Increasing net of projections in von Neumann algebra

Question

In $B(H)$, there exists an increasing sequence of finite rank projections convergent to the identity in SOT.

Is there an analogous statement for a von Neumann algebra $M\subseteq B(H)$?

Answer 1

No, $M$ need not contain any nonzero finite rank projections. For a really simple example, $M$ could consist of only the scalar multiples of the identity. Or, if $H=L^2(\mathbb{R})$, then $M$ could be $L^\infty(\mathbb{R})$, which contains no nonzero finite rank operators since Lebesgue measure is atomless.