Question about von Neumann algebra generated by a complete algebra of projections Hi all, sorry if this is a dumb question, I don't know much about von Neumann algebras except the definition and a few relevant facts I've managed to prove by myself so I expect the answer will turn out to be well known. Anyway, let $\mathcal{H}$ be a Hilbert space, and suppose that $P$ is a commuting set of self-adjoint projections on $\mathcal{H}$, with the additional two properties:
1) $P$ is closed under complements, i.e. if $p \in P$ then so is $1 - p$.
2) $P$ is closed under suprema of arbitrary subsets, i.e. if $S \subseteq P$ then $\sup S \in P$ (here the projections on $\mathcal{H}$ are ordered by defining $p \leq q$ whenever the range of $p$ is contained in the range of $q$).
Now let $V$ denote the smallest von Neumann algebra containing $P$. Suppose that $p \in V$ is a self-adjoint projection. Is $p \in P$?
I know that $p$ is necessarily in the closure (relative to the weak operator topology) of the set of finite sums $\sum_i \lambda_i p_i$, where $p_i \in P$ and $\lambda_i \in \mathbb{R}$. It seems like it may be possible to derive a contradiction from the assumption that $q$ has a strictly smaller range than $p$, where $q \equiv \sup ${$ r \in P | r \leq p $}. But I don't know how to proceed.
 A: The answer "yes" follows from Theorem 2.8 of Bade's "On Boolean algebras of projections and algebras of operators," 1955, which is in the more general context of algebras of operators on a Banach space. 
Bade had previously proven a less general result that still covers your case, dealing with algebras of operators on reflexive spaces, in Theorem 3.4 of "Weak and strong limits of spectral operators," 1954.
A: This is off the cuff, not a complete answer, and I haven't checked through it carefully, but it was getting too long for a comment. I'm making it community wiki in case someone sees a way to iron out the wrinkles, although if they have a better approach they should feel free to write it up as a separate answer.
Anyway. Since $V$ is an abelian von Neumann algebra, it has the form $C(\Omega)$ for some compact, extremally disconnected toppological space. Then your original set $P$ can be regarded as the set of indicator functions of a family $\mathcal F$ of clopen subsets; $\mathcal F$ is closed under taking complements and under formation of arbitrary unions.
Here is the part which I haven't quite nailed down at time of writing: we note/claim that any element of $V$ is the pointwise limit of some bounded net of finite linear combinations of characteristic functions of sets in ${\mathcal F}$, in particular $p$, which must be the indicator function of some clopen $X\subseteq\Omega$, must be approximable in this way. Now we consider the family ${\mathcal C}$ of all $A\in{\mathcal F}$ which  have non-empty intersection with $X$. The approximability assumption should force $\bigcup_{A\in\mathcal C} A$ to be $X$, and thus $X\in\mathcal F$, which is the same as saying that $p\in P$ as required.
