PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Order intervals in Banach lattices and their extreme points

Volume 160 / 2020

Zbigniew Lipecki Colloquium Mathematicum 160 (2020), 119-132 MSC: 46A40, 46B42, 52A07, 28B05, 06E99. DOI: 10.4064/cm7726-5-2019 Published online: 15 January 2020

Abstract

Let $X$ be a Banach lattice with order continuous norm. Then (A) $X$ is atomic if and only if $\operatorname{extr} [0,\,x]$ is weakly closed for every $x\in X_+$ if and only if the weak and strong topologies coincide on $[0,\,x]$ for every $x\in X_+$; (B) $X$ is nonatomic if and only if $\operatorname{extr} [0,\,x]$ is weakly dense in $[0,\,x]$ for every $x\in X_+$. Let, in addition, $X$ have a weak order unit. Then (C) $X^*$ is atomic if and only if $\operatorname{extr} [0,\,x^*]$ is weak$^*$ closed for every $x^*\in X_+^*$; (D) $X^*$ is nonatomic if and only if $\operatorname{extr} [0,\,x^*]$ is weak$^*$ dense in $[0,\,x^*]$ for every $x^*\in X_+^*$.

Authors

  • Zbigniew LipeckiInstitute of Mathematics
    Polish Academy of Sciences
    Wrocław Branch
    Kopernika 18
    51-617 Wrocław, Poland
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image