  We study families of positive and completely positive maps acting on a
bipartite system $\mathbb{C}^M\otimes \mathbb{C}^N$ (with $M\leq N$). The maps
have a property that when applied to any state (of a given entanglement class)
they result in a separable state, or more generally a state of another certain
entanglement class (e.g., Schmidt number $\leq k$). This allows us to derive
useful families of sufficient separability criteria. Explicit examples of such
criteria have been constructed for arbitrary $M,N$, with a special emphasis on
$M=2$. Our results can be viewed as generalizations of the known facts that in
the sufficiently close vicinity of the completely depolarized state (the
normalized identity matrix), all states are separable (belong to "weakly"
entangled classes). Alternatively, some of our results can be viewed as an
entanglement classification for a certain family of states, corresponding to
mixtures of the completely polarized state with pure state projectors,
partially transposed and locally transformed pure state projectors.
