I've been reading recently about various "averaging" tricks in von Neumann algebras. For example, Christensen and Sinclair, in "On von Neumann algebras which are complemented subspaces of $B(H)$." J. Funct. Anal. 122 (1994), no. 1, 91--102. Here we have a von Neumann algebra $M\subseteq B(H)$ and a completely bounded projection $P:B(H)\rightarrow M$. Consider a family $(u_i)$ in $M$ with $\sum_i u_i^* u_i=1$ strongly. Then it's shown that there is a cb map $Q:B(H)\rightarrow M$ which is a right $M$-module map, such that $Q(x)$ is in the w*-closed convex hull of $\sum_i P(xu_i^*)u_i$, for each $x\in B(H)$. Then doing this on the right yields an $M$-bimodule projection, from which it follows that $M$ is injective (and so this solved the problem of whether having a cb projection-- which isn't contractive-- onto $M$ implies that $M$ is injective).

A vaguely similar notion occurs in Dixmier's approximation theorem: For each $a\in M$, the norm closed convex hull of $\{ u^*au: u\in U(M)\}$ meets the centre of $M$ (here $U(M)$ is the unitary group of $M$).

We might wonder if for each $x\in B(H)$, the norm (or even ultraweak) closed convex hull of $\{u^*xu:u\in U(M)\}$ intersects $M'$. But I think, it you consider $M$ acting on $H\otimes K$ for a sufficiently large $K$, this would imply a projection from $B(H)$ onto $M'$, which is proving too much.

So my, slightly vague, question is:

What are some other, similar, examples of "averaging" in this fashion?

I'm not interested in the case when M is assumed injective: that's too much. But I would be interested in special cases, like, if $M$ is a $II_1$-factor. I'd be quite happy just to have references, not detailed answers.