"Averaging" in von Neumann algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T17:27:13Z http://mathoverflow.net/feeds/question/40756 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40756/averaging-in-von-neumann-algebras "Averaging" in von Neumann algebras Matthew Daws 2010-10-01T14:38:37Z 2010-10-01T15:37:02Z <p>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).</p> <p>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 $M$).</p> <p>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.</p> <p>So my, slightly vague, question is:</p> <blockquote> <p>What are some other, similar, examples of "averaging" in this fashion?</p> </blockquote> <p>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.</p> http://mathoverflow.net/questions/40756/averaging-in-von-neumann-algebras/40766#40766 Answer by Owen Sizemore for "Averaging" in von Neumann algebras Owen Sizemore 2010-10-01T15:37:02Z 2010-10-01T15:37:02Z <p>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 $C^*-algebras.$</p> <p>The papers are</p> <p>S. Popa, The relative Dixmier property for inclusions of von Neumann algebras of ¯nite index, Ann. Sci. Ec. Norm. Sup., 32 (1999), 743-767.</p> <p>and</p> <p>S. Popa, On the relative Dixmier property for inclusions of C¤-algebras, Journal of Funct. Analysis, 171 (2000), 139-154.</p> <p>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.</p> <p>For references prob the best place to start would be Popa's ICM talk</p> <p><a href="http://www.math.ucla.edu/~popa/ICMpopafinal.pdf" rel="nofollow">http://www.math.ucla.edu/~popa/ICMpopafinal.pdf</a></p> <p>or appendix C of Stefaan Vaes's Bourbaki seminar.</p> <p><a href="http://arxiv.org/PS_cache/math/pdf/0603/0603434v2.pdf" rel="nofollow">http://arxiv.org/PS_cache/math/pdf/0603/0603434v2.pdf</a></p>