Zhu's definition of type 1 von Neumann algebras

Question

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.

Thanks!

Answer 1

I figured it out by myself and thought I would provide the answer for other people who might have also not seen it immediately.

Let $M$ be ZHU-TYPE-1 and let $p$ be a nonzero projection. Since the central carrier $z_p\geq p\neq 0$ is central there is an abelian nonzero projection $q$ s.t. $q\leq z_p$.

Since $z_p$ is central $z_q\leq z_p$ and since $q\neq 0$ so is $z_q$.

Furthermore we know $z_qz_p=z_q\neq 0$, so there are nonzero projection $\tilde{p}\leq p,\tilde{q}\leq q$ with $\tilde{p}\sim\tilde{q}$.

Since $q$ was abelian so is $\tilde{q}\leq q$ and thus so is $\tilde{p}$ which so turns out to be the desired projection.

If this is not correct I'd appreciate hints and comments!