Maximal theorems of Menchoff--Rademacher type in non-commutative \(L_{q}\)-spaces. (Q1423434)

This paper provides the noncommutative versions of several classical convergence theorems in analysis, including the Menchoff--Rademacher theorem on orthogonal series in \(L_2[0,1]\) and the Bennett--Maurey--Nahoum theorem on unconditionally convergent series in \(L_1[0,1]\). The main tool used is the noncommutative maximal function, already studied in detail in the paper of the second author on the noncommutative Doob inequality [J. Reine Angew. Math. 549, 149--190 (2002; Zbl 1004.46043)]. To prove the previous convergence results, the authors establish the corresponding maximal inequalities in semifinite noncommutative \(L_p\)-spaces.