in Zhu's Book on operator algebras he defines a von Neumann algebra $M$ to be of type 1 iff for every central projection $0\neq p \in M$ there is an abelian projection $0\neq q\leq p$
I'm told this is equivalent to $M$ is of type 1 iff for every (not necessarily central) projection $0\neq p \in M$ there is an abelian projection $0\neq q\leq p$
The one direction is clear, but when I want to prove that Zhu's definition is strong enough and I take an arbitrary nonzero projection $p$ all that comes to my mind is that its central carrier $z_p$ also is non zero and therefore there is an abelian projection $0\neq q\leq z_p$. Can I somehow conclude that $q$ is a subprojection of $p$, or is this way not heeding in the right direction?
Same question goes to the definition of type 2 and type 3 algebras.