Which von Neumann algebras acting on separable Hilbert space $H$ have uncountable antichains of projections? ("Antichain" meaning a set of projections any pair of which has no shared nonzero subprojection.)
I'm pretty sure $\mathcal{B}(H)$ (the algebra of all bounded operators on $H$) does have one. For instance, for any orthogonal $x, y \in H$ and all real $c$, if $P_c$ denotes the unique one-dimensional projection whose range includes $x + cy$, then $\{P_c : c \in \mathbb{R} \}$ is an uncountable antichain.
I'm equally sure that an abelian algebra without minimal projections cannot have one, since its lattice of projections is basically a measure algebra with a countable dense subset.
But with non-abelian type II and type III algebras, I'm not sure where to begin ... even defining what the projections are in concrete cases seems hard; all the examples I've found are defined with intricately-constructed unitary operators rather than with projections. Can anyone help me out here, or point me to a good discussion of the properties of projection lattices of the various types of vN algebras? Thanks!
Unless I'm misreading your definition, this is an antichain of projections in $M_2(\mathbb C)$: $$ P_t=\begin{bmatrix}t&\sqrt{t-t^2}\\ \sqrt{t-t^2}&1-t\end{bmatrix},\ \ t\in[0,1]. $$
This idea can be made to work easily in any type I von Neumann algebra.
For types II and III, a deeper idea is needed, namely halving: given any projection $P_0\in\mathcal M$, there exists $P\leq P_0$ with $P\sim P_0-P$. So there exists $V$ with $V^*V=P$ and $VV^*=P_0-P$. Now, with $Q=P_0-P$, $$ \mathcal M_0=\text{span}\{P,V^*,V,Q\} $$ is a subalgebra of $\mathcal M$, isomorphic to $M_2(\mathbb C)$. Now we can construct the $P_t$ as above, namely $$ P_t=t\,P+\sqrt{t-t^2}\,(V^*+V)+(1-t)\,Q,\ \ \ t\in[0,1]. $$