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 (Formula presented) in a noncommutative quasi-Banach (Formula presented)-bimodule (Formula presented) (here, (Formula presented) stands for a semifinite von Neumann algebra), (Formula presented), having order continuous quasi-norm (1) either satisfies that there exists a constant (Formula presented) such that, for every choice (Formula presented) of scalars, (Formula presented), (Formula presented) (2) or has a subsequence which is equivalent to a sequence of disjoint elements in (Formula presented). As a consequence, we answer a question by Randrianantoanina [J. Aust. Math. Soc. 74 (2003), pp. 331-350] in 2006.