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

Volume 201 / 2010

Małgorzata Marta Czerwińska, Anna Kamińska Studia Mathematica 201 (2010), 253-285 MSC: 46B20, 46B28, 47L05, 47L20. DOI: 10.4064/sm201-3-3

Abstract

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.

Authors

  • Małgorzata Marta CzerwińskaDepartment of Mathematical Sciences
    The University of Memphis
    Memphis, TN 38152, U.S.A.
    e-mail
  • Anna KamińskaDepartment of Mathematical Sciences
    The University of Memphis
    Memphis, TN 38152
    e-mail

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