1 Introduction -- 2 Commutative Theory -- 3 Noncommutative Theory.

The aim of this research is to develop a systematic scheme that makes it possible to transform important parts of the by now classical theory of summation of general orthonormal series into a similar theory for series in noncommutative $L_p$-spaces constructed over a noncommutative measure space (a von Neumann algebra of operators acting on a Hilbert space  together with a faithful normal state on this algebra).