  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.
