  We introduce the concept of completely positive roots of completely positive
maps on operator algebras. We do this in different forms: as asymptotic roots,
proper discrete roots and as continuous one-parameter semigroups of roots. We
present structural and general existence and non-existence results, some
special examples in settings where we understand the situation better, and
several challenging open problems. Our study is closely related to Elfving's
embedding problem in classical probability and the divisibility problem of
quantum channels.
