  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.
