Connes embedding problem

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

Connes' embedding problem asks whether every type II₁ factor on a separable Hilbert space can be embedded into some ${\displaystyle R^{\omega }}$.

A positive solution to the problem would imply that invariant subspaces exist for a large class of operators in type II₁ factors (Uffe Haagerup); all countable discrete groups are hyperlinear. A positive solution to the problem would be implied by equality between free entropy ${\displaystyle \chi ^{*}}$ and free entropy defined by microstates (Dan Voiculescu). In January 2020, a group of researchers[2] claimed to have resolved the problem in the negative, i.e., there exist type II₁ von Neumann factors that do not embed in an ultrapower ${\displaystyle R^{\omega }}$ of the hyperfinite II₁ factor.

The isomorphism class of ${\displaystyle R^{\omega }}$ 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.

The problem admits a number of equivalent formulations.[1]

• Connes' embedding problem and quantum information theory workshop; Vanderbilt University in Nashville Tennessee; May 1–7, 2020 (postponed; TBA)
• The many faceted Connes' Embedding Problem; BIRS, Canada; July 14–19, 2019
• Winter school: Connes' embedding problem and quantum information theory; University of Oslo, January 7–11, 2019
• Workshop on Sofic and Hyperlinear Groups and the Connes Embedding Conjecture; UFSC Florianopolis, Brazil; June 10–21, 2018
• Approximation Properties in Operator Algebras and Ergodic Theory; UCLA; April 30 - May 5, 2018
• Operator Algebras and Quantum Information Theory; Institut Henri Poincare, Paris; December 2017
• Workshop on Operator Spaces, Harmonic Analysis and Quantum Probability; ICMAT, Madrid; May 20-June 14, 2013
• Fields Workshop around Connes Embedding Problem – University of Ottawa, May 16–18, 2008

• Capraro, Valerio (2010). "A Survey on Connes' Embedding Conjecture". arXiv:1003.2076 [math.OA].
• Farah, I.; Hart, B.; Sherman, D. (2013). "Model theory of operator algebras I: stability". Bulletin of the London Mathematical Society. 45 (4): 825–838. arXiv:0908.2790. doi:10.1112/blms/bdt014. S2CID 15024863.
• Ge; Hadwin (2001). "Ultraproducts of C*-algebras". Oper. Theory Adv. Appl. 127: 305–326. doi:10.1007/978-3-0348-8374-0_17. ISBN 978-3-0348-9539-2.
• Collins, Benoıt; Dykema, Ken (2008). "A linearization of Connes' embedding problem" (PDF). New York Journal of Mathematics. 14: 617–641.
• Sherman, David (2008). "Notes on Automorphisms of Ultrapowers of II₁ Factors". arXiv:0809.4439 [math.OA].
• Pisier, Gilles. "Tensor products of C*-algebras and operator spaces: The Connes-Kirchberg problem" (PDF).