy dimension.     

Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on ${\mathbb{R}}^n$ by the map $U?{\left(\mathrm{?}U\right)}_{\mathrm{\#}}{e}^{?U}dx$. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure $e^{?U}dx$ replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.     

Multielement Order Separability in Free Products of Groups

We introduce the notion of multielement order separability and study this property for free groups and free products.     

Free convolution operators and free Hall transform

We define an extension of the polynomial calculus on a W^?$W^{?}$-probability space by introducing an algebra C{_Xi:i∈I}$\mathbb{C}\{X_i:i\in I\}$ which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal–Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N   tends to ∞, of the ?-distribution of the Brownian motion on the linear group _GLN(C)${\mathit{GL}}_N(\mathbb{C})$ to the ?-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N)$U(N)$ to the free Hall transform.     

Graph product Khintchine inequalities and Hecke C⁎-algebras: Haagerup inequalities, (non)simplicity,nuclearity and exactness

Graph products of groups were introduced by Green in her thesis [32]. They have an operator algebraic counterpart introduced and explored in [14]. In this paper we prove Khintchine type inequalities for general C^?-algebraic graph products which generalize results by Ricard and Xu [50] on free products of C^?-algebras. We apply these inequalities in the context of (right-angled) Hecke C^?-algebras, which are deformations of the group algebra of Coxeter groups (see [22]). For these we deduce a Haagerup inequality which generalizes results from [33]. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C^?-algebras. Lastly we characterize exactness and nuclearity of general Hecke C^?-algebras.     

A hydrodynamic exercise in free probability: Setting up free Euler equations

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies vorticity equations and that it produces a family of conserved quantities.     

Short-time dynamics in the presence of wave–particles interactions: A perturbative approach

The self-consistent interaction between a beam of charged particles and a wave is considered, within a Vlasov picture. The model is discussed with reference to the case of a Free Electron Laser. Starting with a spatially bunched water-bag distribution, we derive, via perturbative methods, closed analytical expressions for the time evolution of the main macroscopic observables. Predictions of the theory are shown to agree with direct numerical simulations.     

Mapping resolutions of length three I

We produce some interesting families of resolutions of length three by describing certain open subsets of the spectrum of the generic ring for such resolutions constructed in [6].