论文标题
在不同的订单收敛方式和某些应用程序上
On different modes of order convergence and some applications
论文作者
论文摘要
各种作者考虑了不同的订单收敛概念。与订单融合的每一个概念相关联的拓扑相关，是通过将封闭式设置为poset的子集而定义的，即在他们的订单中没有净收敛到集合之外的点。我们将全面概述这些不同的概念，并对相关拓扑的系统进行系统的比较。然后，在最后一部分中，我们将通过对von Neumann代数的结果进行补充，以\ cite {chhawe}开始。我们表明，对于每个原子von Neumann代数（不一定是$σ$ -finite），将订单拓扑限制到$ m $ $ $ concides的限制与$σ$ -strong-strong Toupology $ s（m，m_ \ ast）的限制相吻合。我们记得\ cite {chhawe}的方法在很大程度上取决于$σ$ - finiteness的假设。除此之外，对于半限定度量空间，我们将全面了解与二元性$ \ langle l^1，l^\ infty \ rangle $及其订单拓扑相关的$ l^\ infty $之间的关系。
Different notions for order convergence have been considered by various authors. Associated to every notion of order convergence corresponds a topology, defined by taking as the closed sets those subsets of the poset satisfying that no net in them order converges to a point that is outside of the set. We shall give a thorough overview of these different notions and provide a systematic comparison of the associated topologies. Then, in the last section we shall give an application of this study by giving a result on von Neumann algebras complementing the study started in \cite{ChHaWe}. We show that for every atomic von Neumann algebra (not necessarily $σ$-finite) the restriction of the order topology to bounded parts of $M$ coincides with the restriction of the $σ$-strong topology $s(M,M_\ast)$. We recall that the methods of \cite{ChHaWe} rest heavily on the assumption of $σ$-finiteness. Further to this, for a semi-finite measure space, we shall give a complete picture of the relations between the topologies on $L^\infty$ associated with the duality $\langle L^1, L^\infty\rangle$ and its order topology.
