Isomorphic matrix algebras with non-isomorphic C*-algebras

Let $A$ and $B$ be two $C^{\ast}$-algebras such that their matrix algebras, $M_2(A)$ and $M_2(B)$, are $\ast$-isomorphic $C^\ast$-algebras.

Question 1: Are $A$ and $B$ isomorphic $C^\ast$-algebras?

In a related case, let $X$ and $Y$ be locally compact topological spaces.

Question 2: Are there non-homeomorphic $X$ and $Y$ such that $M_2\big(C_0(X)\big)$ and $M_2\big(C_0(Y)\big)$ are $\ast$-isomorphic $C^\ast$-algebras?

It is obvious that "2" is not a forced condition.

Solution 1:

(after a couple years, I'm posting a summary of what was said in the comments so that the question does not remain unanswered)

The answer to both questions is no.

In this paper, Rordam shows that there exists a C$^*$-algebra $B$ such that $B$ is not stable while $M_2(B)$ is. So we can take $A=B\otimes K(H)$. Now $$ M_2(A)\simeq M_2(B)\otimes K(H)\simeq M_2(B). $$ But $A\not\simeq B$.

For the second question, if $M_2(C_0(X))\simeq M_2(C_0(Y))$ then both have equal Cuntz semigroup. But since the cuntz semigroup of $M_2(A)$ agrees with that of $A$, we get that $C_0(X)$ and $C_0(Y)$ have equal Cuntz semigroup; being abelian, we obtain $C_0(X)\simeq C_0(Y)$. Then $X$ and $Y$ are homeomorphic.