on the page 44 of Completely Bounded Maps and Operator Algebras by Vern Paulsen , in the proof of Stinespring Dilation Theorem, he uses a matrix factorization result: in $M_n(\mathcal{A}^+)$, the following inequality results satisfied
$$(a_i^*a^*aa_j) \le \|a^*a\|\cdot(a^*_ia_j) $$
Could anyone help me showing this?
If $\mathcal{A}$ is faithfully represented on $H$ and $\xi \in H$, then $$ \langle (a_i^\ast a^\ast a a_j)\xi,\xi\rangle=\sum_{i,j}\langle aa_i \xi_i,aa_j\xi_j\rangle=\left\lVert a\sum_i a_i \xi_i\right\rVert^2\leq \|a\|^2\left\lVert \sum_i a_i\xi_i\right\rVert^2=\|a^\ast a\|\langle (a_i^\ast a_j)\xi,\xi\rangle. $$ Alternatively, let $A\in M_n(\mathcal{A})$ be the matrix with entries $A_{1j}=a_j$ and $A_{ij}=0$ for $i>1$, and let $D\in M_n(\mathcal{A})$ be the diagonal matrix with all diagonal entries equal to $a^\ast a$. By elementary properties of the order in $C^\ast$-algebras one has $$ (a_i^\ast a^\ast a a_j)=A^\ast D A\leq \|D\|A^\ast A=\|a^\ast a\|(a_i^\ast a_j). $$