I have a proof of the following:
Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra and $\{p_k\}^{N}_{k=1}\subset\mathcal{A}$ a set of projections summing to the identity. If $q$ is a projection such that $p_iq\neq qp_i$, then there exists $p_j\neq p_i$ such that $p_jqp_i\neq 0$.
My proof uses the GNS representation, vector states, and what I call state conditioning (see proof of Theorem 3.3 here). While I like the proof I have, mostly because I like using state conditioning, I am thinking there must be a slicker proof (often when I pass to the GNS representation there is).
Anyone have a (much) nicer proof?
This answer was inspired by another MSE user.
Assume that for all $j\neq i$ the $p_jqp_i$ are zero. Add them up to get $(1-p_i)qp_i=0$. This yields $qp_i=p_iqp_i$ which implies, taking adjoints, that $p_i$ commutes with $q$.
Therefore one of the $p_jqp_i$ must be non-zero.