On Some Algebras of Operators. II
On Some Algebras of Operators. II
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DOI：
10.1112/plms/s3-16.1.385
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发表时间：
1965
影响因子：
1.8
通讯作者：
J. Ringrose
中科院分区：
文献类型：
--
作者：
J. Ringrose
本文继续研究在（7）中介绍的称为“套代数”的算子代数，并应用我们的理论回答Kadison和Singer（3）中关于极大三角代数的一些问题。我们的主要结果（定理4.1，4.2，5.4，5.5）是关于这类代数之间的代数同构的;事实证明，在许多情况下，这类同构必然是空间的。对于超可约极大三角代数的特殊情况，我们的一些结果包含在（5）中;下面使用的许多技巧是从（5）中发展而来的。通过极大三角代数之间的代数同构保持超约化的断言，以及通过不存在可分性条件，定理5.4和5.5给出了（5）中没有的信息。然而，§ 5的主要兴趣在于它对某些极大三角代数而不是超可约代数的处理。在本文中，术语Hilbert空间、子空间、投影分别表示复Hilbert空间、闭子空间、正交投影。给定希尔伯特空间H的一个子空间M，我们将PM写为从H到M的投影，HQM写为M在H中的正交补。设{Mu}是H的一个子空间集合，则包含每个Mu的最小子空间记为V-^ a>，包含在每个Mu中的最大子空间记为A-^a-集合包含记为g，我们保留“<=”用于正确包含。
In this paper we continue the study of certain algebras of operators termed'nest algebras', which were introduced in (7); and we apply our theory to answer some questions about the maximal triangular algebras of Kadison and Singer (3). Our main results (Theorems 4.1, 4.2, 5.4, 5.5) are concerned with algebraic isomorphisms between such algebras; it turns out that, in many instances, such isomorphisms are necessarily spatial.For the particular case of hyper-reducible maximal triangular algebras, some of our results are included in (5); and many of the techniques used below are developed from those of (5). By the assertion that hyperreducibility is preserved by algebraic isomorphisms between maximal triangular algebras, and by the absence of a separability condition, Theorems 5.4 and 5.5 give information not contained in (5). However, the main interest of § 5 lies in its treatment of certain maximal triangular algebras other than the hyper-reducible ones. Throughout this paper, the terms Hilbert space, subspace, projection are used to mean complex Hilbert space, closed subspace, orthogonal projection respectively. Given a subspace M of a Hilbert space H, we shall write PM for the projection from H on to M, and HQM for the orthogonal complement of M in H. If {Mu} is a collection of subspaces of H, then the smallest subspace which contains each Ma will be denoted by V-^ a> an d the largest subspace contained in each Ma will be denoted by A-^ a-Set inclusion will be denoted by'g', and we reserve'<='for proper inclusion.