Generalizing Choi-like maps

A problem of further generalization of generalized Choi maps ${\mathrm{\Phi }}_{[a,b,c]}$ acting on ${\mathbb{M}}_3$ introduced by Cho, Kye and Lee is discussed. Some necessary conditions for positivity of the generalized maps are provided as well as some sufficient conditions. Also some sufficient condition for decomposability of these maps is shown.