迈克尔选择定理及其在马雷夏尔拓扑学中的应用

Pierre Fima, François Le Maître, Kunal Mukherjee, Issan Patri
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引用次数: 0

摘要

马歇尔拓扑学(Mar\'echal topology),也叫埃夫罗斯-马歇尔拓扑学(Effros-Mar\'echal topology),是一种可以放在给定冯-诺依曼代数的所有冯-诺依曼子代数空间上的自然拓扑学。本文的主要目标是通过仔细研究可以放在对偶空间的弱-$*$封闭子空间上的拓扑来填补这一空白。我们还指出迈克尔选择定理如何被用作迈向马歇尔定理的一步,以及它如何简化了哈格鲁普和温斯洛对马歇尔拓扑的一个重要选择结果的证明。作为应用,我们证明了有限冯诺伊曼代数空间是完整的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Michael's selection theorem and applications to the Maréchal topology
The Mar\'echal topology, also called the Effros-Mar\'echal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Mar\'echal from 1973 that this topology is Polish as soon as the ambient algebra has separable predual, but the sketch of proof in her research announcement appears to have a small gap. Our main goal in this paper is to fill this gap by a careful look at the topologies one can put on the space of weak-$*$ closed subspaces of a dual space. We also indicate how Michael's selection theorem can be used as a step towards Mar\'echal's theorem, and how it simplifies the proof of an important selection result of Haagerup and Winsl{\o}w for the Mar\'echal topology. As an application, we show that the space of finite von Neumann algebras is $\mathbf\Pi^0_3$-complete.
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