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| 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=εα(x∞)(α ∈ I).推广了交换情形中的相应结果. |
| 关键词: 定向集 非交换鞅 收敛性 一致可积 |
| DOI： |
| 分类号:O177.5 |
| 基金项目:国家自然科学基金资助(11271293) |
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| CONVERGENCE OF NONCOMMUTATIVE MARTINGALES INDEXED BY DIRECTED SETS |
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ZHANG Yan,HOU You-liang
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| Abstract: |
| 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∈I be 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 satisfles 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=εα(x∞)(α ∈ I). It generalizes the corresponding conclusions in the commutative condition. |
| Key words: directed set noncommutative martingale convergence uniform integrability |
