Let $M$ be a von Neumann algebra. There exist pairwise orthogonal central projections

\begin{align*} z_{I},\quad z_{II},\quad z_{III} \in Z(M) \end{align*}

with $z_I+z_{II}+z_{III}=I$ such that $z_I M$ is type I, $z_{II}M$ is type II, and $z_{III}M$ is type III. Refining further, the type I part decomposes into homogeneous type $I_\kappa$ pieces indexed by Hilbert-space dimension cardinals $\kappa$, including the finite pieces $I_n$ and the separable infinite piece $I_\infty$, while the type II part decomposes into finite and properly infinite semifinite pieces.