Bounded sequences in noncommutative bimodules of 𝜏-measurable operators

We establish a noncommutative version of a result due to Lindenstrauss and Tzafriri Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas, Band 92, Springer-Verlag, Berlin–New York, 1977]. Precisely, every bounded sequence x i i = 1 ∞ \xᵢ\₈=₁^ in a noncommutative quasi-Banach M M -bimodule E ⊂ L p (M, τ) + M E L₏ (M, ) + M (here, M M stands for a semifinite von Neumann algebra), p > 0 p>0, having order continuous quasi-norm either satisfies that there exists a constant c > 0 c>0 such that, for every choice a i i = 1 ∞ \aᵢ\₈=₁^ of scalars, ∫ 0 1 ‖ ∑ i = 1