# “Averaging” in von Neumann algebras

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.

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Should I make this community wiki? As I seem to be asking for a list, not one specific "answer"?? – Matthew Daws Oct 1 '10 at 15:47
You should check out Sinclair and Smith's book: Finite von Neumann algebras and MASAs, and, as Owen suggests, Popa's oevre. Between the two of these, there are many applications of "averaging". – Jon Bannon Oct 1 '10 at 16:12
[deleted comment based on misremembering/misthinking] – Yemon Choi Oct 1 '10 at 18:39

As far as the Dixmier property, there is a relative version that Popa has shown holds for for certain inclusions. (i don't have the paper in front of me and i don't quite remember exactly what he shoes). He also has one for $C^*-algebras.$

The papers are

S. Popa, The relative Dixmier property for inclusions of von Neumann algebras of ¯nite index, Ann. Sci. Ec. Norm. Sup., 32 (1999), 743-767.

and

S. Popa, On the relative Dixmier property for inclusions of C¤-algebras, Journal of Funct. Analysis, 171 (2000), 139-154.

Secondly, (and i would say more interestingly for me) is the averaging involved in Popa's intertwining by bi-modules technique. Without going into too much detail, if two subalgebras are close in norm on their unit balls, then by averaging, one can get a partial isometry that intertwines a corner of one into a corner of the other. This is an absolutely crucial element of all the progress in the past 8 years or so in the classification of von Neumann algebras coming from groups, or ergodic group actions.

For references prob the best place to start would be Popa's ICM talk

http://www.math.ucla.edu/~popa/ICMpopafinal.pdf

or appendix C of Stefaan Vaes's Bourbaki seminar.

http://arxiv.org/PS_cache/math/pdf/0603/0603434v2.pdf

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