Some applications of two completely copositive maps

Abstract: 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 $3\times 3$ block matrices are included.