Connes embedding problem

<h2 id="statement">Statement</h2>
Let 
 
 
 
 ω
 
 
 {\displaystyle \omega }
 
 be a <a href="/facts/Ultrafilter/6guRhrEd">free ultrafilter</a> on the <a href="/facts/Natural_number/u65og8JE">natural numbers</a> and let R be the <a href="/facts/Hyperfinite_type_II_factor/TAs9Q0sq">hyperfinite type II1 factor</a> with trace 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
. One can construct the <a href="/facts/Ultrapower/d4cT10hd">ultrapower</a> 
 
 
 
 
 R
 
 ω
 
 
 
 
 {\displaystyle R^{\omega }}
 
 as follows: let 
 
 
 
 
 l
 
 ∞
 
 
 (
 R
 )
 =
 {
 (
 
 x
 
 n
 
 
 
 )
 
 n
 
 
 ⊆
 R
 :
 
 sup
 
 n
 
 
 
 |
 
 
 |
 
 
 x
 
 n
 
 
 
 |
 
 
 |
 
 <
 ∞
 }
 
 
 {\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 
 
 
 
 
 I
 
 ω
 
 
 =
 {
 (
 
 x
 
 n
 
 
 )
 ∈
 
 l
 
 ∞
 
 
 (
 R
 )
 :
 
 lim
 
 n
 →
 ω
 
 
 τ
 (
 
 x
 
 n
 
 
 ∗
 
 
 
 x
 
 n
 
 
 
 )
 
 
 1
 2
 
 
 
 =
 0
 }
 
 
 {\displaystyle I_{\omega }=\{(x_{n})\in l^{\infty }(R):\lim _{n\rightarrow \omega }\tau (x_{n}^{*}x_{n})^{\frac {1}{2}}=0\}}
 
. The quotient 
 
 
 
 
 R
 
 ω
 
 
 =
 
 l
 
 ∞
 
 
 (
 R
 )
 
 /
 
 
 I
 
 ω
 
 
 
 
 {\displaystyle R^{\omega }=l^{\infty }(R)/I_{\omega }}
 
 turns out to be a II1 factor with trace 
 
 
 
 
 τ
 
 
 R
 
 ω
 
 
 
 
 (
 x
 )
 =
 
 lim
 
 n
 →
 ω
 
 
 τ
 (
 
 x
 
 n
 
 
 +
 
 I
 
 ω
 
 
 )
 
 
 {\displaystyle \tau _{R^{\omega }}(x)=\lim _{n\rightarrow \omega }\tau (x_{n}+I_{\omega })}
 
, where 
 
 
 
 (
 
 x
 
 n
 
 
 
 )
 
 n
 
 
 
 
 {\displaystyle (x_{n})_{n}}
 
 is any representative sequence of 
 
 
 
 x
 
 
 {\displaystyle x}
 
.
Connes' embedding problem asks whether every <a href="/facts/Von_Neumann_algebra/8NyWUIRH">type II1 factor</a> on a separable <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> can be embedded into some 
 
 
 
 
 R
 
 ω
 
 
 
 
 {\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 (<a href="/facts/Uffe_Haagerup/zwa7I8nt">Uffe Haagerup</a>); all countable discrete groups are hyperlinear. A positive solution to the problem would be implied by equality between free entropy 
 
 
 
 
 χ
 
 ∗
 
 
 
 
 {\displaystyle \chi ^{*}}
 
 and free entropy defined by <a href="/facts/Microstate_(statistical_mechanics)/ArXuNn08">microstates</a> (<a href="/facts/Dan-Virgil_Voiculescu/os6zf1pI">Dan Voiculescu</a>). In January 2020, a group of researchers<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> 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 
 
 
 
 
 R
 
 ω
 
 
 
 
 {\displaystyle R^{\omega }}
 
 of the hyperfinite II1 factor.
The isomorphism class of 
 
 
 
 
 R
 
 ω
 
 
 
 
 {\displaystyle R^{\omega }}
 
 is independent of the ultrafilter if and only if the <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a> 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.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="conferences-dedicated-to-connes-embedding-problem">Conferences dedicated to Connes' embedding problem</h2>
<ul><li>Connes' embedding problem and quantum information theory workshop; Vanderbilt University in Nashville Tennessee; May 1–7, 2020 (<a href="https://my.vanderbilt.edu/ncgoa20/">postponed; TBA</a>)</li>
<li>The many faceted Connes' Embedding Problem; BIRS, Canada; July 14–19, 2019</li>
<li>Winter school: Connes' embedding problem and quantum information theory; University of Oslo, January 7–11, 2019</li>
<li>Workshop on Sofic and Hyperlinear Groups and the Connes Embedding Conjecture; UFSC Florianopolis, Brazil; June 10–21, 2018</li>
<li>Approximation Properties in Operator Algebras and Ergodic Theory; UCLA; April 30 - May 5, 2018</li>
<li>Operator Algebras and Quantum Information Theory; Institut Henri Poincare, Paris; December 2017</li>
<li>Workshop on Operator Spaces, Harmonic Analysis and Quantum Probability; ICMAT, Madrid; May 20-June 14, 2013</li>
<li>Fields Workshop around Connes Embedding Problem – University of Ottawa, May 16–18, 2008</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Capraro, Valerio (2010). "A Survey on Connes' Embedding Conjecture". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1003.2076">1003.2076</a> [<a href="https://arxiv.org/archive/math.OA">math.OA</a>].</li>
<li>Farah, I.; Hart, B.; Sherman, D. (2013). "Model theory of operator algebras I: stability". Bulletin of the London Mathematical Society. 45 (4): 825–838. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0908.2790">0908.2790</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fblms%2Fbdt014">10.1112/blms/bdt014</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:15024863">15024863</a>.</li>
<li>Ge; Hadwin (2001). "Ultraproducts of C*-algebras". Recent Advances in Operator Theory and Related Topics. Operator Theory: Advances and Applications. Vol. 127. pp. 305–326. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-0348-8374-0_17">10.1007/978-3-0348-8374-0_17</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-0348-9539-2.</li>
<li>Collins, Benoıt; Dykema, Ken (2008). <a href="https://www.emis.de//journals/NYJM/j/2008/14-28.pdf">"A linearization of Connes' embedding problem"</a> (PDF). New York Journal of Mathematics. 14: 617–641.</li>
<li>Sherman, David (2008). "Notes on Automorphisms of Ultrapowers of II1 Factors". <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0809.4439">0809.4439</a> [<a href="https://arxiv.org/archive/math.OA">math.OA</a>].</li>
<li><a href="/facts/Gilles_Pisier/GzRQq0yx">Pisier, Gilles</a>. <a href="https://web.archive.org/web/20210419093748/https://www.math.tamu.edu/~pisier/TPCOS.pdf">"Tensor products of C*-algebras and operator spaces: The Connes-Kirchberg problem"</a> (PDF). Archived from <a href="https://www.math.tamu.edu/~pisier/TPCOS.pdf">the original</a> (PDF) on 2021-04-19. Retrieved 2020-01-25.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">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 href="https://doi.org/10.1090%2FS0002-9939-01-05772-0" target="_blank">https://doi.org/10.1090%2FS0002-9939-01-05772-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP*=RE". arXiv:2001.04383 [quant-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Castelvecchi, Davide (2020). "How 'spooky' is quantum physics? The answer could be incalculable". Nature. 577 (7791): 461–462. Bibcode:2020Natur.577..461C. doi:10.1038/d41586-020-00120-6. PMID 31965099. <a href="https://doi.org/10.1038%2Fd41586-020-00120-6" target="_blank">https://doi.org/10.1038%2Fd41586-020-00120-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Hartnett, Kevin (4 March 2020). "Landmark Computer Science Proof Cascades Through Physics and Math". Quanta Magazine. Retrieved 2020-03-09. <a href="https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/" target="_blank">https://www.quantamagazine.org/landmark-computer-science-proof-cascades-through-physics-and-math-20200304/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (27 September 2020). "Quantum soundness of the classical low individual degree test". arXiv:2009.12982 [quant-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (November 2021). "MIP* = RE". Communications of the ACM. 64 (11): 131–138. doi:10.1145/3485628. S2CID 210165045. <a href="https://doi.org/10.1145%2F3485628" target="_blank">https://doi.org/10.1145%2F3485628</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Isaac Goldbring (October 2022), "The Connes Embedding Problem: A Guided Tour" (PDF), Bulletin of the American Mathematical Society, 58 (4): 503–560, doi:10.1090/bull/1768, S2CID 237940159 <a href="https://www.ams.org/journals/bull/2022-59-04/S0273-0979-2022-01768-5/S0273-0979-2022-01768-5.pdf" target="_blank">https://www.ams.org/journals/bull/2022-59-04/S0273-0979-2022-01768-5/S0273-0979-2022-01768-5.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP*=RE". arXiv:2001.04383 [quant-ph]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">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 href="https://doi.org/10.1090%2FS0002-9939-01-05772-0" target="_blank">https://doi.org/10.1090%2FS0002-9939-01-05772-0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
</ol>

Connes embedding problem open-in-new

Connes embedding problem