Some applications of two completely copositive maps

A linear map $Phi :mathbb{M}_n to mathbb{M}_k$ is called completely copositive if the resulting matrix $[Phi (A_{j,i})]_{i,j=1}^m$ is positive semidefinite for any integer $m$ and positive semidefinite matrix $[A_{i,j}]_{i,j=1}^m$. In this paper, we present some applications of the completely copositive maps $Phi (X)=(mathrm{tr} X)I+X$ and $Psi (X)= (mathrm{tr} X)I-X$. Some new extensions about traces inequalities of positive semidefinite $3times 3$ block matrices are included.