The positive cone of a Banach lattice. Coincidence of topologies and metrizability

Zbigniew Lipecki

Commentationes Mathematicae Universitatis Carolinae (2023)

  • Volume: 64, Issue: 4, page 475-483
  • ISSN: 0010-2628

Abstract

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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 ( Γ ) for a set Γ . 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.

How to cite

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Lipecki, Zbigniew. "The positive cone of a Banach lattice. Coincidence of topologies and metrizability." Commentationes Mathematicae Universitatis Carolinae 64.4 (2023): 475-483. <http://eudml.org/doc/299323>.

@article{Lipecki2023,
abstract = {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.},
author = {Lipecki, Zbigniew},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {normed lattice; Banach lattice; positive cone; AM-space; AL-space; Banach lattice $C(K)$; Banach lattice $l^1(\Gamma )$; strong topology; weak topology; weak$^*$ topology; coincidence of topologies; metrizability; nonatomic measure},
language = {eng},
number = {4},
pages = {475-483},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The positive cone of a Banach lattice. Coincidence of topologies and metrizability},
url = {http://eudml.org/doc/299323},
volume = {64},
year = {2023},
}

TY - JOUR
AU - Lipecki, Zbigniew
TI - The positive cone of a Banach lattice. Coincidence of topologies and metrizability
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2023
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 64
IS - 4
SP - 475
EP - 483
AB - 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.
LA - eng
KW - normed lattice; Banach lattice; positive cone; AM-space; AL-space; Banach lattice $C(K)$; Banach lattice $l^1(\Gamma )$; strong topology; weak topology; weak$^*$ topology; coincidence of topologies; metrizability; nonatomic measure
UR - http://eudml.org/doc/299323
ER -

References

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  9. Lipecki Z., 10.1007/s00229-007-0085-3, Manuscripta Math. 123 (2007), no. 2, 133–146. Zbl1118.28003MR2306629DOI10.1007/s00229-007-0085-3
  10. Lipecki Z., Order intervals in C ( K ) . Compactness, coincidence of topologies, metrizability, Comment. Math. Univ. Carolin. 63 (2022), no. 3, 295–306. MR4542790
  11. Rudin W., Functional Analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973. Zbl0867.46001MR0365062
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  13. Varadarajan V. S., Weak convergence of measures on separable metric spaces, Sankhyā 19 (1958), 15–22. MR0094838

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