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Convergence in Measure and τ -Compactness of τ -Measurable Operators, Affiliated with a Semifinite von Neumann Algebra
Russian Mathematics Pub Date : 2020-06-27 , DOI: 10.3103/s1066369x20050096
A. M. Bikchentaev

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

中文翻译:

与半有限von Neumann代数有关的τ-可测算子的测度和τ-紧致性的收敛

τ为von Neumann代数上的忠实正态半有限迹。我们建立的标志交替系列的莱布尼茨标准τ -measurable运营商和提出的一系列“三明治”系列标准的模拟为τ -measurable运营商。我们证明了针对τ紧凑情况的该标准的改进。根据测度收敛拓扑,建立了任意τ-可测算子的τ-紧致性判据。我们还给出了1的一个充分条件)τ a的换向器的-compactness τ -measurable操作者和投影; 2)τ的收敛可测的算子和投影换向器序列在量度τ上为零算子。
更新日期:2020-06-27
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