Simultaneous decomposition of three projectors

Question

A projector $P$ is a Hermitian matrix satisfying $P^2=P$. For any two projectors, it is easy to show that there exists a unitary matrix $U$ such that both $U^*PU$ and $U^*QU$ are block-diagonal matrices, each block is of size at most $2$.

For arbitrary $k$ projectors, there is no simultaneous decomposition such that the sizes of all blocks are upper bounded by a function of $k$. The answer is given here

Simultaneous Block decomposition of a set of orthogonal projections.

My question is whether there are some special cases we can generalize. The following is a special case I am interested.

Given three projectors $\Pi_1, \Pi_2, \Delta$, where $Range(\Pi_1)\subseteq Range(\Pi_2)$, is there a similar block decomposition such that the size each block is constant?

Answer 1

One way to put these questions is to ask for a classification of finite-dimensional $*$-representations of a universal $C^*$-algebra. In the first case it is the universal $C^*$-algebra generated by two projections, i.e. the unital $C^*$-free product of $\mathbb C^2$ with itself. You observed that the maximal dimension of a irreducible $*$-representation of this algebra is two.

In the second case, you are considering the unital $C^*$-free product of $\mathbb C^3$ with $\mathbb C^2$. The two projections $\Pi_1$ and $\Pi_2$ correspond to $(1,0,0)$ and $(1,1,0)$ in $\mathbb C^3$. This $C^*$-algebra is isomorphic to the universal group $C^*$-algebra of $PSL_2(\mathbb Z)$, being the free product of $\mathbb Z_3$ and $\mathbb Z_2$. It is well-known that the unitary representation theory of $PSL_2(\mathbb Z)$ (which is equivalent to the $*$-representation theory of its universal group $C^*$-algebra) is wild and there is no bound on the dimension of irreducible finite-dimensional representations.