can we find a sequence (or net) of central projection $p_n\in A$ such that $p_n\le e$ for every $n$ and $p_n \uparrow s(f)$?

Question

Let $A\subset B(H)$ be a von Neumann algebra (if necessary, we can assume that $A$ is separable and of type $II_1$) and let $e\in B(H)$ be a projection. Let $f\in A$ be the support of $e$ ($f = 1 -\vee_{f_i\in C} f_i$, where $f_i$ is a projection such that $f_ie=0$ and $C$ is the collection of all such projections). Let $s(f)$ be the central support of $f$ in $A$. Now, we have the relation that $e \le f \le s(f)$.

My question is: can we find a sequence (or net) of central projection $p_n\in A$ such that $p_n\le e$ for every $n$ and $p_n \uparrow s(f)$?

Answer 1

In the case where $e\in A$, the conditions $p_n\leq e$ and $p_n\nearrow s(f)$ imply that $e$ is central. So, whenever $e\in A$ but not central no such sequence can exist.