Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators

Małgorzata Marta Czerwińska¹ ; Anna Kamińska²

¹ Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A.
² Department of Mathematical Sciences The University of Memphis Memphis, TN 38152

Studia Mathematica, Tome 201 (2010) no. 3, pp. 253-285

Résumé

We investigate the relationships between strongly extreme, complex extreme, and complex locally uniformly rotund points of the unit ball of a symmetric function space or a symmetric sequence space $E$, and of the unit ball of the space $E({\cal M},\tau)$ of $\tau$-measurable operators associated to a semifinite von Neumann algebra $(\mathcal{M}, \tau)$ or of the unit ball in the unitary matrix space $C_E$. We prove that strongly extreme, complex extreme, and complex locally uniformly rotund points $x$ of the unit ball of the symmetric space $E(\mathcal{M}, \tau)$ inherit these properties from their singular value function $\mu(x)$ in the unit ball of $E$ with additional necessary requirements on $x$ in the case of complex extreme points. We also obtain the full converse statements for the von Neumann algebra $\mathcal{M}$ with a faithful, normal, $\sigma$-finite trace $\tau$ as well as for the unitary matrix space $C_E$. Consequently, corresponding results on the global properties such as midpoint local uniform rotundity, complex rotundity and complex local uniform rotundity follow.

Détail

DOI : 10.4064/sm201-3-3

Keywords: investigate relationships between strongly extreme complex extreme complex locally uniformly rotund points unit ball symmetric function space symmetric sequence space unit ball space cal tau tau measurable operators associated semifinite von neumann algebra mathcal tau unit ball unitary matrix space prove strongly extreme complex extreme complex locally uniformly rotund points unit ball symmetric space mathcal tau inherit these properties their singular value function unit ball additional necessary requirements complex extreme points obtain full converse statements von neumann algebra mathcal faithful normal sigma finite trace tau unitary matrix space consequently corresponding results global properties midpoint local uniform rotundity complex rotundity complex local uniform rotundity follow