Having a $C^*$ Algebra $\mathcal{A}$ I want to show that $M_n(\mathcal{A})$ is a $*-$Algebra with product and involution:
$$(a_{i,j})\cdot(b_{i,j}) = \sum_{k=1}^n a_{i,k}b_{k,j} $$ $$ (a_{i,j})^* = (a_{j,i}^*)$$
then if I consider the norm,
$$\|(a_{i,j}) \cdot (b_{i,j})\| = max_{i} \sum_{j=1}^n \left \|\sum_{k=1}^na_{i,k}b_{k,j}\right \| \le max_{i} \sum_{j=1}^n \sum_{k=1}^n \|a_{i,k}\| \|b_{k,j}\| \dots ?$$
but I should get $\|(a_{i,j}) \cdot (b_{i,j})\| \le \|(a_{i,j})\| \cdot \|(b_{i,j})\|$
Also how to deal with involution? Thanks