Strong limit theorems in non-commutative probability
type: Item
instance of: scholarly article
title: Strong limit theorems in non-commutative probability
zbMATH Open document ID: 0554.46033
author: Ryszard Jajte
published in: Lecture Notes in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The main purpose of the monograph is to present a self-contained exposition of some results from the theory of probabilities and ergodic theory in the von Neumann algebra context. Chapter 1 (Almost uniform convergence in von Neumann algebras) has a preparatory character. Considering a von Neumann algebra A as a non commutative generalization of the algebra \(L^{\infty}(\Omega,{\mathcal F},\mu)\) of all complex-valued \({\mathcal F}\)-measurable and essentially bounded functions on a probability space \((\Omega,{\mathcal F},\mu)\) the author explains the naturalness of the following analogue of the nearly everywhere convergence. Definition. Let A be a von Neumann algebra with a faithful normal state \(\phi\). We say that a sequence \((x_ n)\) of elements of A converges almost uniformly to an element \(x\in A\) if, for each \(\epsilon>0\), there is a projection \(p\in A\) with \(\phi(1-p)<\epsilon\) and such that \(\|(x_ n-x)p\| \to 0\) as \(n\to \infty\). In this chapter the author observes also various other kinds of ''almost sure'' convergence in von Neumann algebras. All these convergences coincide when the state \(\phi\) is tracial. Chapter 2 (Ergodic theorems) is devoted to non commutative versions of the individual ergodic theorems for *-automorphisms and Markov operators on a von Neumann algebra. In particular the author proves some individual ergodic theorems for normal positive maps of a von Neumann algebra, the non commutative versions of Kingman's subadditive ergodic theorem for *-automorphisms, a random ergodic theorem and a non commutative local ergodic theorem for quantum dynamical semigroups. In Chapter 3 (Convergence of conditional expectations and martingales in von Neumann algebras) the non commutative martingale convergence theorem of Dang-Ngoc and M. S. Goldstein are proved. In Chapter 4 (Strong laws of large numbers in von Neumann algebras) the author gives some results which can be treated as the extensions of well- known classical theorems for …
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 3883037
zbMATH Keywords: probabilities and ergodic theory in the von Neumann algebra context
zbMATH Keywords: Almost uniform convergence
zbMATH Keywords: faithful normal state
zbMATH Keywords: non commutative versions of the individual ergodic theorems for *-automorphisms
zbMATH Keywords: Markov operators
zbMATH Keywords: Kingman's subadditive ergodic theorem
zbMATH Keywords: random ergodic theorem
zbMATH Keywords: non commutative local ergodic theorem for quantum dynamical semigroups
zbMATH Keywords: conditional expectations
zbMATH Keywords: non commutative martingale convergence theorem
zbMATH Keywords: Strong laws of large numbers
zbMATH Keywords: orthogonality relative to a state
zbMATH Keywords: sequences of measurable operators
reviewed by: Sh. A. Ayupov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3883037