# Complemented ideals in von Neumann algebras

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?

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This is a different version of the fact that for a representation of self-adjoint algebra, if there exists a closed subrepresentation, then there exists a complementary closed subrepresentation –  user39158 Apr 28 '14 at 9:52

Yes, that is right, and of course $J$ is also ultraweakly closed.