This article is cited in 5 papers

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

A. M. Bikchentaev^a, A. A. Sabirova<sup>b

a</sup> 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.

UDC: 517.98

Received: 25.02.2011

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

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