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.
出处
《数学杂志》
CSCD
北大核心
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)
关键词
定向集
非交换鞅
收敛性
一致可积
directed set
noncommutative martingale
convergence
uniform integrability
