TAILIEUCHUNG - Báo cáo "Some laws of large numbers in non-commutative probability "

In this report we present some noncommutative weak and strong laws of large numbers. Two case are considered: a von Neumann algebra with a normal faithful state on it and the algebra of measurable operators with normal faithful trace. 1. Introduction and notations One of the problems occurring in noncommutative probability theory concerns the extension of various results centered around limit theorems to the noncommutative context. In this setting the role of a random variable is played by an element of a von Neumann algebra A, and a probability measure is replaced by a normal faithful state on it | VNU Journal of Science Mathematics - Physics 23 2007 159-167 Some laws of large numbers in non-commutative probability Nguyen Van Quang Nguyen Duy Tien Department of Mathematics Mechanics Informatics College of Science VNU 334 Nguyen Trai Hanoi Vietnam Received 15 November 2006 received in revised form 12 September 2007 Abstract. In this report we present some noncommutative weak and strong laws of large numbers. Two case are considered a von Neumann algebra with a normal faithful state on it and the algebra of measurable operators with normal faithful trace. 1. Introduction and notations One of the problems occurring in noncommutative probability theory concerns the extension of various results centered around limit theorems to the noncommutative context. In this setting the role of a random variable is played by an element of a von Neumann algebra A and a probability measure is replaced by a normal faithful state on it. If this state is tracial the von Neumann algebra can be replaced by an algebra consisting of measurable operators possible unbounded . Many results in this area have been obtained by Batty 1 Jajte 2 Luczak 3 . The purpose of this report is to present some noncommutative weak and strong laws of large numbers. Two case are considered a von Neumann algebra with a normal faithful state on it and the algebra of measurable operators with normal faithful trace. Let us begin with some definitions and notations. Throughout of this paper A denote a von Neumann algebra with faithful normal state T. If this state is tracial then the measure topology in A is given by the fundamental system of neighborhoods of zero of the form N e ổ x G A there exists a projection p in A such that xp G A 11xp e and T 1 p Ổ It follows that 1 being the completion of A in the above topology is a topology - algebra see 7 . A is said to be the algebra of measurable operators in Segal-Nelson s sense. The convergence in measure topology is said to be the convergence in measure. Now .

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