定向集指标非交换鞅的几种收敛性  

CONVERGENCE OF NONCOMMUTATIVE MARTINGALES INDEXED BY DIRECTED SETS

作　　者：张艳[1] 侯友良[1] 

机构地区：[1]武汉大学数学与统计学院,湖北武汉430072

出　　处：《数学杂志》2015年第5期1159-1165,共7页Journal of Mathematics

基　　金：国家自然科学基金资助(11271293)

摘　　要：本文研究了定向集指标非交换鞅的几种收敛性.利用非交换鞅的理论,得到了如下结果:设{xα,Mα}α∈I是一个定向集指标的非交换鞅.则{xα}依L1范数收敛(或弱收敛)的充要条件是{xα}一致可积且满足条件(B):对任意的ε>0,存在投影e∈M,使得对任意的y∈M,y≤1及任意的α∈I,有|τ(exαey)|<ε.当1<p<∞时,{xα}依Lp范数收敛(或弱收敛)的充要条件是{xα}在Lp(M)中依Lp范数有界.这也等价于存在一个x∞∈Lp(M),使得xα=Eα(x∞)(α∈I).推广了交换情形中的相应结果.In this paper, we discuss the convergence of noncommutative martingales indexed by directed sets. According to the theory of noncommutative martingales, we come to the following conclusions： Let {xα, Mα}α∈Ibe a noncommutative martingale with a directed index set. Then{xα} converges in L1-norm（or weakly） if and only if {xα} is uniformly integrable and satisfies the condition（B）： for each ε 0 there is a projection e ∈ M such that ｜τ（exαey）｜〈 ε for any y ∈ M, ｜｜y｜｜ ≤ 1 and any α ∈ I. When 1 〈p 〈∞, {xα} converges in Lp-norm（or weakly） if and only if{xα} is Lp-bounded in Lp（M）. It is also equivalent to that there exists an x∞∈ Lp（M） such that xα= Eα（x∞）（α ∈ I）. It generalizes the corresponding conclusions in the commutative condition.

关 键 词：定向集 非交换鞅 收敛性 一致可积 

分 类 号：O177.5[理学—数学]