Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn Operad of Non-commutative Independences Defined by Trees
229
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic 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.
rate research
Read More
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 terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.
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 $mathcal{A}$-valued upper half-plane, such that each $F_t$ is the reciprocal Cauchy transform of an $mathcal{A}$-valued law $mu_t$, such that the mean and variance of $mu_t$ are continuous functions of $t$. We relate $mathcal{A}$-valued Loewner chains to processes with $mathcal{A}$-valued free or monotone independent independent increments just as was done in the scalar case by Bauer (Lowners equation from a non-commutative probability perspective, J. Theoretical Prob., 2004) and Schei{ss}inger (The Chordal Loewner Equation and Monotone Probability Theory, Inf. Dim. Anal., Quantum Probability, and Related Topics, 2017). We show that the Loewner equation $partial_t F_t(z) = DF_t(z)[V_t(z)]$, when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains $F_t$ and vector fields $V_t(z)$ of the form $V_t(z) = -G_{ u_t}(z)$ where $ u_t$ is a generalized $mathcal{A}$-valued law. Based on the Loewner equation, we derive a combinatorial expression for the moments of $mu_t$ in terms of $ u_t$. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws $mu_t$. Finally, we prove a version of the monotone central limit theorem which describes the behavior of $F_t$ as $t to +infty$ when $ u_t$ has uniformly bounded support.
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-commutative geometry: morphisms/categories of spectral triples, categorification of Gelfand duality. We conclude with a summary of the expected applications of categorical non-commutative geometry to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
