This is a question about what happens when you "add" a new projection $p$ to a von Neumann algebra $\mathcal{R}$ to generate a larger v.N. algebra $(\mathcal{R} \cup \{p\})''$.
Suppose that $\mathcal{R}$ is a non-type-I factor acting on Hilbert space $H$. When $p \not \in \mathcal{R}$ is any (bounded) projection operator on $H$, call $p$ "$\mathcal{R}$-finite" if it is the join of finitely many minimal projections in $(\mathcal{R} \cup \{p\})''$.
If $p$ and $q$ are $\mathcal{R}$-finite, must their join $p \vee q$ also be $\mathcal{R}$-finite? ($p \vee q$ meaning the projection onto the closed span of $p$ and $q$.)
$p \vee q$ will be trivially $\mathcal{R}$-finite if the ranges of $p, q$ are finite-dimensional subspaces of $H$, so assume this is not the case. (For example $p$ could be a minimal projection in some type I algebra $\mathcal{S} \supseteq \mathcal{R}$ that is not the whole algebra $B(H)$ of all bounded operators on $H$, and $q$ could be a minimal projection in another such algebra.)
[this is reposted from https://math.stackexchange.com/questions/2711943/adding-a-projection-to-von-neumann-algebra -- the issue keeps coming up in a project I'm working on and I've had trouble even finding a promising angle from which to view the issue.]
