Suppose $W$ is a proper von Neumann Algebra contained in $B(H)$ and the identity in $W$ is the identity mapping of $H$ (namely, $W$ does not have non-trivial null space).


1. Given a self-adjoint $T\in W$, does there exists a projection $P$ that is not in $W$ such that one of $TP, PT, PTP$ is not in $W$? (we definitely do not want all of them to be zero) Can we find a sufficient condition for this to happen?
2. Conversely, given a self-adjoint $T$ that is not in $W$, does there exists a projection $Q$ that is in $W$ such that one of $TQ, QT, QTQ$ is in $W$? (again we do not want all of them to be zero) Can we find a sufficient condition for this to happen?


In general, if $T$ is seld-adjoint, then there exists a projection in $W^\*(T)$ (the von Neumann Algebra generated by $T$) such that the range of that projection is the closure of the range of $T$ and that projection corresponds to the characteristic function of $\sigma(T)$. So far, this is the only thing I know about this question. Besides, given $T$ self-adjoint and not in $W$, can $W^\*(T)\cap W\neq\emptyset$? If we have, say $f(T)\in W^\*(T)\cap W$, can we always recover the entire $W^\*(T)$ using $f(T)$ (even when $f$ is not the characteristic function of $\sigma(T)$)?



"Project" an operator outside of a von Neumann Algebra into it
These aren't research-level questions, but anyway. Let $W$ be a von Neumann algebra unitally contained in $B(H)$ and let $T \in W$ be nonzero (doesn't have to be self-adjoint). Suppose for every projection $P \in B(H)$ we have $PT \in W$. Taking linear combinations and weak\* limits, we get $ST \in W$ for all $S \in B(H)$.


Since $T \neq 0$, we can find nonzero vectors $v,w \in H$ with $Tv = w$. Then for any $u \in H$, taking $S = u\otimes w$ so that $ST$ is a scalar multiple of $u \otimes v$, we get $u \otimes v \in W$. Taking adjoints, we also have $v\otimes u \in W$ for all $u \in H$. Then $(u\otimes v)(v\otimes u') = u\otimes u' \in W$ for all $u,u' \in H$ and this implies that $W = B(H)$. Conclusion: if $W \neq B(H)$ then there is some projection $P \in B(H)$ with $PT \not\in W$.


The second question trivially fails with $W = \mathbb{C}\cdot I$, as long as ${\rm dim}(H) > 1$.


