In this paper we give a construction of operators satisfying q-CCR relations for q > 1: $A\left(f\right)A*\left(g\right)-A*\left(g\right)A\left(f\right)=q^N⟨f,g⟩I$ and also q-CAR relations for q < -1: $B\left(f\right)B*\left(g\right)+B*\left(g\right)B\left(f\right)={\left|q\right|}^N⟨f,g⟩I$, where N is the number operator on a suitable Fock space ${ℱ}_q\left(\right)$ acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

This paper consists of two parts. The first part is devoted to the study of continuous diagrams and their connections with the boolean convolution. In the second part we investigate the rectangular Young diagrams and respective discrete measures. We recall the definition of Kerov's α-transformation of diagrams, define the α-transformation of finitely supported discrete measures and generalize the notion of the α-transformation.

We give a necessary and sufficient criterion for a normal CP-map on a von Neumann algebra to admit a restriction to a maximal commutative subalgebra. We apply this result to give a far reaching generalization of Rebolledo's sufficient criterion for the Lindblad generator of a Markov semigroup on ℬ(G).

Belinschi and Nica introduced a composition semigroup of maps on the set of probability measures. Using this semigroup, they introduced a free divisibility indicator, from which one can know quantitatively if a measure is freely infinitely divisible or not. In the first half of the paper, we further investigate this indicator: we calculate how the indicator changes with respect to free and Boolean powers; we prove that free and Boolean 1/2-stable laws have free divisibility indicators equal to infinity;...

Given an orthogonal projection P and a free unitary Brownian motion $Y={\left(Yₜ\right)}_{t\ge 0}$ in a W*-non commutative probability space such that Y and P are *-free in Voiculescu’s sense, we study the spectral distribution νₜ of Jₜ = PYₜPYₜ*P in the compressed space. To this end, we focus on the spectral distribution μₜ of the unitary operator SYₜSYₜ*, S = 2P - 1, whose moments are related to those of Jₜ via a binomial-type expansion already obtained by Demni et al. [Indiana Univ. Math. J. 61 (2012)]. In this connection,...

From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can...

We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then ${\left(\mu ₁⨄\mu ₂\right)}^s=\mu {₁}^s⨄\mu {₂}^s$ and if ν₁,ν₂ are symmetric then ${\left(\nu ₁⨄\nu ₂\right)}^{\left(2\right)}=\nu {₁}^{\left(2\right)}⨄\nu {₂}^{\left(2\right)}$. Finally we investigate necessary and sufficient conditions under which the latter equality holds.

A stronger version of almost uniform convergence in von Neumann algebras is introduced. This "bundle convergence" is additive and the limit is unique. Some extensions of classical limit theorems are obtained.

We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by $\mu {⊞}_T\nu =T^{-1}\left(T\mu ⊞T\nu \right)$. We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution power ${\mu }^{{⊞}_{T}s}$ for...