Narrow Results

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 , and the Brown-measure of , where Y is a circular element *-free from T for ∊>0. Moreover we give a new proof of Śniady's formula for the moments τ(((T*)^kT^k)ⁿ) for k,n∈ℕ.

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy χ_orb for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties, the relation of χ_orb with the usual free entropy χ is shown. Moreover, the dimension counterpart δ_0,orb of χ_orb is discussed, and we obtain the relation between δ_0,orb and the original free entropy dimension δ₀ together with applications to δ₀ itself.

The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used again by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy (MOE) for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the MOE.

We follow the guiding line offered by canonical operators on the full Fock space, in order to identify what kind of cumulant functionals should be considered for the concept of bi-free independence introduced in the recent work of Voiculescu. By following this guiding line we arrive to consider, for a general noncommutative probability space (𝒜, φ), a family of "(ℓ, r )-cumulant functionals" which enlarges the family of free cumulant functionals of the space. In the motivating case of canonical operators on the full Fock space we find a simple formula for a relevant family of (ℓ, r )-cumulants of a (2d)-tuple (A₁,…,A_d, B₁,…,B_d), with A₁,…,A_d canonical operators on the left and B₁,…,B_d canonical operators on the right. This extends a known one-sided formula for free cumulants of A₁,…,A_d, which establishes a basic operator model for the R-transform of free probability.

Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to the identity in the group. This is related to the probability of return for walks on the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index-free subgroup (using a result of Dunwoody). In this work, we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then ${lim sup}_n{a}_n^{1/n}$ is either $1$, $2$ or at least $2\sqrt{2}$.

We study the asymptotics of the reducible representations of the wreath products G≀S_q = G^q ⋊ S_q for large q, where G is a fixed finite group and S_q is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀S_q as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

We examine the free entropy and free entropy dimension for projections, and obtain a sufficient condition for the factoriality of the von Neumann algebra generated by projections in terms of their free entropy dimension. This corresponds to Voiculescu's result for self-adjoint elements.

In this paper, we firstly characterize the class of free self-decomposable distributions as a class of limiting distributions of suitably normalized partial sums of free independent random variables. Secondly, we introduce nested classes between the class of free self-decomposable distributions and that of free stable distributions, characterize them and show that the limit of the nested classes coincides with the closure of the class of free stable distributions. All results here are analogues of the results known in classical probability theory.

We show that some C*-dynamical systems obtained by free Fock quantization of classical ones, enjoy ergodic properties much stronger than their boson or fermion analogous. Namely, if the classical dynamical system (X, T, μ) is ergodic but not weakly mixing, then the resulting free quantized system (𝔊, α) is uniquely ergodic (w.r.t. the fixed point algebra) but not uniquely weak mixing. The same happens if we quantize a classical system (X, T, μ) which is weakly mixing but not mixing. In this case, the free quantized system is uniquely weak mixing but not uniquely mixing. Finally, a free quantized system arising from a classical mixing dynamical system, will be uniquely mixing. In such a way, it is possible to exhibit uniquely weak mixing and uniquely mixing C*-dynamical systems whose Gelfand–Naimark–Segal representation associated to the unique invariant state generates a von Neumann factor of one of the following types: I_∞, II₁, III_λ where λ ∈ (0, 1]. The resulting scenario is then quite different from the classical one. In fact, if a classical system is uniquely mixing, it is conjugate to the trivial one consisting of a singleton. For the sake of completeness, the results listed above are extended to the q-Commutation Relations, provided . The last result has a self-contained meaning as we prove that the involved C*-dynamical systems based on the q-Commutation Relations are conjugate to the corresponding one arising from the free case (i.e. q = 0), at least if .

In this paper, we show the Basel problem via the beta distributions, which include the free Poisson distribution and positive arcsine law. This is a generalization of Ref. 2 by Fujita. We also obtain special values of an extension of generalized Hurwitz–Lerch zeta function, which was introduced by Gang, Jain and Kalla.

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

We prove that the binomial sequence is positive definite if and only if either p ≥ 1, -1 ≤ r ≤ p - 1 or p ≤ 0, p - 1 ≤ r ≤ 0 and that the Raney sequence is positive definite if and only if either p ≥ 1, 0 ≤ r ≤ p or p ≤ 0, p - 1 ≤ r ≤ 0 or else r = 0. The corresponding probability measures are denoted by ν(p, r) and μ(p, r) respectively. We prove that if p > 1 is rational and -1 < r ≤ p - 1 then the measure ν(p, r) is absolutely continuous and its density V_p,r(x) can be represented as Meijer G-function. In some cases V_p,r is an elementary function. We show that for p > 1 the measures ν(p,-1) and ν(p,0) are certain free convolution powers of the Bernoulli distribution.

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group S_n and the orthogonal group O_n as well as Wang's quantum permutation group and his free orthogonal quantum group . In this paper, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another paper.

It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free Lévy processes as high-dimensional limits of classical matricial Lévy processes.

We will focus here on one specific such construction, discussing and generalizing the work done previously by Biane in Ref.2, who has shown that the (classical) Brownian motion on the Unitary group U(d) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free Lévy process on the dual group (in the sense of Voiculescu) U〈n〉. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of which Biane's result constitutes the case n = 1.

In Ref. 5, O. Bauer interpreted the chordal Loewner equation in terms of noncommutative probability theory. We follow this perspective and identify the chordal Loewner equations as the non-autonomous versions of evolution equations for semigroups in monotone and anti-monotone probability theory. We also look at the corresponding equation for free probability theory.

In this paper, we develop the notion of free-Boolean independence in an amalgamated setting. We construct free-Boolean cumulants and show that the vanishing of mixed free-Boolean cumulants is equivalent to our free-Boolean independence with amalgamation. We also provide a characterization of free-Boolean independence by conditions in terms of mixed moments. In addition, we study free-Boolean independence over a $C^{\ast }$-algebra and prove a positivity property. A central limit law for our free-Boolean independence with amalgamation is also studied.

Recently, Brannan and Vergnioux showed that the orthogonal free quantum group factors ${ℒ}𝔽O_M$ have Jung’s strong $1$-boundedness property, and hence are not isomorphic to free group factors. We prove an analogous result for the other unimodular case, where the parameter matrix is the standard symplectic matrix in $2N$ dimensions $J_{2N}$. We compute free derivatives of the defining relations by introducing self-adjoint generators through a decomposition of the fundamental representation in terms of Pauli matrices, resulting in $1$-boundedness of these generators. Moreover, we prove that under certain conditions, one can add elements to a $1$-bounded set without losing $1$-boundedness. In particular, this allows us to include the character of the fundamental representation, proving strong $1$-boundedness.

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional $\mathrm{\Phi }$ are naturally described in terms of Motzkin paths identified with reduced Motzkin words $M$. Using this fact, we demonstrate that the mixed moments of order $n$ of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to $\mathrm{\Phi }$ with summation extending over $M_n$, the set of reduced Motzkin paths of length $n$. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice $\mathrm{\text{NC}}(n)$ into sublattices ${ℳ}(w)$ of partitions which are monotonically adapted to colors in $w\in M_n$. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.

For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the global existence for a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion.

After a short review of the quantum mechanics canonically associated with a classical real valued random variable with all moments, we begin to study the quantum mechanics canonically associated to the standard semi-circle random variable$X$, characterized by the fact that its probability distribution is the semi-circle law $\mu$ on $[-2,2]$. We prove that, in the identification of $L^2([-2,2],\mu )$ with the $1$-mode interacting Fock space ${\mathrm{\Gamma }}_{\mu }$, defined by the orthogonal polynomial gradation of $\mu$, $X$ is mapped into position operator and its canonically associated momentum operator $P$ into $i$ times the $\mu$-Hilbert transform $H_{\mu }$ on $L^2([-2,2],\mu )$. In the first part of the present paper, after briefly describing the simpler case of the $\mu$-harmonic oscillator, we find an explicit expression for the action, on the $\mu$-orthogonal polynomials, of the semi-circle analogue of the translation group $e^{itP}$ and of the semi-circle analogue of the free evolution $e^{it{P}^2/2}$, respectively, in terms of Bessel functions of the first kind and of confluent hyper-geometric series. These results require the solution of the inverse normal order problem on the quantum algebra canonically associated to the classical semi-circle random variable and are derived in the second part of the present paper. Since the problem to determine, with purely analytic techniques, the explicit form of the action of $e^{-t{H}_{\mu }}$ and $e^{-it{H}_{\mu }^2/2}$ on the $\mu$-orthogonal polynomials is difficult, the above mentioned results show the power of the combination of these techniques with those developed within the algebraic approach to the theory of orthogonal polynomials.