finite and properly infinite projections

Question

Let $M$ be a von Neumann algebras and $p$ be a projection in $M$.

$Q1:$I want to prove that there is a central projection $z \in M$ such that $pz$ is finite and $P(1-z)$ is properly infinite.

$Q2:$ $z$ is unique if $p=1$

Answer 1

Let $\mathcal Z$ be the set $$ \mathcal Z=\{ (z_j):\ \forall j\ne k,\ z_j\in \mathcal P(\mathcal Z(M)),\ z_jz_k=0,\ z_jp \text{ is finite }\} $$ i.e. the set of pairwise orthogonal families of central projections $z$ such that $zp$ is finite.

Via Zorn's Lemma we can get a family $(z_j)\in\mathcal Z$, maximal. Let $z=\sum z_j$. The maximality of $(z_j)$ guarantees that $p(1-z)$ is properly infinite. And $zp$ is finite because the projections $z_j$ are central.

When $p=1$, the projection $z$ is a maximal finite projection in the center of $M$. It has to be unique, because if there were another maximal finite projection $z'$, then $z\vee z'$ would be a finite projection bigger than $z$ and $z'$.