Structure in Reflexive Subspace Lattices
Structure in Reflexive Subspace Lattices
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DOI：
10.1112/jlms/s2-26.1.117
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发表时间：
1982-08
影响因子：
1.2
通讯作者：
F. Gilfeather;D. Larson
中科院分区：
文献类型：
--
作者：
F. Gilfeather;D. Larson
作用在希尔伯特空间上的投影（或子空间）的自反格的形式理论是由PR Halmos [9]在1970年提出的，此后许多作者在几个方向上取得了实质性的进展。本文的目的是解决在反身性研究中出现的某些结构性问题。这项研究的动机来自于一个愿望，以一个易于处理的方式来描述的不变子空间格的一个嵌套子代数的冯诺依曼代数（nsva）。定理3.5表明，包含一个套和一个与套成对交换的自反（不一定是交换的）正交可补格的最小完备格是自反格。然后，详细分析了从基本成分构造联格的元素的方式。Arveson证明了作用在可分Hilbert空间上的每个交换子空间格都是自反的。众所周知，完备性、强闭包和自反性对于可分离作用交换格是等价的，因此这样的子空间格可以被看作是由至多可数的成对交换套族生成的完备格。提出的问题是，是否可以用更一般的自反格来代替“巢”这个词。在这个方向上，我们的研究表明，答案是肯定的情况下，一个单一的巢和一个单一的自反正交格。一般的问题似乎是开放的，即使是两个成对交换的自反正交格。
A formal theory of reflexive lattices of projections (or subspaces) acting on Hilbert space was initiated by PR Halmos [9] in 1970, and substantial progress has since been made by a number of authors in several directions. The purpose of this paper is to address certain structural questions that arise in the study of reflexivity. Motivation for this study came from a desire to characterize the invariant subspace lattice of a nest subalgebra of a von Neumann algebra (nsva) in a tractable fashion. Theorem 3.5 shows that the smallest complete lattice containing a nest and a reflexive (not necessarily commutative) orthogonally complemented lattice commuting pairwise with the nest is a reflexive lattice. A detailed analysis of the manner in which elements of the join lattice can be constructed from elementary constituents is then undertaken.In [2] W. Arveson proved that every commutative subspace lattice acting on separable Hilbert space is reflexive. It is known that completeness, strong closure, and reflexivity are equivalent for separably acting commutative lattices, and so such a subspace lattice can be viewed as the complete lattice generated by an at most countable family of pairwise commuting nests. The question is raised as to whether the word" nests" can be replaced by more general reflexive lattices. In this direction, our study shows that the answer is affirmative in the case of a single nest and a single reflexive ortholattice. The general question seems to be open even for two pairwise commuting reflexive ortholattices.