Let $\mathcal{G} = \mathbb{M}\_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}\_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ divides $n$. Then is it true that there exists another subalgebra of $\mathcal{G}$ (let's call it $\mathcal{B}$) such that


1) $\mathcal{G} \cong \mathcal{A} \otimes \mathcal{B}$.


2) Every element in $\mathcal{A}$ commutes with every element in $\mathcal{B}$.


3) The intersection of $\mathcal{A}$ and $\mathcal{B}$ are multiples of the identity.


If it is not true, what are the necessary conditions?



Existence of tensor product of subalgebras
Yes, those are true and are consequences of the Double Centralizer Theorem for central simple algebras. See, for example, section 12.7 in Pierce's *Associative Algebras*.


