How to show matrix algebras are Nuclear

Question

How would I show that $M_n(\mathbb{C})$ is nuclear? I know that $M_n(B)$ already is a C*-algebra with its obvious norm why would that show that then the min and max norm are the same?

Answer 1

Because $M_n(B)\simeq M_n(\mathbb C)\otimes B$ is canonically isomorphic to the algebraic tensor product. As the algebraic tensor product is already complete, you can only "complete it" in a single way. They key fact is that a C$^*$-algebra admits a unique norm.