A projection in a von Neumann algebra $A$ is central if it commutes with every element of $A$.
We say that $A$ is Type I if every non-zero central projection in $A$ majorizes a non-zero abelian projection in $A$. We say that A is Type II if it has no non-zero abelian projections and every non-zero central projection majorizes a non-zero finite projection.
What dose the terminology “majorize” mean？If $\operatorname{ran}p\subset \operatorname{ran}q$, can we say $p$ is majorized by $q$? where $p$ and $q$ are projections in $A$.
“$p$ majorizes $q$” is simply a way to pronounce $p \leq q$. ($p \leq q$ iff $q-p = c^*c$ for some c. If represented on some Hilbert space $H$, this is equivalent to $\langle p x,x\rangle \leq \langle q x,x \rangle$ for every vector $x \in H$.)