I would like a source for some Artin-Wedderburn type facts about these algebras which seem to have easy proofs, and are probably written somewhere.


Let $\mathcal{A} \subset M\_n(\mathbb{C})$ be an algebra which is closed under taking adjoints. (That is, $X \in \mathcal{A} \Rightarrow X^\*\in \mathcal{A}$)


Then,
 $\mathcal{A}\cong \bigoplus\_i M\_{n\_i}(\mathbb{C})$ for some finite sequence of integers $n\_i.$ Furthermore if $\pi:\bigoplus\_i M\_{n\_i}(\mathbb{C}) \rightarrow \mathcal{A}$ is a $\*$-isomorphism, then there is a unitary $U$ and integers $m\_i$ such that for every $ \bigoplus\_i X\_i\in\bigoplus\_i M\_{n\_i}(\mathbb{C}),$ we can write the homomorphism via the formula
$\pi(\bigoplus\_i X\_i) = U^\* \bigoplus\_i (X\_i \otimes I\_{m\_i}) U.$



Structure theorem for finite dimensional $C^*$-algebras and their representations
$\def\CC{\mathbb C}$Suppose $A$ is a $\*$-closed subalgebra of $M\_n(\CC)$. $\CC^n$ is a left $M\_n(\CC)$-module in a canonical way, and therefore also an $A$-module. Using the fact that $A$ is $\*$-closed it is easy to see that if $S\subseteq\CC^n$ is an $A$-submodule, then $S^\perp$ is also an $A$-submodule. This implies that $\CC^n$ is a semisimple $A$-module.


$\CC^n$ is then a semisimple faithful $A$-module: it follows at once that $A$ itself is a semisimple algebra (because the Jacobson radical acts by zero on every semisimple module). The decomposition you want then follows from the usual A-W theorem.


