This book not only examines the constructions and free-probabilistic properties of semicircular elements, as defined within the text, but also considers certain Banach-space operators acting on these semicircular elements and shows how they deform (i.e., preserve-or-distort) the semicircular law induced by orthogonal projections.
FEATURES
Suitable for graduate students and professional researchers in operator theory and/or analysis
Numerous applications in related scientific fields and areas
Author(s): Ilwoo Cho
Series: Monographs and Research Notes in Mathematics
Publisher: CRC Press/Chapman & Hall
Year: 2023
Language: English
Pages: 198
City: Boca Raton
Cover
Half Title
Series Page
Title Page
Copyright Page
Table of Contents
Author
Chapter 1 ■ Introduction
1.1 Free Probability
1.2 Semicircular Elements
1.3 Some Free Random Variables Induced by Semicircular Elements
1.3.1 Circular Elements
1.3.2 Free Poisson Elements
1.4 Joint Free Moments of Multi Semicircular Elements
References
Chapter 2 ■ Semicircular Elements Induced by p-Adic Number Fields Q[sub(p)]
2.1 Analysis of the p-Adic Number Fields Q[sub(P)]
2.2 Statistical Models Induced by Q[sub(P)]
2.3 Representations of (M[sub(p)], φ[sub(P)])
2.4 Measure-Theoretic Models On {M[sub(p)]}[sub(p∊P)]
2.5 C*-Subalgebras S[sub(p)] of (M[sub(p)], φ[sub(P,j)])
2.6 Statistical Models for Banach *-Algebras LS[sub(p)] (j)
2.7 On the Free Adelic Filterization LS
2.7.1 Weighted-Semicircular Elements
2.7.2 Weighted-Semicircular Elements of LS
2.7.3 The Adelic Semicircular Filterization LS
2.8 Circular Elements of LS
2.9 Free Poisson Elements of LS Induced by U ∪ U
2.10 Banach-Space Operators Acting On LS
2.11 Pi-Shift Operators On LS
References
Chapter 3 ■ Semicircular Elements Induced by Orthogonal Projections
3.1 Some Banach *-Algebras Induced by Projections
3.2 Weighted-Semicircular Elements Induced by Q
3.3 Free-Distributional Data On L[sub(Q)]
3.4 Circular Elements of L[sub(Q)]
3.5 Free Poisson Elements of L[sub(Q)] Induced by U ∪ U
3.6 Certain *-Isomorphisms Acting on L[sub(Q)]
3.7 Group-Dynamical System On L[sub(Q)]
3.7.1 Discrete-Group Dynamical Systems
3.7.2 The Group-Dynamical System (B, L[sub(Q)], α)
3.7.3 The Dynamical Semicircular *-Probability Space L[sub(Q)]
3.8 Banach *-Probability Spaces L[sub(Q)][N]
References
Chapter 4 ■ Certain Banach-Space Operators on L[sub(Q)] [N]
4.1 The Integer-Shift-Operator Algebra B On L[sub(Q)]
4.2 Restricted-Shift-Operator Algebras B[sub(N)] On L[sub(Q)][N]
4.3 Free Probability On L[sub(Q)][N] Deformed by B[sub(N)]
4.4 Conditional-Nilpotent-Property On B[sub(N)]
4.5 Deformed Circular Laws On L[sub(Q)][N] by B[sub(N)]
4.6 Deformed Free Poisson Distributions On L[sub(Q)][N] by B[sub(N)]
4.7 The Semicircular ∞-Filterization L[sub(Q)][∞]
4.8 More About L[sub(Q)][∞]
References
Chapter 5 ■ Discussion
References
Index
