Generalized master equations leading to completely positive dynamics

We provide a general construction of quantum generalized master equations with memory kernel leading to well defined, that is completely positive and trace preserving, time evolutions. The approach builds on an operator generalization of memory kernels appearing in the description of non-Markovian classical processes, and puts into evidence the non uniqueness of the relationship arising due to the typical quantum issue of operator ordering. The approach provides a physical interpretation of the structure of the kernels, and its connection with the classical viewpoint allows for a trajectory description of the dynamics. Previous apparently unrelated results are now connected in a unified framework, which further allows to phenomenologically construct a large class of non-Markovian evolutions taking as starting point collections of time dependent maps and instantaneous transformations describing the microscopic interaction dynamics.