Norm matrix over C*algebras

Question

I have a little question about matrix with entries in $A$ a $C^*$-algebra.

One knows that we can endowed $M_n(A)$ with a unique norm that satisfies the $C^*$-algebra axioms.

But suppose that we have a matrix $T\in M_n(\mathbb{C})$ and $1$ is the unit of $A$.

Does the norm of the matrix in $M_n(A)$ of $\lVert [T_{i,j}.1]\rVert_{M_n(A)}$ equals to the operator norm of $\lVert T\rVert_{op}$ in $M_n(\mathbb{C})$.

Thanks for your help.

Answer 1

Yes.

The tensor norm is a cross norm. So, in $M_n(\mathbb C)\otimes A$, $$ \|T\otimes I_A\|=\|T\|\,\|I_A\|=\|T\|. $$

Or you can think that $M_n(\mathbb C)$ embeds in $M_n(\mathbb C)\otimes A$ as $M_n(\mathbb C)\otimes 1$. A $*$-monomorphism is isometric, so the norm in $M_n(\mathbb C)$ is the same as the norm in $M_n(\mathbb C)\otimes 1$.