On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure
- 作者: Bikchentaev A.1
-
隶属关系:
- Lobachevskii Institute of Mathematics and Mechanics
- 期: 卷 98, 编号 3 (2018)
- 页面: 545-548
- 栏目: Mathematics
- URL: https://journals.rcsi.science/1064-5624/article/view/225577
- DOI: https://doi.org/10.1134/S1064562418070037
- ID: 225577
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详细
For a von Neumann algebra with a faithful normal semifinite trace, the properties of operator “intervals” of three types for operators measurable with respect to the trace are investigated. The first two operator intervals are convex and closed in the topology of convergence in measure, while the third operator interval is convex for all nonnegative operators if and only if the von Neumann algebra is Abelian. A sufficient condition for the operator intervals of the second and third types not to be compact in the topology of convergence in measure is found. For the algebra of all linear bounded operators in a Hilbert space, the operator intervals of the second and third types cannot be compact in the norm topology. A nonnegative operator is compact if and only if its operator interval of the first type is compact in the norm topology. New operator inequalities are proved. Applications to Schatten–von Neumann ideals are obtained. Two examples are considered.
作者简介
A. Bikchentaev
Lobachevskii Institute of Mathematics and Mechanics
编辑信件的主要联系方式.
Email: airat.bikchentaev@kpfu.ru
俄罗斯联邦, Kazan, 420008 Tatarstan