Article
| Title: | The positive cone of a Banach lattice. Coincidence of topologies and metrizability (English) |
| Author: | Lipecki, Zbigniew |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 475-483 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Let $X$ be a Banach lattice, and denote by $X_+$ its positive cone. The weak topology on $X_+$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1(\Gamma)$ for a set $\Gamma$. The weak$^*$ topology on $X_+^*$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C(K)$-space, where $K$ is a metrizable compact space. (English) |
| Keyword: | normed lattice |
| Keyword: | Banach lattice |
| Keyword: | positive cone |
| Keyword: | AM-space |
| Keyword: | AL-space |
| Keyword: | Banach lattice $C(K)$ |
| Keyword: | Banach lattice $l^1(\Gamma)$ |
| Keyword: | strong topology |
| Keyword: | weak topology |
| Keyword: | weak$^*$ topology |
| Keyword: | coincidence of topologies |
| Keyword: | metrizability |
| Keyword: | nonatomic measure |
| MSC: | 46B42 |
| MSC: | 46E05 |
| MSC: | 54E35 |
| DOI: | 10.14712/1213-7243.2024.004 |
| . | |
| Date available: | 2024-11-05T11:50:31Z |
| Last updated: | 2024-11-05 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152626 |
| . | |
| Reference: | [1] Aliprantis C. D., Burkinshaw O.: Positive Operators.Pure and Applied Mathematics, 119, Academic Press, Orlando, 1985. Zbl 1098.47001, MR 0809372 |
| Reference: | [2] Barroso C. S., Kalenda O. F. K., Lin P.-K.: On the approximate fixed point property in abstract spaces.Math. Z. 271 (2012), no. 3–4, 1271–1285. MR 2945607, 10.1007/s00209-011-0915-6 |
| Reference: | [3] Bauer H.: Measure and Integration Theory.De Gruyter Studies in Mathematics, 26, Walter de Gruyter, Berlin, 2001. MR 1897176 |
| Reference: | [4] Dilworth S. J.: A weak topology characterization of $l_1(m)$.Geometry of Banach Spaces, Strobl 1989, London Math. Soc. Lecture Note Ser., 158, Cambridge University Press, Cambridge, 1990, pages 89–94. MR 1110188 |
| Reference: | [5] Dunford N., Schwartz J. T.: Linear Operators. I. General Theory.Pure and Applied Mathematics, 7, Interscience Publishers, New York, 1958. MR 0117523 |
| Reference: | [6] Edgar G. A., Wheeler R. F.: Topological properties of Banach spaces.Pacific J. Math. 115 (1984), no. 2, 317–350. MR 0765190, 10.2140/pjm.1984.115.317 |
| Reference: | [7] Engelking R.: General Topology.Monografie Matematyczne, 60, PWN---Polish Scientific Publishers, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference: | [8] Halmos P. R.: A Hilbert Space Problem Book.D. van Nostrand Co., Princeton, 1967. MR 0208368 |
| Reference: | [9] Lipecki Z.: On compactness and extreme points of some sets of quasi-measures and measures. IV.Manuscripta Math. 123 (2007), no. 2, 133–146. Zbl 1118.28003, MR 2306629, 10.1007/s00229-007-0085-3 |
| Reference: | [10] Lipecki Z.: Order intervals in $C(K)$. Compactness, coincidence of topologies, metrizability.Comment. Math. Univ. Carolin. 63 (2022), no. 3, 295–306. MR 4542790 |
| Reference: | [11] Rudin W.: Functional Analysis.McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. Zbl 0867.46001, MR 0365062 |
| Reference: | [12] Semadeni Z.: Banach Spaces of Continuous Functions. Vol. I.Monografie Matematyczne, 55, PWN---Polish Scientific Publishers, Warszawa, 1971. MR 0296671 |
| Reference: | [13] Varadarajan V. S.: Weak convergence of measures on separable metric spaces.Sankhyā 19 (1958), 15–22. MR 0094838 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
