So, I am looking for some kind of 'product' $\bullet$ on the category of (unital?) $C^*$-algebras satisfying that
$M_n(\mathbb{C})\bullet M_m(\mathbb{C}) = M_{m+n}(\mathbb{C})$
where $M_n(\mathbb{C})$ denotes the complex $n\times n$ matrices. If such a thing indeed exists...
Motivation: I am studying a certain groupoud $G$ with a sub-groupoid $H$ and their corresponding groupoid $C^*$-algebras $C_r^*(G)$ and $C_r^*(H)$. By taking the reductions to a certain subset $A$ of the unit space of the groupoids, I get that $C_r^*(G|_A)\simeq M_{n+m}(\mathbb{C})$ and $C_r^*(H|_A)\simeq M_{n}(\mathbb{C})\oplus M_{m}(\mathbb{C})$. If there was some kind of general relationship between the algebras of the two reduction, perhaps it would shed light on the relationship between the two larger algebras as well.