Let $n$ be a positive integer and consider of finite set $S \subset M_n(\mathbb{C})$ such that $S^* = S$ (i.e. if $a \in S$ then $a^* \in S$). The algebra generated by $S$ is a finite dimensional $*$-algebra over $\mathbb{C}$, so is isomorphic to a direct sum of matrix algebras, i.e. there are $n_1 \le n_2 \le \dots \le n_r$ such that:
$$\langle S \rangle \simeq \bigoplus_{i=1}^r M_{n_i}(\mathbb{C})$$

Question: How to compute the change of basis $p$ such that for all $a \in S$, we have $p^{-1}ap$ block-diagonal as for the above decomposition?

We can see each $M_{n_i}(\mathbb{C})$ above as $End(V_i)$ with $V_i$ an irreducible representation of $\langle S \rangle$.

If the matrices commute over each other then (using $S^*=S$) we get that the matrices are normal, so diagonalizable, and then what we ask in this case is just a simultaneous diagonalization (ok). So in general, what I am looking for is how to compute a (thinnest) simultaneous block-diagonalization.

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    $\begingroup$ Vrej Zarikian wrote a preprint that solves this problem (and a couple others) called "Algorithms for Operator Algebra Calculations" in 2003. However, I am not able to find a public copy of this paper anywhere online, so I am hesitant to share it without permission. Maybe you could e-mail him and ask? usna.edu/Users/math/zarikian/index.php $\endgroup$ – Nathaniel Johnston Sep 17 '19 at 11:42

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