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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 …

Let $R \subset S$ be distinct non-type-I von Neumann factors; say two projections $P, Q \in S$ "meet" if they have a common non-null subprojection (i.e. if $P \wedge Q \neq 0$), and call $P$ "$R$-disj …

Question here about how the projections of a von Neumann algebra $\mathcal{R}$ might be arranged, relative to a projection that is not in $\mathcal{R}$. Stipulate the following:     $H$ is a Hilbert …

H. A. Dye showed that a type II or III factor $R$ is determined, up to *-algebraic isomorphism or anti-isomorphism, by the ortholattice-isomorphism type of its projection lattice ("ortholattice-isomor …

Let $R$ be a non-type-I factor acting on a separable Hilbert space. Let $P(R)$ be the set of $R$'s projections with the usual ordering ($x \leq y \iff$ range$(x) \subseteq$ range$(y)$) under which it …

I believe I've shown (see answer to related stackoverflow question at https://math.stackexchange.com/a/4223869/250373 ) that there is such an embedding, on the supposition that every nontrivial lattic …