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Edit: according to Narutaka Ozawa, question 3) is still open in the type $\mathrm{II}_1$ case. In other terms, it is not known whether every topologically complemented type $\mathrm{II}_1$ factor in $B(H)$ is injective.

Let $M$ be a von Neumann algebra sitting in $B(H)$. I will say that $M$ is B-complemented (resp. CB-complemented, 1-complemented) if there exists a bounded (resp. completely bounded, norm $1$) linear idempotent from $B(H)$ onto $M$. Of course, 1-complemented is just a synonym for injective. Pisier (Corollaire 5) and Christensen-Sinclair proved that CB-complemented implies 1-complemented. The converse follows from Tomiyama. So CB-complemented=1-complemented=injective.

1) Haagerup-Pisier proved that the free group factors $L(F_n)$ ($2\leq n\leq \infty$) are not B-complemented in $B(H)$. Are there other known examples among finite factors? Other than those in which the factors $L(F_n)$ are B-complemented. And other than non injective McDuff factors.

2) If I am not mistaken, their Corollary 4.6 also implies that every non injective McDuff factor and every non injective properly infinite von Neumann algebra is not B-complemented. Was this known before by other means?

3) Are there examples of B-complemented not 1-complemented (injective) von Neumann algebras?

4) If no to 3), what is known about B-complemented implies 1-complemented in general?

5) More generally, if $M$ is semi-finite or finite and sits in a von Neumann algebra $N$, Pisier proved that $M$ CB-complemented in $N$ implies 1-complemented in $N$. He says it should not be too hard, but has it been shown that the assumption semi-finite or finite can be removed?

6) When $M$ is B or CB-complemented, what can be said about the complement other than it is a closed subspace of $B(H)$?

I apologize if all this is well-known, but I couldn't find the answers in the literature I am aware of. Sorry also for all these questions in one, but I thought it would be inappropriate to post several questions on such related topics. Thank you.

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up vote 9 down vote accepted

(i) Obviously, a vN subalgebra $N$ of a B-complemented vN algebra $M$ is also B-complemented if there exists a conditional expectation from $M$ onto $N$. (ii) Haagerup and Pisier's proof actually says if $N$ is a B-complemented vN algebra which contains a (possibly non-unital) copy of $M_2(N)$ with a conditional expectation, then $N$ is CB-complemented and hence is injective. (iii) Existence of a conditional expectation is automatic if the vN algebras are finite. With these observations, one can see that "almost all" known non-injective vN algebras are not B-complemented, but it is still open whether that holds for all vN algebras. For Question (5), Christensen and Sinclair (Proc LMS 1995) proved the theorem in that generality. There is another proof by Haagerup (unpublished).

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Thank you very much. I don't know all the known non-injective factors well enough to see what your "almost all" means. What would be some of these rare examples for which it is not known whether they are B-complemented? Also, since B-complemented implies CB-complemented is open, does it relate anyhow to bounded implies CB (Kadison similarity problem) for homomorphisms? –  1015 Jul 11 '13 at 15:08
    
Only type II_1 case remains open. It is not known whether there exists a non-injective type II_1 vN algebra which does not contain the free group factor $LF_2$. (Von Neumann algebraic von Neumann's problem.) Anyway, the only candidates I know of are the group vN algebras of non-amenable groups without rank 2 free subgroups. Even for this case, sometimes (e.g. free Burnside groups) the Gaboriau--Lyons theorem helps to solve it. As to Kadison's similarity problem, I don't see any real connection. I suspect the finite injective vN algebra $\prod_n M_n$ is a counterexample to that problem. –  Narutaka OZAWA Jul 11 '13 at 23:34
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