I have two basic questions on von Neumann algebras :
1) If a von Neumann algebra $M$ is simple (only trivial ideals), is it a factor (i.e. $M\cap M'=\mathbb C \cdot 1_M$ ?).
2) If the reduced group algebra of a discrete group $C_r(\Gamma)$ (i.e. the completion of $C[\Gamma]$ with respect to the norm induced by the left regular representation) is simple, is it true that the associated von Neumann algebra $L(\Gamma)=C_r(\Gamma)''$ is a factor ? Thanks a lot, sorry fot these naive questions.
The answer to your first question is yes.
Suppose that $M$ is not a factor; it must contain a nontrivial central projection $p \in M \cap M'$. But then $pMp$ is a nontrivial ideal in $M$. So if $M$ is simple, it must be a factor.
The answer to your second question is also yes.
Let $\Gamma$ be a countably infinite discrete group. Suppose that the reduced group $C^*$-algebra $C^*_r(\Gamma)$ is simple. Then the group $\Gamma$ must be i.c.c.. That is, each of its conjugacy classes different from $\{1\}$ is infinite. But then the group von Neumann algebra $L(\Gamma)$ is a $II_1$-factor.
"Simplicity implies i.c.c." follows from Proposition 3 in http://arxiv.org/abs/math/0509450, as noted in Section X of that paper.
That "i.c.c. implies factor" goes back as far as Murray and von Neumann.