Projections in Type III von Neumann agebra

Question

If $M$ is a type III von Neumann algebra, I wonder how many projections does $M$ possess? Countably infinite or uncountably?

Answer 1

The only von Neumann algebras that do not have uncountably many projections are the commutative finite-dimensional ones, which have finitely many ones. There is no von Neumann algebra with countably many projections.

Already in $M_2(\mathbb C)$, you have the uncountable family $$ \begin{bmatrix} t&\sqrt{t-t^2}\\ \sqrt{t-t^2}&1-t\end{bmatrix},\qquad t\in[0,1]. $$ In any factor other than $\mathbb C$, take a nontrivial projection $p$, and consider equivalent projections $q_1\leq p$, $q_2\leq 1-p$. If $v$ is the partial isometry realizing the equivalence, $\operatorname{span}\{q_1,v,v^*,q_2\}$ is isomorphic to $M_2(\mathbb C)$ and so the above example applies.

The above can also be done in an arbitrary non-commutative von Neumann algebra.

And if $M$ is commutative and infinite-dimensional, then there is a copy of $\ell^\infty(\mathbb N)$ inside, so there are also uncountably many projections.