45:
developing his free entropy theory found that Connes' embedding problem is related to the existence of microstates. Some results of von
Neumann algebra theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to
438:
74:
that implies a negative answer to Connes' embedding problem. However, an error was discovered in
September 2020 in an earlier result they used; a new proof avoiding the earlier result was published as a preprint in September. A broad outline was published in
600:
293:
822:
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.
502:
298:
816:
782:
698:
640:
170:
740:
107:
143:
1020:
Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (27 September 2020). "Quantum soundness of the classical low individual degree test".
511:
660:
175:
1198:
996:
743:
81:
in
November 2021, and an article explaining the connection between MIP*=RE and the Connes Embedding Problem appeared in October 2022.
1083:
835:
Connes' embedding problem and quantum information theory workshop; Vanderbilt
University in Nashville Tennessee; May 1β7, 2020 (
1290:
443:
433:{\displaystyle I_{\omega }=\{(x_{n})\in l^{\infty }(R):\lim _{n\rightarrow \omega }\tau (x_{n}^{*}x_{n})^{\frac {1}{2}}=0\}}
49:
The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems:
1285:
848:
Workshop on Sofic and
Hyperlinear Groups and the Connes Embedding Conjecture; UFSC Florianopolis, Brazil; June 10β21, 2018
750:). In January 2020, a group of researchers claimed to have resolved the problem in the negative, i.e., there exist type II
1251:
703:
A positive solution to the problem would imply that invariant subspaces exist for a large class of operators in type II
118:
77:
71:
845:
Winter school: Connes' embedding problem and quantum information theory; University of Oslo, January 7β11, 2019
857:
Workshop on
Operator Spaces, Harmonic Analysis and Quantum Probability; ICMAT, Madrid; May 20-June 14, 2013
747:
42:
959:
819:
60:
860:
Fields
Workshop around Connes Embedding Problem – University of Ottawa, May 16–18, 2008
666:
38:
41:
theory. During that time, the problem was reformulated in several different areas of mathematics.
1233:
1169:
1151:
1128:
1106:
1064:
1021:
926:
904:
794:
760:
712:
676:
605:
148:
854:
Operator
Algebras and Quantum Information Theory; Institut Henri Poincare, Paris; December 2017
66:
The predual of any (separable) von
Neumann algebra is finitely representable in the trace class.
851:
Approximation Properties in Operator Algebras and Ergodic Theory; UCLA; April 30 - May 5, 2018
718:
1194:
977:
925:
Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (2020). "MIP*=RE".
92:
1186:
1161:
1098:
1054:
1041:
Ji, Zhengfeng; Natarajan, Anand; Vidick, Thomas; Wright, John; Yuen, Henry (November 2021).
967:
894:
1142:
Farah, I.; Hart, B.; Sherman, D. (2013). "Model theory of operator algebras I: stability".
595:{\displaystyle \tau _{R^{\omega }}(x)=\lim _{n\rightarrow \omega }\tau (x_{n}+I_{\omega })}
128:
17:
963:
715:. A positive solution to the problem would be implied by equality between free entropy
645:
288:{\displaystyle l^{\infty }(R)=\{(x_{n})_{n}\subseteq R:\sup _{n}||x_{n}||<\infty \}}
1209:
1279:
1247:
1110:
1068:
708:
1173:
34:
1252:"Tensor products of C*-algebras and operator spaces: The Connes-Kirchberg problem"
899:
882:
1190:
110:
1185:. Operator Theory: Advances and Applications. Vol. 127. pp. 305β326.
972:
947:
70:
In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in
1258:
755:
54:
46:
the realization that it also has important implications in computer science.
981:
1165:
842:
The many faceted Connes' Embedding Problem; BIRS, Canada; July 14β19, 2019
1102:
908:
1127:
Capraro, Valerio (2010). "A Survey on Connes' Embedding Conjecture".
836:
1059:
1042:
1228:
Sherman, David (2008). "Notes on Automorphisms of Ultrapowers of II
1026:
997:"Landmark Computer Science Proof Cascades Through Physics and Math"
948:"How 'spooky' is quantum physics? The answer could be incalculable"
931:
1238:
1156:
1133:
295:
be the von Neumann algebra of norm-bounded sequences and let
826:
The problem admits a number of equivalent formulations.
673:
on a separable Hilbert space can be embedded into some
497:{\displaystyle R^{\omega }=l^{\infty }(R)/I_{\omega }}
1183:
Recent Advances in Operator Theory and Related Topics
818:
is independent of the ultrafilter if and only if the
797:
763:
721:
679:
648:
608:
514:
446:
301:
178:
151:
131:
95:
1181:Ge; Hadwin (2001). "Ultraproducts of C*-algebras".
830:Conferences dedicated to Connes' embedding problem
810:
776:
734:
692:
654:
634:
594:
496:
432:
287:
164:
137:
101:
27:Mathematical problem in von Neumann algebra theory
887:Proceedings of the American Mathematical Society
545:
360:
237:
1210:"A linearization of Connes' embedding problem"
1091:Bulletin of the American Mathematical Society
1084:"The Connes Embedding Problem: A Guided Tour"
665:Connes' embedding problem asks whether every
8:
754:von Neumann factors that do not embed in an
427:
315:
282:
201:
1144:Bulletin of the London Mathematical Society
1237:
1155:
1132:
1058:
1025:
971:
930:
898:
802:
796:
768:
762:
726:
720:
684:
678:
647:
626:
616:
607:
583:
570:
548:
524:
519:
513:
488:
479:
464:
451:
445:
410:
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385:
363:
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325:
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251:
246:
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221:
211:
183:
177:
156:
150:
130:
94:
920:
918:
870:
1208:Collins, BenoΔ±t; Dykema, Ken (2008).
876:
874:
711:); all countable discrete groups are
7:
37:in the 1970s, is a major problem in
145:. One can construct the ultrapower
642:is any representative sequence of
465:
342:
279:
184:
25:
883:"A Noncommutative Moment Problem"
1082:Isaac Goldbring (October 2022),
995:Hartnett, Kevin (4 March 2020).
829:
1217:New York Journal of Mathematics
113:on the natural numbers and let
53:Kirchberg's QWEP conjecture in
623:
609:
589:
563:
552:
538:
532:
476:
470:
407:
378:
367:
353:
347:
331:
318:
272:
267:
252:
247:
218:
204:
195:
189:
1:
946:Castelvecchi, Davide (2020).
900:10.1090/S0002-9939-01-05772-0
63:in quantum information theory
1191:10.1007/978-3-0348-8374-0_17
742:and free entropy defined by
811:{\displaystyle R^{\omega }}
777:{\displaystyle R^{\omega }}
693:{\displaystyle R^{\omega }}
635:{\displaystyle (x_{n})_{n}}
165:{\displaystyle R^{\omega }}
18:Connes embedding conjecture
1307:
973:10.1038/d41586-020-00120-6
1047:Communications of the ACM
791:The isomorphism class of
735:{\displaystyle \chi ^{*}}
78:Communications of the ACM
72:quantum complexity theory
31:Connes' embedding problem
102:{\displaystyle \omega }
812:
778:
736:
694:
656:
636:
596:
498:
434:
289:
166:
139:
103:
1291:Disproved conjectures
813:
784:of the hyperfinite II
779:
737:
695:
657:
637:
597:
499:
435:
290:
167:
140:
138:{\displaystyle \tau }
104:
1286:Von Neumann algebras
881:Hadwin, Don (2001).
820:continuum hypothesis
795:
761:
719:
677:
646:
606:
512:
504:turns out to be a II
444:
299:
176:
149:
129:
93:
1166:10.1112/blms/bdt014
964:2020Natur.577..461C
395:
119:hyperfinite type II
61:Tsirelson's problem
39:von Neumann algebra
808:
774:
732:
690:
652:
632:
592:
559:
508:factor with trace
494:
430:
381:
374:
285:
245:
162:
135:
99:
1200:978-3-0348-9539-2
1103:10.1090/bull/1768
958:(7791): 461β462.
655:{\displaystyle x}
544:
418:
359:
236:
16:(Redirected from
1298:
1272:
1270:
1269:
1263:
1257:. Archived from
1256:
1243:
1241:
1224:
1214:
1204:
1177:
1159:
1138:
1136:
1114:
1113:
1088:
1079:
1073:
1072:
1062:
1038:
1032:
1031:
1029:
1017:
1011:
1010:
1008:
1007:
992:
986:
985:
975:
943:
937:
936:
934:
922:
913:
912:
902:
893:(6): 1785β1791.
878:
817:
815:
814:
809:
807:
806:
783:
781:
780:
775:
773:
772:
741:
739:
738:
733:
731:
730:
699:
697:
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691:
689:
688:
661:
659:
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638:
633:
631:
630:
621:
620:
601:
599:
598:
593:
588:
587:
575:
574:
558:
531:
530:
529:
528:
503:
501:
500:
495:
493:
492:
483:
469:
468:
456:
455:
439:
437:
436:
431:
420:
419:
411:
405:
404:
394:
389:
373:
346:
345:
330:
329:
311:
310:
294:
292:
291:
286:
275:
270:
265:
264:
255:
250:
244:
226:
225:
216:
215:
188:
187:
172:as follows: let
171:
169:
168:
163:
161:
160:
144:
142:
141:
136:
111:free ultrafilter
108:
106:
105:
100:
33:, formulated by
21:
1306:
1305:
1301:
1300:
1299:
1297:
1296:
1295:
1276:
1275:
1267:
1265:
1261:
1254:
1246:
1231:
1227:
1212:
1207:
1201:
1180:
1141:
1126:
1123:
1121:Further reading
1118:
1117:
1086:
1081:
1080:
1076:
1060:10.1145/3485628
1053:(11): 131β138.
1040:
1039:
1035:
1019:
1018:
1014:
1005:
1003:
1001:Quanta Magazine
994:
993:
989:
945:
944:
940:
924:
923:
916:
880:
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872:
867:
832:
798:
793:
792:
787:
764:
759:
758:
753:
722:
717:
716:
706:
680:
675:
674:
670:
644:
643:
622:
612:
604:
603:
579:
566:
520:
515:
510:
509:
507:
484:
460:
447:
442:
441:
440:. The quotient
406:
396:
337:
321:
302:
297:
296:
256:
217:
207:
179:
174:
173:
152:
147:
146:
127:
126:
122:
91:
90:
87:
28:
23:
22:
15:
12:
11:
5:
1304:
1302:
1294:
1293:
1288:
1278:
1277:
1274:
1273:
1248:Pisier, Gilles
1244:
1229:
1225:
1205:
1199:
1178:
1150:(4): 825β838.
1139:
1122:
1119:
1116:
1115:
1097:(4): 503β560,
1074:
1033:
1012:
987:
938:
914:
869:
868:
866:
863:
862:
861:
858:
855:
852:
849:
846:
843:
840:
837:postponed; TBA
831:
828:
805:
801:
785:
771:
767:
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748:Dan Voiculescu
729:
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683:
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629:
625:
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615:
611:
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586:
582:
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573:
569:
565:
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505:
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463:
459:
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450:
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426:
423:
417:
414:
409:
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399:
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384:
380:
377:
372:
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366:
362:
358:
355:
352:
349:
344:
340:
336:
333:
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324:
320:
317:
314:
309:
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284:
281:
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249:
243:
239:
235:
232:
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224:
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43:Dan Voiculescu
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1264:on 2021-04-19
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1266:. Retrieved
1259:the original
1220:
1216:
1182:
1147:
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1094:
1090:
1077:
1050:
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1036:
1015:
1004:. Retrieved
1000:
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951:
941:
890:
886:
825:
790:
702:
664:
114:
88:
76:
69:
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35:Alain Connes
30:
29:
1043:"MIP* = RE"
744:microstates
713:hyperlinear
125:with trace
1280:Categories
1268:2020-01-25
1232:Factors".
1223:: 617β641.
1027:2009.12982
1006:2020-03-09
932:2001.04383
865:References
756:ultrapower
55:C*-algebra
1239:0809.4439
1157:0908.2790
1134:1003.2076
1111:237940159
1069:210165045
804:ω
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85:Statement
1174:15024863
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602:, where
960:Bibcode
909:2669132
667:type II
117:be the
1197:
1172:
1109:
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952:Nature
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123:factor
57:theory
1262:(PDF)
1255:(PDF)
1234:arXiv
1213:(PDF)
1170:S2CID
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1065:S2CID
1022:arXiv
927:arXiv
905:JSTOR
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1187:doi
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.