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](https://mathoverflow.net/questions/26515/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? 



Simultaneous decomposition of three projectors 
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. 


