Families of positivity preserving but not completely positive superoperators

Superoperators (supops for short) which are positivity preserving (PP), but not completely positive (CP), serve to test the separability or distillability of quantum states. The two main test supops, 'transposition' and 'reduction', are of definite symmetry under a certain involution of supops, which we call # conjugation. These symmetries are utilized to build families of PP but not CP test superoperators.