Connes embedding problem

In von Neumann algebras, the Connes embedding problem or conjecture, due to Alain Connes, asks whether every type II₁ factor on a separable Hilbert space can be embedded into the ultrapower of the hyperfinite type II₁ factor by a free ultrafilter. The problem admits a number of equivalent formulations.^[1]

Statement

Let ω be a free ultrafilter on the natural numbers and let R be the hyperfinite type II₁ factor with trace τ. One can construct the ultrapower R^ω as follows: let $l^\infty(R)=\{(x_n)_n\subseteq R:sup_n||x_n||<\infty\}$ be the von Neumann algebra of norm-bounded sequences and let $I_\omega=\{(x_n)\in l^\infty(R):lim_{n\rightarrow\omega}\tau(x_n^*x_n)^{\frac{1}{2}}=0\}$. The quotient $l^\infty(R)/I_\omega$ turns out to be a II₁ factor with trace $\tau_{R^\omega}(x)=lim_{n\rightarrow\omega}\tau(x_n+I_\omega)$, where (x_n)_n is any representative sequence of x.

Connes' Embedding Conjecture asks whether every type II₁ factor on a separable Hilbert space can be embedded into some R^ω.

The isomorphism class of R^ω depends on the ultrafilter if and only if continuum hypothesis is true (Ge-Hadwin and Farah-Hart-Sherman), but such an embedding property does not depend on the ultrafilter because von Neumann algebras acting on separable Hilbert spaces are, roughly speaking, very small.

References

• Fields Workshop around Connes' Embedding Problem – University of Ottawa, May 16–18, 2008^[2]
• Survey on Connes' Embedding Conjecture, Valerio Capraro^[3]
• Model theory of operator algebras I: stability, I. Farah - B. Hart - D. Sherman^[4]
• Ultraproducts of C*-algebras, Ge and Hadwin, Oper. Theory Adv. Appl. 127 (2001), 305-326.
• A linearization of Connes’ embedding problem, Benoıt Collins and Ken Dykema^[5]
• Notes On Automorphisms Of Ultrapowers Of II₁ Factors, David Sherman, Department of Mathematics, University of Virginia^[6]

Notes

1. ^ http://www.jstor.org/pss/2669132
2. ^ http://www.fields.utoronto.ca/programs/scientific/07-08/embedding/abstracts.html#brown
3. ^ http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.2076v1.pdf
4. ^ http://people.virginia.edu/~des5e/papers/2009c30-stable-appl.pdf
5. ^ http://www.emis.de/journals/NYJM/j/2008/14-28.pdf
6. ^ http://people.virginia.edu/~des5e/papers/sm-autultra.pdf