Let $A$ be the C*-algebra and $M_{n}(A)$ be the C*-algebra of $n\times n$ matrices with entries in $A$. We use $(a_{ij})$ to denote the element of $M_{n}(A)$.
My question is:
For every $a\in A$, can we verifies that$(a_{i}^{*}a^{*}aa_{j})\leq||a^{*}a||(a_{i}^{*}a_{j})$. Here, we fix $\{a_{i}\}_{i=1}^{n}\in A$.
Note that $M_n(\mathcal{A})$ is a $C^*$-algebra, so we have $$ A\leq B\implies C^*AC\leq C^*BC $$ for all $A,B,C\in M_n(\mathcal{A})$. Since $A\leq \Vert A\Vert 1_{M_n(\mathcal{A})}$ for all $A\in M_n(\mathcal{A})_+$ we get $$ C^*AC\leq \Vert A\Vert C^*C $$ Now consider $$ A= \begin{pmatrix} a^*a & 0 & \ldots & 0\\ 0 & a^*a & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & a^*a\\ \end{pmatrix} \qquad C= \begin{pmatrix} a_1 & a_2 & \ldots & a_n\\ 0 & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & 0\\ \end{pmatrix} $$