Ultrapowers of matrix algebras

Question

I have been working with ultrapowers a bit and was simply wondering whether the following statement is true.

Suppose $\omega$ denotes some free ultrafilter on $\mathbb{N}$ and let $A$ be any C$^\ast$-algebra. Then $\text{M}_n(A_\omega)$ may be identified with $(\text{M}_n(A))_\omega$.

Any hint or suggestion would be appreciated

Answer 1

First a general remark: If $J$ is a closed ideal in a $C^*$-algebra $A$ and $B$ is a nuclear $C^*$-algebra, then $$ \frac{A \otimes B}{J \otimes B} \cong (A / J) \otimes B. $$ Now apply this to $(A,J,B) = (\ell^\infty(A),c_0^\omega(A),M_n(\mathbb C)).$