Abstract
In the Banach space L 1(M, τ) of operators integrable with respect to a tracial state τ on a von Neumann algebra M, convergence is analyzed. A notion of dispersion of operators in L 2(M, τ) is introduced, and its main properties are established. A convergence criterion in L 2(M, τ) in terms of the dispersion is proposed. It is shown that the following conditions for X ∈ L 1(M, τ) are equivalent: (i) τ(X) = 0, and (ii) ‖I + zX‖1 ≥ 1 for all z ∈ C. A.R. Padmanabhan’s result (1979) on a property of the norm of the space L 1(M, τ) is complemented. The convergence in L 2(M, τ) of the imaginary components of some bounded sequences of operators from M is established. Corollaries on the convergence of dispersions are obtained.
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Original Russian Text © A.M. Bikchentaev, 2016, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 293, pp. 73–82.
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Bikchentaev, A.M. Convergence of integrable operators affiliated to a finite von Neumann algebra. Proc. Steklov Inst. Math. 293, 67–76 (2016). https://doi.org/10.1134/S0081543816040052
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DOI: https://doi.org/10.1134/S0081543816040052