Geodesic
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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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/