Let $I$ be an ultraweakly closed ideal in a von Neumann algebra $M$. For example, this can be the kernel of an ultraweakly continuous homomorphism. Is it true that there is another ideal $J\subset M$ such that $M=I\oplus J$?
It seems like yes: for $I$, there is a central projection $p\in M$ such that $I=Mp=pM$, then put $J=(1-p)M$ - this is an ideal and $M=I\oplus J$. Is this right?
Yes, that is right, and of course $J$ is also ultraweakly closed.