Definition 37.8.7 (Type $\vnII$).label Let $H$ be a complex Hilbert space and $A \subset B(H)$ be a von Neumann algebra, then $A$ is of type $\vnII$ if:

1. (1)

   $A$ has no non-zero abelian projections.

2. (2)

   For every non-zero central projection $P \in \text{Proj}(Z(A))$, there exists a non-zero finite projection $Q \in \text{Proj}(A)$ with $P \ge Q$.