arXiv subjects

Serban T. Belinschi

Publications and source records attributed to Serban T. Belinschi.

14 recordsLinked to original sources

arXiv2023

Iteration theory of noncommutative maps

This note aims to study the iteration theory of noncommutative self-maps of bounded matrix convex domains. We prove a version of the Denjoy-Wolff theorem for the row ball and the maximal quantization of the unit ball of $\mathbb{C}^d$. For more general bounded matrix convex sets, we prove a version of Wolff's theorem inspired by the results of Abate. Lastly, we use iteration and fixed point theory to generalize the commutative results of Davidson, Ramsey, and Shalit to quotients of the free semigroup algebra by WOT closed ideals.

math.OA↗

arXiv2022

Upgrading subordination properties in free probability theory

The existence of Voiculescu's subordination functions in the context of non-tracial operator-valued C*-probability spaces has been established using analytic function theory methods. We use a matrix construction to show that the subordination functions thus obtained also satisfy an appropriately modified form of subordination for conditional expectations.

math.OA↗

arXiv2022

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

math.OA↗

arXiv2022

Noncommutative Christoffel-Darboux Kernels

We introduce from an analytic perspective Christoffel-Darboux kernels associated to bounded, tracial noncommutative distributions. We show that properly normalized traces, respectively norms, of evaluations of such kernels on finite dimensional matrices yield classical plurisubharmonic functions as the degree tends to infinity, and show that they are comparable to certain noncommutative versions of the Siciak extremal function. We prove estimates for Siciak functions associated to free products of distributions, and use the classical theory of plurisubharmonic functions in order to propose a notion of support for noncommutative distributions. We conclude with some conjectures and numerical experiments.

math.OA↗

arXiv2014

Almost one bit violation for the additivity of the minimum output entropy

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that the maximum is attained on a vector of shape (a,b,...,b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any eps > 0, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as log 2 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.

math-ph↗

arXiv2012

Operator-valued free multiplicative convolution: analytic subordination theory and applications to random matrix theory

We give an explicit description, via analytic subordination, of free multiplicative convolution of operator-valued distributions. In particular, the subordination function is obtained from an iteration process. This algorithm is easily numerically implementable. We present two concrete applications of our method: the product of two free operator-valued semicircular elements and the calculation of the distribution of $dcd+d^2cd^2$ for scalar-valued $c$ and $d$, which are free. Comparision between the solution obtained by our methods and simulations of random matrices shows excellent agreement.

math.OA↗

arXiv2012

Free Infinite Divisibility for Ultrasphericals

We prove that the integral powers of the semicircular distribution are freely infinitely divisible. As a byproduct we get another proof of the free infnite divisibility of the classical Gaussian distribution.

math.PR↗

arXiv2011

Infinite divisibility and a non-commutative Boolean-to-free Bercovici-Pata bijection

We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.

math.OA↗

arXiv2011

Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution μone can define convolution powers with respect to free additive convolution and with respect to Boolean convolution, where the exponent considered in the power is a suitably chosen linear map ηfrom B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever ηis completely positive, while the free additive convolution power is defined whenever η- 1 is completely positive (where 1 stands for the identity map on B). In connection to these convolution powers we define an evolution semigroup related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to the free additive convolution powers with exponent η. One of the results concerns analytic subordination for B-valued Cauchy-Stieltjes transforms. The other gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion.

math.OA↗

arXiv2010

On the operator-valued analogues of the semicircle, arcsine and Bernoulli laws

We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these measures are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that generally the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.

math.OA↗

arXiv2010

The normal distribution is $\boxplus$-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

math.OA↗

arXiv2008

On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect to free additive convolution and then the 1/(1+t) power with respect to Boolean convolution of the given element. We show that the family of maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism for the operation of free multiplicative convolution. We prove that for t=1 the transformation $\mathbb B_1$ coincides with the canonical bijection $\mathbb B : M \to M_{inf-div}$ discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M_{inf-div} stands for the set of probability distributions in M which are infinitely divisible with respect to free additive convolution. As a consequence, we have that $\mathbb B_t(μ)$ is infinitely divisible with respect to free additive convolution for any for every $μ$ in M and every t greater than or equal to one. On the other hand we put into evidence a relation between the transformations $\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations $\mathbb B_t$ as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to free multiplicative convolution, and always reaches infinite divisibility with respect to free additive convolution by the time t=1.

math.OA↗

arXiv2007

Free Brownian motion and evolution towards boxplus-infinite divisibility for k-tuples

Let D be the space of non-commutative distributions of k-tuples of selfadjoints in a C*-probability space (for a fixed k). We introduce a semigroup of transformations B_t of D, such that every distribution in D evolves under the B_t towards infinite divisibility with respect to free additive convolution. The very good properties of B_t come from some special connections that we put into evidence between free additive convolution and the operation of Boolean convolution. On the other hand we put into evidence a relation between the transformations B_t and free Brownian motion. More precisely, we introduce a transformation Phi of D which converts the free Brownian motion started at an arbitrary distribution m in D into the process B_t (Phi(m)), t>0.

math.OA↗

arXiv2007

Eta-series and a Boolean Bercovici-Pata bijection for bounded k-tuples

On the space of (non-commutative) distributions of k-tuples of selfadjoint elements in a $C^*$-probability space $D_c(k)$, one has an operation $\freeplus$ of free additive convolution, and one can consider the subspace $D_c^{inf-div}$ of distributions which are infinitely divisible with respect to this operation. The linearizing transform for free additive convolution is the R-transform. Thus, one has $R_{μ\freeplusν}=R_μ+R_ν$. The eta-series $η_μ$ is the counterpart of $R_μ$ in the theory of Boolean convolution. We prove that the space of eta-series of distributions belonging to $D_c(k)$ coincides with the space of R-transforms of distributions which are infinitely divisible with respect to free additive convolution. As a consequence of this fact, one can define a bijection $B : D_c(k) \to D_c^{inf-div}$ via the formula $R_{B(μ)} = η_μ$, for all distributions $μ$ in $D_c(k)$. We show that $B$ is a multi-variable analogue of a bijection studied by Bercovici and Pata for k=1, and we prove a theorem about convergence in moments which parallels the Bercovici-Pata result. On the other hand we prove the formula $B(μ\freetimesν) = B(μ) \freetimes B(ν),$ with $μ,ν$ considered in a space $D^{alg}(k)$ containing $D_c (k)$ where the operation of free multiplicative convolution $\freetimes$ always makes sense. An equivalent reformulation for this equality is that $η_{μ\freetimesν}=η_μ \freestar η_ν,$ for all $μ,ν\in D^{alg}(k)$. This shows that, in a certain sense, eta-series behave in the same way as R-transforms in connection to the operation of multiplication of free k-tuples of non-commutative random variables.

math.OA↗