Japanese Journal of Mathematics

, Volume 8, Issue 1, pp 147–183 | Cite as

About the Connes embedding conjecture

Algebraic approaches
Original Articles


In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.

Keywords and phrases

Connes embedding conjecture Kirchberg’s conjecture Tsirelson’s problem semi-pre-C*-algebras noncommutative real algebraic geometry 

Mathematics Subject Classification (2010)

16W80 46L89 81P15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BT.
    Bakonyi M., Timotin D.: Extensions of positive definite functions on free groups. J. Funct. Anal., 246, 31–49 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. Ba.
    A. Barvinok, A Course in Convexity, Grad. Stud. Math., 54, Amer. Math. Soc., Providence, RI, 2002.Google Scholar
  3. Be1.
    Bekka M.B.: On the full C*-algebras of arithmetic groups and the congruence subgroup problem. Forum Math., 11, 705–715 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. Be2.
    Bekka M.B.: Operator-algebraic superridigity for \({{SL}_n({\mathbb Z})}\), n ≥ 3. Invent. Math., 169, 401–425 (2007)MathSciNetCrossRefMATHGoogle Scholar
  5. Bo.
    Boca F.: Completely positive maps on amalgamated product C*-algebras. Math. Scand., 72, 212–222 (1993)MathSciNetMATHGoogle Scholar
  6. BO.
    N.P. Brown and N. Ozawa, C*-Algebras and Finite-Dimensional Approximations, Grad. Stud. Math., 88, Amer. Math. Soc., Providence, RI, 2008.Google Scholar
  7. C+.
    J.R. Carrión, M. Dadarlat and C. Eckhardt, On groups with quasidiagonal C*-algebras, preprint, arXiv:1210.4050.Google Scholar
  8. CE.
    Choi M.D., Effros E.G.: Injectivity and operator spaces. J. Functional Analysis, 24, 156–209 (1977)MathSciNetCrossRefMATHGoogle Scholar
  9. Ci.
    Cimprič J.: A representation theorem for Archimedean quadratic modules on *-rings. Canad. Math. Bull., 52, 39–52 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. Co.
    Connes A.: Classification of injective factors. Ann. of Math. (2), 104, 73–115 (1976)MathSciNetCrossRefMATHGoogle Scholar
  11. DJ.
    Dykema K., Juschenko K.: Matrices of unitary moments. Math. Scand., 109, 225–239 (2011)MathSciNetMATHGoogle Scholar
  12. EL.
    Exel R., Loring T.A.: Finite-dimensional representations of free product C*-algebras. Internat. J. Math., 3, 469–476 (1992)MathSciNetCrossRefMATHGoogle Scholar
  13. FH.
    Fack T., de la Harpe P.: Sommes de commutateurs dans les algèbres de von Neumann finies continues. Ann. Inst. Fourier (Grenoble), 30, 49–73 (1980)MathSciNetCrossRefMATHGoogle Scholar
  14. F+.
    D. Farenick, A.S. Kavruk, V.I. Paulsen and I.G. Todorov, Operator systems from discrete groups, preprint, arXiv:1209.1152.Google Scholar
  15. FP.
    D. Farenick and V.I. Paulsen, Operator system quotients of matrix algebras and their tensor products, arXiv:1101.0790; Math. Scand., to appear.Google Scholar
  16. Fr.
    T. Fritz, Tsirelson’s problem and Kirchberg’s conjecture, Rev. Math. Phys., 24 (2012), 1250012, 67 pp.Google Scholar
  17. HM.
    Helton J.W., McCullough S.A.: A Positivstellensatz for non-commutative polynomials. Trans. Amer. Math. Soc., 356, 3721–3737 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. J+.
    M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V.B. Scholz and R.F. Werner, Connes embedding problem and Tsirelson’s problem, J. Math. Phys., 52 (2011), 012102, 12 pp.Google Scholar
  19. JP.
    Juschenko K., Popovych S.: Algebraic reformulation of Connes embedding problem and the free group algebra. Israel J. Math., 181, 305–315 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. Ka.
    A.S. Kavruk, The weak expectation property and Riesz interpolation, preprint, arXiv:1201.5414.Google Scholar
  21. K+.
    Kavruk A.S., Paulsen V.I., Todorov I.G., Tomforde M.: Tensor products of operator systems. J. Funct. Anal., 261, 267–299 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. Ki1.
    Kirchberg E.: On nonsemisplit extensions, tensor products and exactness of group C*-algebras. Invent. Math., 112, 449–489 (1993)MathSciNetCrossRefMATHGoogle Scholar
  23. Ki2.
    Kirchberg E.: Discrete groups with Kazhdan’s property T and factorization property are residually finite. Math. Ann., 299, 551–563 (1994)MathSciNetCrossRefMATHGoogle Scholar
  24. KS.
    Klep I., Schweighofer M.: Connes’ embedding conjecture and sums of Hermitian squares. Adv. Math., 217, 1816–1837 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. LS.
    A. Lubotzky and Y. Shalom, Finite representations in the unitary dual and Ramanujan groups, In: Discrete Geometric Analysis, Contemp. Math., 347, Amer. Math. Soc., Providence, RI, 2004, pp. 173–189.Google Scholar
  26. Mc.
    McCullough S.: Factorization of operator-valued polynomials in several non-commuting variables. Linear Algebra Appl., 326, 193–203 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. NT.
    T. Netzer and A. Thom, Real closed separation theorems and applications to group algebras, arXiv:1110.5619; Pacific J. Math., to appear.Google Scholar
  28. Oz1.
    Ozawa N.: About the QWEP conjecture. Internat. J. Math., 15, 501–530 (2004)MathSciNetCrossRefMATHGoogle Scholar
  29. Oz2.
    N. Ozawa, Tsirelson’s problem and asymptotically commuting unitary matrices, arXiv:1211.2712; J. Math. Phys., accepted.Google Scholar
  30. Pa.
    Paulsen V.: Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., 78. Cambridge Univ. Press, Cambridge (2002)Google Scholar
  31. Pi1.
    Pisier G.: Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., 294. Cambridge Univ. Press, Cambridge (2003)Google Scholar
  32. Pi2.
    Pisier G.: Grothendieck’s theorem, past and present. Bull. Amer. Math. Soc. (N.S.), 49, 237–323 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. Pu.
    Putinar M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J., 42, 969–984 (1993)MathSciNetCrossRefMATHGoogle Scholar
  34. Ru.
    Rudin W.: The extension problem for positive-definite functions. Illinois J. Math., 7, 532–539 (1963)MathSciNetMATHGoogle Scholar
  35. Sd.
    Scheiderer C.: Sums of squares on real algebraic surfaces. Manuscripta Math., 119, 395–410 (2006)MathSciNetCrossRefMATHGoogle Scholar
  36. Sm.
    K. Schmüdgen, Noncommutative real algebraic geometry—some basic concepts and first ideas, In: Emerging Applications of Algebraic Geometry, IMA Vol. Math. Appl., 149, Springer-Verlag, 2009, pp. 325–350.Google Scholar
  37. Ta.
    M. Takesaki, Theory of Operator Algebras. II, Encyclopedia Math. Sci., 125, Springer-Verlag, 2002.Google Scholar
  38. Ts.
    B.S. Tsirelson, Some results and problems on quantum Bell-type inequalities, In: Fundamental Questions in Quantum Physics and Relativity, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1993, pp. 32–48.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2013

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations