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.

Statement

Let $\omega$ be a free ultrafilter on the natural numbers and let R be the hyperfinite type II₁ factor with trace $\tau$. One can construct the ultrapower $R^{\omega }$ 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 $R^{\omega }$.

The isomorphism class of $R^{\omega }$ is independent of the ultrafilter if and only if the 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
• Survey on Connes' Embedding Conjecture, Valerio Capraro
• Model theory of operator algebras I: stability, I. Farah - B. Hart - D. Sherman
• 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
• Notes On Automorphisms Of Ultrapowers Of II₁ Factors, David Sherman, Department of Mathematics, University of Virginia