k-Extreme Points in Symmetric Spaces of Measurable Operators

Let Μ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace Τ and E be a strongly symmetric Banach function space on [0, τ (1)). We show that an operator x in the unit sphere of Ε (Μ, τ) is k-extreme, k ∈ Ν, whenever its singular value function μ(x) is k-extreme and one of the following conditions hold (i) μ(∞, x) =lim_t→∞ μ (t, x)=0 or (ii) n(x) Μn (x*)=0 and ⌈x⌉ ≥ μ (∞, x) s(x), where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever Μ is non-atomic. The global k-rotundity property follows, that is if Μ is non-atomic then E is k-rotund if and only if Ε (Μ, 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 µ (f) is k-extreme and ⌈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 Ω (g), g ∈ L₁ + L∞, or Ω' (g), g ∈ L₁ [0, α), α < ∞, if and only if µ(f) =µ (g) and ⌈f⌉≥ (∞, f). From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.