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 Sibirsk. Mat. Zh., 2012, Volume 53, Number 2, Pages 258–270 (Mi smj2304)

Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators

A. M. Bikchentaeva, A. A. Sabirovab

a Research Institute of Mathematics and Mechanics, Kazan State University, Kazan
b Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, Kazan

Abstract: Consider a von Neumann algebra $\mathcal M$ with a faithful normal semifinite trace $\tau$. We prove that each order bounded sequence of $\tau$-compact operators includes a subsequence whose arithmetic averages converge in $\tau$. We also prove a noncommutative analog of Pratt's lemma for $L_1(\mathcal M,\tau)$. The results are new even for the algebra $\mathcal{M=B(H)}$ of bounded linear operators with the canonical trace $\tau=\mathrm{tr}$ on a Hilbert space $\mathcal H$. We apply the main result to $L_p(\mathcal M,\tau)$ with $0<p\le1$ and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of $\tau$-compactness of the dominant.

Keywords: Hilbert space, von Neumann algebra, normal semifinite trace, measurable operator, topology of convergence in measure, spectral theorem, Banach space, Banach–Saks property, arithmetic average.

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English version:
Siberian Mathematical Journal, 2012, 53:2, 207–216

Bibliographic databases:

UDC: 517.98

Citation: A. M. Bikchentaev, A. A. Sabirova, “Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators”, Sibirsk. Mat. Zh., 53:2 (2012), 258–270; Siberian Math. J., 53:2 (2012), 207–216

Citation in format AMSBIB
\Bibitem{BikSab12} \by A.~M.~Bikchentaev, A.~A.~Sabirova \paper Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators \jour Sibirsk. Mat. Zh. \yr 2012 \vol 53 \issue 2 \pages 258--270 \mathnet{http://mi.mathnet.ru/smj2304} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2975933} \transl \jour Siberian Math. J. \yr 2012 \vol 53 \issue 2 \pages 207--216 \crossref{https://doi.org/10.1134/S0037446612020036} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000303357900003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84860359795} 

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This publication is cited in the following articles:
1. Bikchentaev A.M., “On Sets of Measurable Operators Convex and Closed in Topology of Convergence in Measure”, Dokl. Math., 98:3 (2018), 545–548
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