Noncommutative stationary processes
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 products of Hilbert spaces; 3.5 Nonlinear prediction errors; 3.6 The adjoint of an adapted isometry. Chapter 4. Examples and applications:4.1 Commutative stationarity; 4.2 Prediction errors for commutative processes; 4.3 Low-dimensional examples; 4.4 Clifford algebras and generalizations; 4.5 Tensor products of matrices;4.6 Noncommutative extension of adaptedness. Appendix. Some facts about unital completely positive maps: A.1 Stochastic maps; A.2 Representation theorems; A.3 The isometry \(v\); A.4 The preadjoints \(C\) and \(D\); A.5 Absorbing vector states. References. Index. Chapter 1 treats normal unital completely positive maps on von Neumann algebras, called stochastic maps. The purpose of this chapter is to study the problem of finding normal completely positive extensions acting on all bounded operators of the GNS-Hilbert spaces. It is shown that there exists a duality relating this problem to a dilation problem on the commutants. The concept of a weak tensor dilation is introduced to formulate a one-to-one correspondence between certain extensions and dilations. Some examples are given, where the extensions can be explicitly computed. In Chapter 2, first of all different notions of noncommutative Markov processes are reviewed with some emphasis on the structure of a coupling representation, and related results on Cuntz algebra representations and on endomorphisms are derived. In particular, a conjugacy closely connected to Kümmerer-Maassen scattering theory is proved. The extension theory of Chapter 1 applied to the transition operator of the Markov processes can be used in a new criterion for asymptotic completeness. An interpretation in terms of entangled states is also given. Chapter 3 is devoted to an axiomatic approach to time evolutions of stationary processes which are non-Markovian in general but adapted to a given filtration. This is called an adapted endomorphism. Adapted isometries are defined, and the author undertakes a detailed study of them in the situation where the filtration can be factorized as a tensor product. Then it turns out that the same ergodic properties used in Chapter 2 in order to determine asymptotic completeness now determine the asymptotics of nonlinear prediction errors for the implemented process. This leads to an answer to the problem of unitarity of an adapted isometry. In Chapter 4, some interesting examples are given to illustrate an abstract theory of noncommutative stationary processes based on the concept of an adapted endomorphism. In particular, the author shows that, by choosing suitable noncommutative filtrations and adapted endomorphisms, his criteria can be used to determine whether the range of an adapted endomorphism is a proper subfactor of the hyperfinite \(II_1\)-factor, namely, he gives an answer to a question about subfactors in the theory of von Neumann algebras. Lastly, the author closes this work with some surprising observations about extensions of adapted endomorphisms, exhibiting phenomena typical of non-Markovian cases.
scientific article; zbMATH DE number 2060671