Complex rotundities and midpoint local uniform rotundity in symmetric spaces of measurable operators
Volume 201 / 2010
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.
