Matrix amplification of a $C^*$-algebras

Question

This is probably a very basic question. When discussing the $n-$fold amplification of a $C^*$-algebra, $M_n(A)$, we frequently denote it by the tensor product $A\otimes M_n(\mathbb{C})$. Why do we have this identification?

Does this follow from the distributive properties of tensor products (of modules over commutative rings)? And, if so, how?

Thank you very much

Answer 1

This is checked directly. Define $\gamma:M_n(A)\to A\otimes M_n(\mathbb C)$ by $$ \gamma:[A_{kj}]\longmapsto \sum_{kj}A_{kj}\otimes E_{kj}. $$ It is trivial to verify that this is a $*$-isomorphism.