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 withrespect 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.
Keywords: quasi-Banach lattice, maximal quasi-normed extension, Fatou property, Levi property vector measure, space of weakly integrable functions
For citation: Kusraev A. G., Tasoev B. B. Maximal quasi-normed extension of quasi-normed lattices // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 41-50.
DOI 10.23671/VNC.2017.3.7111
1. Abramovich Yu. A. On maximal normed extension of a
semi-ordered normed spaces. Izv. Vyssh. Uchebn. Zaved. Mat., 1970, vol.3, pp. 7-17.
2. Abramovich Y. A. and Aliprantis C. D.
An Invitation to Operator Theory. Providence (R.I.): Amer. Math. Soc,
2002. iv+530 p. (Graduate Stud. in Math., vol. 50).
3. Aliprantis C. D. and Burkinshaw O. Positive
Operators. London etc.: Acad. Press Inc., 1985. xvi+367 p.
4. Calabuig J. M., Delgado O., Juan M. A., and Sanchez Perez E. A. On the Banach lattice
structure of \(L^1_w\) of a vector measure on a \(\delta\)-ring. Collect. Math. 2014, vol. 65, pp. 67-85.
5. Curbera G. P. and Ricker W. J. Vector measures, integration, applications,
Positivity. Basel: Birkhauser, 2007, pp. 127-160 (Trends Math.).
6. Hyers D. H. A note on linear topological spaces. Bull. Amer.
Math. Soc., 1938, vol. 44, № 2, pp. 76-80.
7. Juan A. M. and Sanchez-Perez E. A. Maurey-Rosenthal domination for abstract Banach
lattices. J. Ineq. and Appl., 2013, vol. 213, pp. 1-12.
8. Kalton N. J. Convexity conditions for non-locally convex lattices. Glasgow Math. J., 1984, vol. 25, pp. 141-142.
9. Kalton N. J. Quasi-Banach spaces. Handbook of the Geometry of Banach Spaces
(Eds. W. B. Johnson and J. Lindenstrauss). Amsterdam a.o.: Elsevier, 2003. P. 1118-1130.
10. Kusraev A. G. and Tasoev B. B. Kantorovich-Wright integration
and representation of quasi-Banach lattices. Dokl. Math. 2017, vol. 474, pp. 15-18.
11. Kusraev A. G. and Tasoev B. B. Kantorovich-Wright integration
and representation of vector lattices, J. Math. Anal. Appl. 2017, vol. 455, pp. 554-568.
12. Kusraev A. G. and Tasoev B. B. Kantorovich-Wright
integration and representation of quasi-Banach lattices. J. Math. Anal. Appl. (to appear).
13. Luxemburg W. A. J. and Zaanen A. C. Riesz Spaces.
Vol. 1. Amsterdam-London: North-Holland, 1971. 514 pp.
14. Lewis D. R. On integration and summability in vector spaces. Illinois J.
Math. 1972, vol. 16, pp. 294-307.
15. Maligranda L. Type, cotype and convexity properties of quasi-Banach spaces. Proc.
of the International Symposium on Banach and Function Spaces Kitakyushu. Japan, 2003, pp. 83-120.
16. Masani P. R. and Niemi H. The integration theory of Banach space valued measures and the
Tonelli-Fubini theorems. II. Pettis integration. Adv. Math., 1989, vol. 75, pp. 121-167.
17. Meyer-Nieberg P. Banach Lattices. Berlin etc.: Springer,
1991. xvi+395 p.
18. Okada S., Ricker W. J., and Sanchez-Perez E. A. Optimal Domain and Integral
Extension of Operators Acting in Function Spaces. Basel: Birkhauser, 2008. (Oper. Theory Adv. Appl.,
vol. 180).
20. Sanchez Perez E. A. and Tradacete P. Bartle-Dunford-Schwartz integration for positive
vector measures and representation of quasi-Banach lattices.
J. Nonlin. and Conv. Anal. 2016, vol. 17, no. 2, pp. 387-402.
21. Veksler A. I. On realization of Archimedean \(K\)-lineals. Sib.
Math. J., 1962, vol. 3, no. 1, pp. 7-16.
22. Veksler A. I. The concept of a linear lattice which is normal in itself, and certain applications
of this concept to the theory of linear and linear normed lattices.
Isv. Vuzov. Math., 1966, vol. 4, pp. 13-22.
23. Veksler A. I. Interval completeness and intervally complete normability of
\(KN\)-lineals. Isv. Vuzov. Math., 1970, vol. 4, pp. 36-46.
24. Vulikh B. Z. Introduction to the Theory of Partially Ordered
Spaces. M.: Fizmatgiz, 1961 (in Russian).
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
