发表机构
University of Houston; NWU（休斯顿大学; 西北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了von Neumann代数上态的正交投影可数可加性与序列弱*连续性等价的问题，并给出了Gleason定理的新变体及对Haagerup问题的部分解答。
AI 中文摘要
我们首先在ZFC中解决了第一作者和Weaver十年前提出的一个问题：von Neumann代数上的一个态在正交投影上是可数可加的，当且仅当它是序列弱*连续的，等价地，序列正规的。这样的态可以被视为量子（可数可加）概率测度，或者更确切地说，是它们到非交换积分的扩展。为此所用的主要工具是20世纪70年代直接积分理论中的已知思想。实际上，我们证明了关于sigma-有限von Neumann代数乘积上的态的分解定理。一个适应于给定序列的分解将我们主要结果中的连续性断言归结为经典的支配收敛定理。该论证不需要可分性假设。我们给出了几个应用。例如，我们给出了Gleason定理的一个新变体，描述了任何没有类型$I_2$直和项的von Neumann代数上的可数可加投影测度。然后我们转向权重情形，讨论序列正规权重以及著名的相关Haagerup问题1.11，给出了一些部分结果（这些结果在某种意义上可能被证明是最佳的）。例如，如果我们的权重是强或严格半有限的，或者对于某些类别的von Neumann代数没有限制，我们就解决了Haagerup的问题。我们对权重给出了我们主要结果的几个应用。在即将进行的工作中，我们考虑对“量子测度和积分理论”的许多应用，例如von Neumann代数的Lebesgue支配收敛定理的变体。
英文摘要
We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools for this are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. A decomposition adapted to a given sequence reduces the continuity assertion in our main result to the classical dominated convergence theorem. The argument needs no separability hypothesis. We give several applications. For example we give a new variant of Gleason theorem, describing the countably additive projection measures for any von Neumann algebra with no type $I_2$ direct summand. We then turn to the weight case, discussing sequentially normal weights and the famous related Haagerup's Problem 1.11, giving some partial results (which may conceivably may turn out to be best possible in a certain sense). For example we solve Haagerup's problem if our weight is strongly or strictly semifinite, or with no restrictions for certain classes of von Neumann algebras. We give several applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras.
