ℓ₂-strictly singular operators on the predual of a hyperfinite von Neumann algebra

Abstract

We show that every ℓ 2 \ell _2 -strictly singular operator on the predual of any atomless hyperfinite finite von Neumann algebra M \mathcal {M} is Dunford–Pettis, which extends a Rosenthal’s theorem for the case of commutative algebra M = L ∞ ( 0 , 1 ) \mathcal {M}=L_\infty (0,1) . We also apply our result to the study of noncommutative symmetric spaces X = E ( M , τ ) X=E(\mathcal {M},\tau ) for which every ℓ 2 \ell _2 -strictly singular operator from L p ( 0 , 1 ) L_p(0,1) into X X is narrow.