On the bundle convergence of double orthogonal series in noncommutative L-2-spaces

The notion of bundle convergence in von Neumann algebras and their L-2-spac es for single (ordinary) sequences was introduced by Hensz, Jajte, and Pasz kiewicz in 1996. Bundle convergence is stronger than almost sure convergenc e in von Neumann algebras. Our main result is the extension of the two-para meter Rademacher-Men'shov theorem from the classical commutative case to th e noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series. Our method of pr oof is different from the classical one, because of the lack of the triangl e inequality in a noncommutative von Neumann algebra. In this context, bundle convergence resembles the regular convergence intro duced by Hardy in the classical case. The noncommutative counterpart of con vergence in Pringsheim's sense remains to be found.