A simple question on infinite dimensional von Neumann algebra

Question

Recall a projection $p\in N$ is called abelian if $pNp$ is an abelian algebra.

If $N$ is a von Neumann algebra without abelian projections, then can we conclude that $N$ must be infinite dimensional?

Answer 1

Yes. Because a finite-dimensional von Neumann algebra will always have minimal projections, and minimal projections are abelian.