This is an announcement for the paper "Uo-convergence and its applications to Ces`aro means in Banach lattices" by Niushan Gao, Vladimir G. Troitsky, and Foivos Xanthos.
Abstract: A net $(x_\alpha)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_\alpha-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Ces`aro means in Banach lattices. In particular, we establish an intrinsic version of Koml'os' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].
Archive classification: math.FA
Remarks: 45 pages
Submitted from: foivos@ryerson.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1509.07914
or