# 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

## 1 Answer

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

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Thank you; isn't it strange that such a nice thing is not mentioned in textbooks (Kadison & Ringrose, for example)? –  Yulia Kuznetsova Jan 5 '12 at 9:51
@YuliaKuznetsova: You're welcome. Yes, I would think that it would always be mentioned in a textbook after the result characterizing the ultraweakly closed 2-sided ideals as principal ideals generated by central projections, but it often isn't. It is mentioned in Blackadar's book on operator algebras, however. –  Jonas Meyer Jan 5 '12 at 9:54