Bhat [1] characterizes the family of linear maps defined on B(H) which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study k-positivity of linear maps defined on finite-dimensional matrix algebras. In [5] Choi showed that n-positivity is different from (n − 1)-positivity for the linear maps defined on n by n matrix algebras. In this paper, we present a parametric family of linear maps Φ α,β,n : Mn(C) → M n 2 (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 Φ α,β,n is positive for any natural number n ≥ 3. We focus on the case of n = 3, that is, Φ α,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 Φ α,β,3 is 2-positive and not completely positive.

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