A Family of Completely PPT Maps

Given $\alpha \in \mathbb{R}$ and $B_1,\dots ,B_q\in M_{m,n}$, consider the map $${\mathrm{\Phi }}_{\alpha ,\mathfrak{B}}^{\pm }(X)=\alpha tr(X)I_m\pm \sum _{s=1}^q{B}_{s}X{B}_s^{\ast }forallX\in M_n\mathrm{.}$$ In this study, necessary and sufficient conditions on $\alpha$ and $B_1,\dots ,B_q$ are given that guarantee ${\mathrm{\Phi }}_{\alpha ,\mathfrak{B}}^{\pm }$ is $k$-positive, $k$-copositive, and $k$-PPT, respectively. As application, known results about the map $\alpha tr(X)I_n\pm X$ are recovered. Furthermore, singular value inequalities associated with these positivity constraints are considered.