Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von Neumann algebra
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2020), pp. 89-93
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Let $ \tau $ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of $ \tau $-measurable operators. An analogue of the criterion of “sandwich” convergence of series for $ \tau $-measurable operators is obtained. We prove a refinement of this criterion for the $ \tau $-compact case. In terms of measure convergence topology, the criterion of $ \tau $-compactness of an arbitrary $ \tau $-measurable operator is established. We also give a sufficient condition of 1) $ \tau $-compactness of the commutator of a $ \tau $-measurable operator and a projection; 2) convergence of $ \tau$-measurable operator and projection commutator sequences to the zero operator in the measure $ \tau $.
Keywords:
Hilbert space, von Neumann algebra, normal trace, measurable operator, topology of convergence in measure, series of operators, $ \tau $-compact operator.
@article{IVM_2020_5_a8,
author = {A. M. Bikchentaev},
title = {Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von {Neumann} algebra},
journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
pages = {89--93},
publisher = {mathdoc},
number = {5},
year = {2020},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/IVM_2020_5_a8/}
}
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%0 Journal Article %A A. M. Bikchentaev %T Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von Neumann algebra %J Izvestiâ vysših učebnyh zavedenij. Matematika %D 2020 %P 89-93 %N 5 %I mathdoc %U http://geodesic.mathdoc.fr/item/IVM_2020_5_a8/ %G ru %F IVM_2020_5_a8
A. M. Bikchentaev. Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von Neumann algebra. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2020), pp. 89-93. http://geodesic.mathdoc.fr/item/IVM_2020_5_a8/