k-Extreme Points in Symmetric Spaces of Measurable Operators

Abstract

Let ${\mathcal{M}}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace ${\tau}$ and E be a strongly symmetric Banach function space on ${[0,\tau({\bf 1}))}$ . We show that an operator x in the unit sphere of ${E(\mathcal{M}, \tau)}$ is k-extreme, ${k \in {\mathbb{N}}}$ , whenever its singular value function ${\mu(x)}$ is k-extreme and one of the following conditions hold (i) ${\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}$ or (ii) ${n(x)\mathcal{M}n(x^*) = 0}$ and ${|x| \geq \mu(\infty, x)s(x)}$ , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever ${\mathcal{M}}$ is non-atomic. The global k-rotundity property follows, that is if ${\mathcal{M}}$ is non-atomic then E is k-rotund if and only if $E(\mathcal{M}, \tau)$ is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement ${\mu(f)}$ is k-extreme and ${|f| \geq \mu(\infty,f)}$ . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit ${\Omega(g),\,g \in L_1 + L_{\infty}}$ , or ${\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}$ , if and only if ${\mu(f) = \mu(g)}$ and ${|f| \geq \mu(\infty, f)}$ . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.