M-embedded symmetric operator spaces and the derivation problem

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals

We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional ϕ fails the condition that if (Hardy-Littlewood-Polya submajorization) and 0 ≤ x, y, then 0 ≤ ϕ(x) ≤ ϕ(y). We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.