Let $A$ be a unital $\mathrm{C}^*$-algebra and $p,\,q$ non-zero projections in $A$. Consider two faithful representations of $A$ on Hilbert spaces $\mathsf{H}_1,\,\mathsf{H}_2$.
Is it possible for the intersections of the ranges of the projections to be zero in one representation and non-zero in the other?
Let $H$ be a Hilbert space with orthonormal basis $(e_n)$. Let $p\in B(H)$ be the projection onto the span of the vectors $e_{2n}$ and let $q\in B(H)$ be the projection onto the span of the vectors $e_{2n}+2^{-n}e_{2n+1}$.
Note now that $p-q$ is a compact operator (it can be written as the norm limit of the finite rank operators $p_n-q_n$ where $p_n$ and $q_n$ are the corresponding projections onto the spans of just the first $n$ of the basis vectors). So, $p$ and $q$ are equal and nonzero projections in the quotient $B(H)/K(H)$. Now take a faithful representation of $B(H)/K(H)$ on some Hilbert space $H'$, so $p$ and $q$ act by the same nonzero projection on $H'$. Then $B(H)$ has a natural faithful representation on $H\oplus H'$, and the ranges of $p$ and $q$ have nontrivial intersection in this representation. However, in the faithful representation on just $H$, the ranges of $p$ and $q$ have trivial intersection.