von Neumann algebra factors 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.
 A: 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.
A: To complement Tom's answer, it is interesting to know that the following statements are equivalent:

*

*$M$ is simple as a von Neumann algebra (i.e., no sot-closed ideals)


*$M$ is simple as a C$^*$-algebra (i.e., no norm-closed ideals)


*When $M$ is finite, it is a factor
Indeed, if $M$ is not a factor then its centre is a nontrivial von Neumann algebra and so it has a nontrivial projection $p$; thus $pM$ is a nontrivial (sot-closed, norm-closed) ideal of $M$, so $M$ is not simple neither as a von Neumann algebra nor as a C$^*$-algebra. This shows that either of the first two assertions implies the last one.
That C$^*$-simplicity implies von Neumann simplicity is trivial. So the only nontrivial assertion is that a factor is simple as a C$^*$-algebra. This follows from Dixmier's Approximation Theorem (see, for instance, section 8.3 in Kadison-Ringrose), which says that for any $x\in M$, there exists $y\in M\cap M'$ such that
$$
y\in\overline{\text{conv}\,\{uxu^*:\ u\in\mathcal U(M)\}}.
$$
So if you take a norm closed ideal $J\subset M$ and $M$ is a factor, take $x\in J$ nonzero, and now by Dixmier's Approximation Theorem there exists $\lambda I$ that belongs to the norm-closure of the convex hull of the unitary orbit of $x$, which is in $J$. So $I\in J$ and thus $J=M$ (it is easy to see that $\lambda\ne0$ by using the faithful trace).
Edit: This last argument requires the $M$ is finite (or a similar condition) to guarantee that $\lambda$ above is not zero. For instance if you take $M=B(H)$, which has a proper ideal, and $x$ a rank-one projection, then the $y$ from Dixmier's Theorem is zero. Thanks Søren for noticing.
