Matrix norm inequality for C*-Algebras [closed]

Question

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$ are completely positive. With this I want to show the inequality $\left\Vert \left(a_{ij}\right)\right\Vert \leq n\left\Vert \sum_{i\text{, }j}a_{ij}\right\Vert $ for all $\left(a_{ij}\right)\in M_{n}\left(A\right)$. I'm not sure how to make use of the first part. Has someone an idea?

Answer 1

This inequality is wrong. Consider $A=M_2(\mathbb{C})$ and $(a_{ij})=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$.