Let $\mathfrak{M}$ be a vN-Alg. Let $\theta\in \text{Aut}(\mathfrak{M})$. Let $p\in\mathfrak{M}$ be a projection, so that $\theta(p)\neq p$. Is there a projection $q\in\mathfrak{M}$ with $0\neq q\leq p$, so that $\theta(q)q=0$ (equivalently: so that $\theta(q)$ and $q$ are orthogonal projections). If not, why? If so, how does one show this? Is there a lemma/exercise somewhere for this?
Should this fail in general, does it at least hold, when either:
- $\frak{M}$ has faithful normal tracial state, $\tau$, and $\theta\in\text{Aut}(\frak{M},\tau)$;
- or furthermore $\mathfrak{M}=$ the hyperfinite II${}_{1}$-factor viewed with the usual trace?