An Operad of Non-commutative Independences Defined by Trees

Abstract

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.

arXiv comment114 pages, 3 figures, updated with newer references and more examples

References in corpus (11)

• Free Probability for Pairs of Faces I
• The Monotone Cumulants
• Relations between cumulants in noncommutative probability
• Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures
• Boolean convolution of probability measures on the unit circle
• Complex analysis methods in noncommutative probability
• Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals
• Multiplicative Monotone Convolution
• Free-Boolean independence for pairs of algebras
• Some properties of the free stable distributions
• Berry-Esseen type estimate and return sequence for parabolic iteration in the upper half-plane

Cited by in corpus (4)

• Boolean Cumulants and Subordination in Free Probability
• Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants
• Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices
• Tree convolution for probability distributions with unbounded support