中文摘要：

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L^1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L^p范数收敛（或弱收敛）的充要条件是{xα）在L^p（M）中依L^p范数有界．这也等价于存在一个x∞∈L^p（M），使得xα=∈α（x∞）（α∈I）．推广了交换情形中的相应结果．

英文摘要：

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.