The Gelfand–Phillips and Dunford–Pettis type properties in bimodules of measurable operators

Abstract

We fully characterize noncommutative symmetric spaces E ( M , τ ) E(\mathcal {M},\tau ) affiliated with a semifinite von Neumann algebra M \mathcal {M} equipped with a faithful normal semifinite trace τ \tau on a (not necessarily separable) Hilbert space having the Gelfand–Phillips property and the WCG-property. The complete list of their relations with other classical structural properties (such as the Dunford–Pettis property, the Schur property and their variations) is given in the general setting of noncommutative symmetric spaces.