Noncommutative stationary processes
type: Item
instance of: scholarly article
title: Noncommutative stationary processes
zbMATH Open document ID: 1047.46001
author: Rolf Gohm
published in: Lecture Notes in Mathematics
publication date: 2004-03-18T00:00:00Z
review text: One of the main concerns common in both quantum probability and the theory of operator algebras consists in the study of noncommutative dynamics. This book treats recent results of research on noncommutative stationary processes with discrete-time parameter, which lead to an interesting interplay between operator algebraic and probabilistic topics. Focussing on noncommutative stationary processes, this book provides a systematic presentation of various related topics, including extensions and dilations of completely positive maps, Markov property and adaptedness, endomorphisms of operator algebras and the applications arising from the interplay of these themes. Most of the material is new and much effort has been made to elaborate presentation, so it gives a quick access to the heart of the matter and to new results as directly as possible. Clearly this is no substitute for the study of the older sources in these fields, however, the Appendix compensates for its lack of basic material with a review of the facts on unital completely positive maps. It should be accessible even to the reader who is equipped only with basic knowledge of noncommutative probability and operator algebras. The following is the contents of the book: Introduction. Chapter 1. Extensions and dilations: 1.1 An example with \(2 \times 2\)-matrices; 1.2 An extension problem; 1.3 Weak tensor dilations; 1.4 Equivalence of weak tensor dilations; 1.5 Duality; 1.6 The automorphic case; 1.7 Examples. Chapter 2. Markov processes: 2.1 Kümmerer's approach; 2.2 Bhat's approach; 2.3 Coupling representation on a Hilbert space; 2.4 Cuntz algebra representations; 2.5 Cocycles and coboundaries; 2.6 Kümmerer-Maassen scattering theory; 2.7 Restrictions and extensions; 2.8 An interpretation using entanglement. Chapter 3. Adaptedness: 3.1 A motivation: Hessenberg form of an isometry; 3.2 Adapted endomorphisms - an abstract view; 3.3 Adapted endomorphisms and stationary processes; 3.4 Adapted isometries on tensor pro…
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46L55
zbMATH DE Number: 2060671
zbMATH Keywords: noncommutative stationary processes
zbMATH Keywords: extensions
zbMATH Keywords: dilations
zbMATH Keywords: unital completely positive maps
reviewed by: Isamu Dôku
MaRDI profile type: Publication
full work available at URL: b95349
OpenAlex ID: W587396073
DOI: 10.1007/B95349
description: scientific article; zbMATH DE number 2060671