The $2$-co-lacunary subsequence principle for arbitrary noncommutative $L_1$-spaces

Abstract

For every von Neumann algebra \(\M\), we prove that each bounded sequence in the Haagerup space \(L_1(\M)=\M_*\) admits either a norm-convergent subsequence or a \(2\)-co-lacunary subsequence. This answers an open problem of Parcet and Randrianantoanina \cite[Remark~3.11]{ParcetRandrianantoanina2006} and removes both hyperfiniteness and semifiniteness.