Abstract
Consider a von Neumann algebra M with a faithful normal semifinite trace τ. We prove that each order bounded sequence of τ-compact operators includes a subsequence whose arithmetic averages converge in τ. We also prove a noncommutative analog of Pratt’s lemma for L 1(M, τ). The results are new even for the algebra M = B(H) of bounded linear operators with the canonical trace τ = tr on a Hilbert space H. We apply the main result to L p (M, τ) with 0 < p ≤ 1 and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of τ-compactness of the dominant.
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Original Russian Text Copyright © 2012 Bikchentaev A. M. and Sabirova A. A.
The authors were supported by the Ministry for Education and Science (State Contract 02.740.11.0193).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 53, No. 2, pp. 258–270, March–April, 2012.
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Bikchentaev, A.M., Sabirova, A.A. Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators. Sib Math J 53, 207–216 (2012). https://doi.org/10.1134/S0037446612020036
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DOI: https://doi.org/10.1134/S0037446612020036