Given any state of a type III$_1$ factor $\omega$ and any $\epsilon>0$ is there always a state of infinite multiplicity $\varphi$ such that $\|\omega-\varphi\|\leq \epsilon$?
A related question: Given a projector $e$ in the algebra and a state $\varphi$, is there always a state near $\varphi$ that has $e$ in its centralizer?