The following is a proposition of Takesaki's Operator Theory:
My questions are:
1- He claims for two sided ideal $\cal m$, $e \in M\cap M'$. While I think for $\sigma -$ weakly closed two sided ideal we can conclude it. Also I think he uses of $\sigma-$ weakly closed ideal in the proof. Am I right?
2- He claims if $M$ is a factor, then every nonzero two-sided ideal is $\sigma-$weakly dense in it. I think in this situation, there is just a nonzero $\sigma-$ weak closed two sided ideal because the central projections are $1, -1$ and ${\cal m}_1=M(1) = M(-1) ={\cal m}_2$, but there may be different two sided ideals with equal $\sigma- $ weak clusers. If this is true, I would like to see an emample about it.
$\sigma- $weak means ultraweak topology. Thanks in advance.
niki, he keeps the assumption that that $\mathfrak{m}$ is $\sigma$-weakly closed. So the third line should read:
Of course, when you deal with norm-closed ideals it is very easy to come up with a counter-example: take the compact operators inside $B(H)$. This is also an illustration to your second question: compact operators are $\sigma$-weakly dense in $B(H)$ and $B(H)$ is a factor.
$II_1$-factors are algebraically simple: they have no non-zero proper ideals.