k-Extreme Points in Symmetric Spaces of Measurable Operators

Abstract

Let $${\mathcal{M}}$$M be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $${\tau}$$τ and E be a strongly symmetric Banach function space on $${[0,\tau({\bf 1}))}$$[0,τ(1)). We show that an operator x in the unit sphere of $${E(\mathcal{M}, \tau)}$$E(M,τ) is k-extreme, $${k \in {\mathbb{N}}}$$k∈N , whenever its singular value function $${\mu(x)}$$μ(x) is k-extreme and one of the following conditions hold (i) $${\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}$$μ(∞,x)=limt→∞μ(t,x)=0 or (ii) $${n(x)\mathcal{M}n(x^*) = 0}$$n(x)Mn(formula presented.)=0 and $${|x| \geq \mu(\infty, x)s(x)}$$|x|≥μ(∞,x)s(x) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever $${\mathcal{M}}$$M is non-atomic. The global k-rotundity property follows, that is if $${\mathcal{M}}$$M is non-atomic then E is k-rotund if and only if $$E(\mathcal{M}, \tau)$$E(M,τ) 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 $${\mu(f)}$$μ(f) is k-extreme and $${|f| \geq \mu(\infty,f)}$$|f|≥μ(∞,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 $${\Omega(g),\,g \in L_1 + L_{\infty}}$$Ω(g),g∈L1+L∞ , or $${\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}$$(formula presentd.)(g),g∈L1[0,α),α<∞ , if and only if $${\mu(f) = \mu(g)}$$μ(f)=μ(g) and $${|f| \geq \mu(\infty, f)}$$|f|≥μ(∞,f). From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.

Publication Title

Integral Equations and Operator Theory

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