**Connes' embedding problem**, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathematics. Dan Voiculescu developing his free entropy theory found that Connes’ embedding problem is related to the existence of microstates. Some results of von Neumann algebras theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in computer science.

The problem admits a number of equivalent formulations.^{[1]} Notably, it is equivalent to the following long standing problems:

- Kirchberg's QWEP conjecture in C*-algebra theory
- Tsirelson's problem in quantum information theory
- The predual of any (separable) von Neumann algebra is finitely representable in the trace class.

In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in quantum complexity theory^{[2]} that implies a negative answer to Connes' embedding problem.^{[3]}^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}

## Statement

Let be a free ultrafilter on the natural numbers and let *R* be the hyperfinite type II_{1} factor with trace . One can construct the ultrapower as follows: let be the von Neumann algebra of norm-bounded sequences and let . The quotient turns out to be a II_{1} factor with trace , where is any representative sequence of .

Connes' embedding problem asks whether every type II_{1} factor on a separable Hilbert space can be embedded into some .

Positive solution to the problem would imply that invariant subspaces exist for a large class of operators in II-1-factors (Uffe Haagerup); all countable discrete groups are hyperlinear. A positive solution to the problem would be implied by equality between free entropy 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 II_{1} von Neumann factors that do not embed in an ultrapower of the hyperfinite II_{1} factor.

The isomorphism class of 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]}

## Conferences dedicated to Connes' embedding problem

- 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 07-11, 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

## References

- ^
^{a}^{b}Hadwin, Don (2001). "A Noncommutative Moment Problem".*Proceedings of the American Mathematical Society*.**129**(6): 1785–1791. doi:10.1090/S0002-9939-01-05772-0. JSTOR 2669132. - ^
^{a}^{b}Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP*=RE". arXiv:2001.04383. Bibcode:2020arXiv200104383J. Cite journal requires`|journal=`

(help) **^**Castelvecchi, Davide (2020). "How 'spooky' is quantum physics? The answer could be incalculable".*Nature*.**577**(7791): 461–462. doi:10.1038/d41586-020-00120-6.**^**Kalai, Gil (2020-01-17). "Amazing: Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen proved that MIP* = RE and thus disproved Connes 1976 Embedding Conjecture, and provided a negative answer to Tsirelson's problem".*Combinatorics and more*. Retrieved 2020-03-06.**^**Barak, Boaz (2020-01-14). "MIP*=RE, disproving Connes embedding conjecture".*Windows On Theory*. Retrieved 2020-03-06.**^**Aaronson, Scott (16 January 2020). "MIP*=RE".*Shtetl-Optimized*. Retrieved 2020-03-06.**^**Regan, Kenneth W. (2020-01-15). "Halting Is Poly-Time Quantum Provable".*Gödel's Lost Letter and P=NP*. Retrieved 2020-03-06.**^**Vidick, Thomas (2020-01-14). "A Masters project".*MyCQstate*. Retrieved 2020-03-06.**^**Hartnett, Kevin. "Landmark Computer Science Proof Cascades Through Physics and Math".*Quanta Magazine*. Retrieved 2020-03-09.

## Further reading

- 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. - Ge; Hadwin (2001). "Ultraproducts of C*-algebras".
*Oper. Theory Adv. Appl*.**127**: 305–326. doi:10.1007/978-3-0348-8374-0_17. - 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 II
_{1}Factors" (PDF). Department of Mathematics, University of Virginia. arXiv:0809.4439. Cite journal requires`|journal=`

(help) - Pisier, Gilles. "Tensor products of C*-algebras and operator spaces: The Connes-Kirchberg problem" (PDF). CS1 maint: discouraged parameter (link)