Cluster labeling

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 $l^{\mathrm{\infty }}(R)=\{(x_n{)}_n\subseteq R:su{p}_n\left|\right|x_n\left|\right|<\mathrm{\infty }\}$ be the von Neumann algebra of norm-bounded sequences and let $I_{\omega }=\{(x_n)\in l^{\mathrm{\infty }}(R):li{m}_{n\to \omega }\tau (x_n^{*}{x}_n{)}^{\frac{1}{2}}=0\}$. The quotient $l^{\mathrm{\infty }}(R)/I_{\omega }$ turns out to be a II₁ factor with trace ${\tau }_{R^{\omega }}(x)=li{m}_{n\to \omega }\tau (x_n+I_{\omega })$, 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

Notes

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