局部希尔伯特空间的直接积分和解拆

Chaitanya J. Kulkarni, Santhosh Kumar Pamula
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引用次数: 0

摘要

在这项工作中,我们通过使用局部标准度量空间(类似于经典设置中定义的标准度量空间)的概念,引入了局部希尔伯特空间直接积分的概念,我们通过考虑可度量空间的严格归纳系统和有限度量的投影系统得到了这个概念。接下来,我们定义一个局部希尔伯特空间,它由一个局部希尔伯特空间族的直接积分给出。之后,我们引入了可分解的局部有界算子和可对角化的局部有界算子。此外,我们还证明了可对角局部有界算子类是一个无边局部冯-诺依曼代数,这可以看作是可分解局部有界算子的换元。最后,我们讨论下面的反向问题:对于一个局部希尔伯特空间 $\mathcal{D}$ 和一个无边局部冯-诺依曼代数 $\mathcal{M}$ ,是否存在一个局部标准度量空间和一个局部希尔伯特空间族,使得 (1) 局部希尔伯特空间 $\mathcal{D}$ 与局部希尔伯特空间族的直接积分相一致;(2) $\mathcal{M}$ 的非等边局部冯-诺依曼代数与所有可对角局部有界运算符的非等边局部冯-诺依曼代数相一致?对于某类无性局部冯-诺依曼代数,我们给出了肯定的答案。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Direct Integral and Decompoisitions of Locally Hilbert spaces
In this work, we introduce the concept of direct integral of locally Hilbert spaces by using the notion of locally standard measure space (analogous to standard measure space defined in the classical setup), which we obtain by considering a strictly inductive system of measurable spaces along with a projective system of finite measures. Next, we define a locally Hilbert space given by the direct integral of a family of locally Hilbert spaces. Following that we introduce decomposable locally bounded and diagonalizable locally bounded operators. Further, we show that the class of diagonalizable locally bounded operators is an abelian locally von Neumann algebra, and this can be seen as the commutant of decomposable locally bounded operators. Finally, we discuss the following converse question: For a locally Hilbert space $\mathcal{D}$ and an abelian locally von Neumann algebra $\mathcal{M}$, does there exist a locally standard measure space and a family of locally Hilbert spaces such that (1) the locally Hilbert space $\mathcal{D}$ is identified with the direct integral of family of locally Hilbert spaces; (2) the abelian locally von Neumann algebra $\mathcal{M}$ is identified with the abelian locally von Neumann algebra of all diagonalizable locally bounded operators ? We answer this question affirmatively for a certain class of abelian locally von Neumann algebras.
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