(Non-)Dunford–Pettis operators on noncommutative symmetric spaces

Abstract

We fully characterize the class of von Neumann algebras M (on a separable Hilbert space) such that any non-Dunford–Pettis operator on the predual M⁎ fixes a copy of ⊕ℓ1(ℓ2), which extends results due to Bourgain [9] and Rosenthal [49]. We also fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann algebras, which have the Dunford–Pettis property. This extends earlier results for the special cases (e.g., for symmetric function spaces, and von Neumann algebras and their preduals), by Bunce [11], by Chu and Iochum [15], and by Kaminska and Mastylo [36]. Finally, we supply new examples of symmetric spaces with the DPP on general (non-resonant) σ-finite measure spaces.