Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements

Ilwoo Cho

• Publisher: Chapman and Hall/CRC
• Year: 2022
• Print ISBN: 9781032199016
• eBook ISBN: 9781000600711
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Banach-Space Operators On C*-Probability Spaces Generated by Multi Semicircular Elements introduces new areas in operator theory and operator algebra, in connection with free probability theory. In particular, the book considers projections and partial isometries distorting original free-distributional data on the C?-probability spaces.

Features

• Suitable for graduate students and professional researchers in operator theory and/or analysis
• Numerous applications in related scientific fields and areas

Chapter 1 Introduction

DOI: 10.1201/9781003263487-1

The main purposes of this monograph are (i) to consider the C^*-probability space 𝔛 = (𝔛, φ) generated by the free semicircular family X = {x_j}_j∈ℤ, and its free-probabilistic substructures, 𝔛_N = (𝔛_N , φ_N ) generated by free semicircular sub-families $X_N={\left\{x_j\right\}}_{j=1}^N$, for

$N\in {\mathbb{N}}_{>1}^{\infty }\stackrel{def}{=}(\mathbb{N}\backslash \{1\left\}\right)\cup \left\{\infty \right\},$

(ii) to construct suitable ∗-isomorphisms λ = {β^k}_k∈ℤ acting on 𝔛, and their restrictions λ_N = {β^k |_𝔛N}_k∈ℤ, which are ∗-homomorphisms acting on 𝔛_N, and the adjointable Banach-space operators generated by λ and λ_N, respectively, for $N\in {\mathbb{N}}_{>1}^{\infty }$, (iii) to study certain types of adjointable Banach-space operators of (ii), and to consider how they deform the free probability on 𝔛, and those on 𝔛_N, (iv) to characterize the deformations on the C^∗-probability spaces from the Banach-space operators of (iii), and (v) to study dynamics induced by our Banach-space operators.

In particular, we are interested in certain types of “adjointable” Banach-space operators, induced by λ_N acting on 𝔛_N, especially, where they are projections or partial isometries. Different from the Banach-space operators induced by λ acting on 𝔛, such operators from λ_N distort the free probability on 𝔛_N, in general. We characterize such distortions on 𝔛_N occurred by projections, and partial isometries.

1.1 Motivation and Background

The study of semicircular elements is one of the major topics (e.g., [20,21,29,30]), not only in both commutative function theory and noncommutative free probability theory but also in various applied fields, including quantum statistical physics (e.g., [5,6,7 and 8,10,11,12]). The semicircular law, which is the free distributions of semicircular elements, is well characterized analytically, and combinatorially, in classical function theory and in free probability theory (e.g., [1,17,18,21,28,29 and 30]). In particular, it is playing a key role in free-probabilistic operator algebra theory (and hence, in quantum physics) by the (free) central limit theorem(s) (e.g., see [2,17,19,28,29 and 30]), i.e., it becomes a noncommutative analytic analogue of the classical Gaussian (or the normal) distribution (in commutative function theory).

From combinatorial approaches (e.g., [17,22,23]), the semicircular law is universally characterized by the Catalan numbers ${\left\{c_k\right\}}_{k=1}^{\infty }$, where

$c_n=\frac{1}{n+1}\left(\begin{array}{c}\hfill 2n\hfill \\ \hfill n\hfill \end{array}\right)=\left(\frac{1}{n+1}\right)\left(\frac{\left(2n\right)!}{n!(2n-n)!}\right)=\frac{\left(2n\right)!}{n!(n+1)!},$

for all n ∈ ℕ₀ = ℕ ∪ {0}. i.e., the semicircular law is characterized by the free-moment sequence,

${\left({\omega }_n{c}_n\right)}_{n=1}^{\infty }=(0,c_1,0,c_2,0,c_3,\dots ),$(1.1.1)

where

${\omega }_n\stackrel{def}{=}\{\begin{array}{cc}\begin{array}{c}1\\ 0\end{array}& \begin{array}{c}ifn\text{is even}\\ ifn\text{is odd},\end{array}\hfill \end{array}$

and c_k are the Catalan numbers, for all k ∈ ℕ.

From the analysis on p-adic number fields ℚ_p (e.g., [26,27]), one can construct semicircular elements (e.g., [5,12]). By generalizing the constructions of [5,12], semicircular elements are constructed whenever there are |ℤ|-many orthogonal projections in a C^∗-algebra (e.g., [6,7 and 8,10] and [11]), different from earlier works (e.g., [20,25,29,30]). In this new approach, the semicircular elements are understood as Banach-space operators acting on a given C^∗-algebra, by regarding the C^∗-algebra as a Banach space equipped with its C^∗-norm (e.g., [13,14]).

Independently, the joint free distributions of mutually free, multi semicircular elements were re-characterized in [9] (See...

Table of contents

• Cover
• Half Title Page
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• Author
• Chapter 1 ▪ Introduction
• Chapter 2 ▪ Preliminaries
• Chapter 3 ▪ Joint Free Distributions of Multi Semicircular Elements
• Chapter 4 ▪ A C*-Probability Space of |ℤ|-Many Semicircular Elements
• Chapter 5 ▪ C*-Probability Spaces (𝔛N, φN)
• Chapter 6 ▪ Adjointable Banach-Space Operators Acting on 𝔛N
• Chapter 7 ▪ Free-Probabilistic Information on 𝔛N Affected by Partial Isometries
• Chapter 8 ▪ Application I. Shift Operators Acting on 𝔛∞
• Chapter 9 ▪ Application II. The Circular Law
• Chapter 10 ▪ Application III. Free Poisson Distributions
• Chapter 11 ▪ Examples
• Chapter 12 ▪ The Group-Dynamical System Γ
• Chapter 13 ▪ On The Γ-Semicircular *-Probability Space 𝕏
• Chapter 14 ▪ Operator-Theoretic Properties on 𝕏
• Chapter 15 ▪ Free Probability on 𝔛N = (𝔛N ⊗C ΛN, τN)
• Chapter 16 ▪ Operator-Theoretic Properties on 𝔛N
• Chapter 17 ▪ Summary
• Bibliography
• Index