Asymptotically isometric and isometric copies of \(\ell_1\) in some Banach function lattices

Anna Kamińska ; Mieczysław ~Mastyło

Commentationes Mathematicae, Tome 53 (2013) no. 2, p. 383−400.

Résumé

We identify the class of Caldern-Lozanovskii spaces that do not contain an asymptotically isometric copy of \(\ell_1\), and consequently we obtain the corresponding characterizations in the classes of Orlicz-Lorentz and Orlicz spaces equipped with the Luxemburg norm. We also give a~complete description of order continuous Orlicz-Lorentz spaces which contain (order) isometric copies of \(\ell_1^{(n)}\) for each integer \(n \geq 2\).~As an application we provide necessary and sufficient conditions for order continuous Orlicz-Lorentz spaces to contain an (order) isometric copy of \(\ell_1\).~In particular we give criteria in Orlicz and Lorentz spaces for (order) isometric containment of \(\ell_1^{(n)}\) and \(\ell_1\).~The results are applied to obtain the~description of universal Orlicz-Lorentz spaces for all two-dimensional normed spaces.

Détail

DOI : 10.14708/cm.v53i2.798

Mots-clés : Caldern-Lozanovskii spaces, Orlicz-Lorentz spaces, Orlicz spaces, Lorentz spaces, Marcinkiewicz spaces, isometric copies of \(\ell_1^{(n)}\) and \(\ell_1\), asymptotically isometric copies of \(\ell_1\), fixed point property