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We study $N$-ary non-commutative notions of independence, which are given by trees and which generalize free, Boolean, and monotone independence. For every rooted subtree $mathcal{T}$ of the $N$-regular tree, we define the $mathcal{T}$-free product of $N$ non-commutative probability spaces and we define the $mathcal{T}$-free additive convolution of $N$ non-commutative laws. These $N$-ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities (such as the relation between free, monotone, and subordination convolutions studied by Lenczewski) can be reduced to combinatorial manipulations of trees. We also develop a theory of $mathcal{T}$-free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. In particular, we study the case where the root vertex of $mathcal{T}$ has $n$ children and each other vertex has $d$ children, and we relate the $mathcal{T}$-free convolution powers to free and Boolean convolution powers and the Belinschi-Nica semigroup.
In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in ter
We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a $C^*$-algebra $mathcal{A}$. We define an $mathcal{A}$-valued chordal Loewner chain as a subordination chain of analytic self-maps of the $mathcal{A
The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.
After an introduction to some basic issues in non-commutative geometry (Gelfand duality, spectral triples), we present a panoramic view of the status of our current research program on the use of categorical methods in the setting of A.Connes non-com
In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters $0<p,qleq infty$. Furthermore, we establish R{e}nyi entropic uncertainty principles for subfactor planar algebras.