Semisimplicity and global dimension of a finite von Neumann algebra

Abstract

We prove that a finite von Neumann algebra $\mathcal{A}$ is semisimple if the algebra of affiliated operators $\mathcal{U}$ of $\mathcal{A}$ is semisimple. When $\mathcal{A}$ is not semisimple, we give the upper and lower bounds for the global dimensions of $\mathcal{A}$ and $\mathcal{U}.$ This last result requires the use of the Continuum Hypothesis.