Strong limit theorems in non-commutative probability.

*(English)*Zbl 0554.46033
Lecture Notes in Mathematics. 1110. Berlin etc.: Springer-Verlag. VI, 152 p. DM 26,50 (1985).

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 sequences of independent or uncorrelated random variables. Instead of independence (which is a too restrictive condition) one can consider the much less restrictive condition of orthogonality relative to a state. In the case when the state is tracial there are some results for sequences of measurable operators.

The prerequisites for reading this book are the fundamentals of functional analysis and probability theory. The elements of the theory of von Neumann algebras are collected in the Appendix which is almost self-contained and can be also read separately, before studying the main chapters. The book is written mainly for a reader familiar with the theory of probabilities but may be of interest for those who are interested in some techniques of operator algebras.

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 sequences of independent or uncorrelated random variables. Instead of independence (which is a too restrictive condition) one can consider the much less restrictive condition of orthogonality relative to a state. In the case when the state is tracial there are some results for sequences of measurable operators.

The prerequisites for reading this book are the fundamentals of functional analysis and probability theory. The elements of the theory of von Neumann algebras are collected in the Appendix which is almost self-contained and can be also read separately, before studying the main chapters. The book is written mainly for a reader familiar with the theory of probabilities but may be of interest for those who are interested in some techniques of operator algebras.

Reviewer: Sh.A.Ayupov

##### MSC:

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L55 | Noncommutative dynamical systems |

28D05 | Measure-preserving transformations |

60F15 | Strong limit theorems |

60G48 | Generalizations of martingales |