**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.