Suppose we have a C$^{*}$-algebra $E$ and distinct projections $p$ and $q$ in $E$. I want to find all possible irreducible representations of $A:=C^{*}(p,q)$ (up to unitary equivalence). (This is part of the last exercise in Rordam's book). Suppose that $\pi\colon A\to B(H)$ is an irreducible representation of $A$.
I think that $H$ must have dimension one, since every closed vector subspace of $H$ should be invariant under $\pi(A)$ (as $A$ is generated by projections) and because $\pi$ is irreducible. Thus, $B(H)\cong\mathbb{C}$. So we have $\pi(p)=:\lambda_{p}$ and $\pi(q)=:\lambda_{q}$, where $\lambda_{p},\lambda_{q}\in\{0,1\}$.
Therefore, we have four possibilities: $\lambda_{p}=\lambda_{q}=1$ or $\lambda_{p}=1,\lambda_{q}=0$ or $\lambda_{p}=0,\lambda_{q}=1$ or $\lambda_{p}=\lambda_{q}=0$.
I'm not sure if this proof is correct. There is a hint given in the exercise which I am not using at all.
Thanks.
Every irreducible representation of a pair of projections $p,q$ is either $1$- or $2$-dimensional. The 2-dimensional representations are given by $$p=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix},\ q=\begin{bmatrix} c^2 & cs \\ cs & s^2 \end{bmatrix},$$ where $c,s\in\mathbb R,\ c^2+s^2=1$.
The proof is not trivial. See e.g. the paper by Halmos "Two subspaces" (1969).