On a family of linear maps from Mn(C) to Mn2(C)

In this paper, we present a parametric family of linear maps Φα,β,n:Mn(C)→Mn2(C) and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters α and β for which the family of maps Φα,β,n is positive for any natural number n≥3. We focus on the case of n=3, that is, Φα,β,3 and study the properties of 2-positivity, completely positivity and decomposability. In particular, we give values of parameters α and β for which the family of maps Φα,β,3 is 2-positive and not completely positive.