Statement/proof that a matrix algebra cannot be *-isomorphic to the direct sum of smaller matrix algebras.

Question

Let $n \in \mathbb{N}$ and $k_1, k_2, \ldots, k_p \in \mathbb{N}$ with all $k_i < n$.

I'm writing a proof that asserts $\mathfrak{gl}(n,\mathbb{C})$ cannot be *-isomorphic to any subalgebra of $\mathcal{A}$ where $$\mathcal{A} = \mathfrak{gl}(k_1,\mathbb{C}) \oplus \mathfrak{gl}(k_2,\mathbb{C}) \oplus \cdots \oplus \mathfrak{gl}(k_p,\mathbb{C})$$

More specifically, I'm asserting that $\not\exists \;\mathcal{B}\subseteq \mathcal{A}, U$ such that $$U\mathcal{B}U^\dagger = \mathfrak{gl}(n,\mathbb{C}) \oplus 0_{m\times m}$$ where $U$ is a unitary matrix and $m = \sum_i k_i - n$.

To me, this result seems obvious based on the observation that the matrix blocks of $\mathcal{A}$ can never fully "cover" $\mathfrak{gl}(n,\mathbb{C})$. It's been suggested that I might find a proof of it (or something similar) in a book on Morita equivalence. My problem is that I know little about operator theory and even less about Morita equivalence. I'm simply in need of a reference (ideally, a book name and chapter/page number) to back up this assertion.

Of course, if my intuition is wrong and the assertion is false, please set me straight.

Thanks for any assistance.

Answer 1

I can't find a reference for a proof, but a proof isn't too hard to obtain, as long as you know that $M(n,\mathbb C)$ is simple:

We can proceed by induction on $p$. If $p=1$, this is obvious just by counting dimensions. Suppose the result holds for some $p$, and $$\mathcal A\cong M(k_1,\mathbb C)\oplus\cdots\oplus M(k_{p+1},\mathbb C),$$ with $k_i<n$ for $1\leq i\leq p+1$. If $M(n,\mathbb C)\subset\mathcal A$, then $M(n,\mathbb C)\cap(0\oplus M(k_{p+1},\mathbb C))$ is an ideal in $M_n(\mathbb C)$, and since $k_{p+1}<n$ we must have $M(n,\mathbb C)\cap(0\oplus M(k_{p+1},\mathbb C))=0$. It follows that $$M(n,\mathbb C)\subset M(k_1,\mathbb C)\oplus\cdots\oplus M(k_p,\mathbb C)\oplus 0,$$ which is impossible.