Norm of a matrix in a C$^*-$algebra

Question

For the C$^*$-algebra $\ $ A=Mat$_{n\times n}(\mathbb{C})$ and $a\in A$, is the norm of A $$\sup \{ \lVert Ax\rVert \ :\ \lVert x\rVert \leq 1 \} $$ equal to $\left( \sum_{i,j}|a_{i,j}|^2\right)^{\frac{1}{2}} $.

I think this is true. but I have no proof.

Answer 1

Not true.

For example, $n=2$, $a=\left(\begin{array}{cc} 1&0\\0&2\end{array}\right)$.

Then, $\|a\|=2$, while $\Big(\sum_{i,j}|a_{i,j}|^2\Big)^{1/2}=\sqrt{5}$.