I know that provided a von Neumann Algebra acting on Hilbert $(\mathcal{M},\mathcal{H})$ a projection $e \in P(\mathcal{M})$ is said to be abelian if $e\mathcal{M} e$ is $\textbf{abelian}$ and the same is said to be $\textbf{finite}$ if $f \le e$ and $f \sim e \Rightarrow f = e$ and I should prove the following statements:

1)Any abelian projection is finite

2)if $e$ is abelian and $f \preceq e$ then f is abelian ($\preceq$ means that $f$ is equivalent to a subprojection of e)

3)if $e$ finite and $f \preceq e$ then $f$ is finite

Can you give me any hints? For the first one I was thinking that if $e$ is abelian then $e\mathcal{M} e = \mathcal{Z}(\mathcal{M})$...


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