This article is cited in 1 paper

Maximal quasi-normed extension of quasi-normed lattices

A. G. Kusraev^ab, B. B. Tasoev<sup>c

a</sup> North Ossetian State University, 44-46 Vatutin Street, Vladikavkaz, 362025, Russia
^b Vladikavkaz Science Center of the RAS, 22 Markus Street, Vladikavkaz, 362027, Russia
^c Southern Mathematical Institute — the Affiliate of Vladikavkaz Science Center of the RAS, 22 Markus street, Vladikavkaz, 362027, Russia

Abstract: The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension $X^\varkappa$ of a Dedekind complete quasi-normed lattice $X$ with the weak $\sigma$-Fatou property is a quasi-Banach lattice if and only if $X$ is intervally complete. Moreover, $X^\varkappa$ has the Fatou and the Levi property provided that $X$ is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions with respect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.

Key words: quasi-Banach lattice, maximal quasi-normed extension, Fatou property, Levi property vector measure, space of weakly integrable functions.

UDC: 517.98

MSC: 46A16, 46B42, 46E30, 46G10, 47B38, 47G10

Received: 14.07.2017

Language: English