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 ∞ \{x_i\}ᵢ₌₁^∞ in a noncommutative quasi-Banach M M -bimodule E ⊂ L p ( M , τ ) + M E⊂ Lₚ(M,τ )+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 ∞ \{a_i\}ᵢ₌₁^∞ of scalars, ∫ 0 1 ‖ ∑ i = 1