Given a von Neumann algebra that is a type III$_1$ factor with the state $\omega$ and any $\epsilon>0$ is it always possible to find a projection or a partial isometry in the algebra such that its Averson spectrum with respect to the modular flow contained is in the interval $(-\epsilon,\epsilon)$?
Side note: It is easy to construct arbitrary operators in the algebra with Averson spectrum of modular flow in this range: Take any function $f(t)$ that only has Fourier modes in the range $\lambda\in (-\epsilon,\epsilon)$ then $\int dt \: f(t)\: \alpha^\omega_t(x)$ with $\alpha^\omega_t(a)$ the modular flow of $a$ has $\text{Sp}_\alpha(x)\in (-\epsilon,\epsilon)$. The question is can I tune this operator such that it is a projection or a partial isometry?