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 $\omega$ 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^[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

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[2] ttp://www.fields.utoronto.ca/programs/scientific/07-08/embedding/abstracts.html#brown

[3] ttps://arxiv.org/PS_cache/arxiv/pdf/1003/1003.2076v1.pdf

[4] ttp://people.virginia.edu/~des5e/papers/2009c30-stable-appl.pdf%5Bpermanent+dead+link%5D

[5] ttp://www.emis.de/journals/NYJM/j/2008/14-28.pdf

[6] ttp://people.virginia.edu/~des5e/papers/sm-autultra.pdf