A Survey of Noncommutative Dynamical Entropy

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 126)


With the success of entropy in classical ergodic theory it became a natural problem to extend the entropy concept to operator algebras.


Relative Entropy Free Product Topological Entropy Bernoulli Shift Approximation Entropy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-F]
    Alicki, R. and Fannes, M. Defining quantum dynamical entropy, Lett. Math. Phys. 32 (1994), 75–82.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [A]
    Araki, H. Relative entropy for states of von Neumann algebras II, Publ. RIMS Kyoto Univ. 13 (1977), 173–192.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [Ar]
    Arveson, W. Subalgebras of C*-algebras, Acta Math. 123 (1969), 141–224.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [Av]
    Avitzour, D. Free products of C*-algebras and von Neumann algebras, Trans. Amer. Math. Soc. 217 (1982), 423–435.MathSciNetGoogle Scholar
  5. [B]
    Besson, O. On the entropy of quantum Markov states, Lecture Notes in Math. 1136 (1985), 81–89, Springer-Verlag.Google Scholar
  6. [B-G]
    Bezuglyi, S.I. and Golodets, V.Ya. Dynamical entropy for Bogoliubov actions of free abelian groups on the CAR-algebra, Ergod. Th. & Dynam. Sys. 17 (1997), 757–782.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Bo-Go]
    Boca, F. and Goldstein, P. Topological entropy for the canonical endomorphism of Cuntz-Krieger algebras, Bull. London Math. Soc. 32 (2000), 345–352.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [BKRS]
    Bratteli, O., Kishimoto, A., Rørdam, M. and Størmer, E. The crossed product of a UHF-algebra by a shift, Ergod. Th. & Dynam. Sys. 13 (1993), 615–626.zbMATHGoogle Scholar
  9. [B-R]
    Bratteli, O. and Robinson, D.W. Operator algebras and quantum statistical mechanics II. Springer-Verlag 1981.zbMATHGoogle Scholar
  10. [Brl]
    Brown, N. Topological entropy in exact C*-algebras, Math. Ann. 314 (1999), 347–367.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Br2]
    Brown, N. A note on topological entropy, embeddings and unitaries in nuclear quasidiagonal C*-algebras, Proc. Amer. Math. Soc. 128 (2000), 2603–2609.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Br3]
    Brown, N. Characterizing type I C*-algebras via entropy, preprint 2000..Google Scholar
  13. [B-C]
    Brown, N. and Choda, M. Approximation entropies in crossed products with an application to free shifts, Pacific J. Math., to appear.Google Scholar
  14. [BDS]
    Brown, N., Dykema, K. and Shlyakhtenko, D. Topological entropy of free product automorphisms, preprint 2000.Google Scholar
  15. [C1]
    Choda, M. Entropy for *-endomorphisms and relative entropy for subalgebras, J. Operator Th. 25 (1991), 125–140.MathSciNetzbMATHGoogle Scholar
  16. [C2]
    Choda, M. Entropy for canonical shifts, Trans. Amer. Math. Soc. 334 (1992), 827–849.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [C3]
    Choda, M. Reduced free products of completely positive maps and entropy for free products of automorphismes, Publ. RIMS Kyoto Univ. 32 (1996), 371–382.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [C4]
    Choda, M. Entropy of Cuntz’s canonical endomorphism, Pacific J. Math. (1999), 235–245.Google Scholar
  19. [C5]
    Choda, M. A dynamical entropy and applications to canonical endomorphisms, J. Funct. Anal. 173 (2000), 453–480.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [C6]
    Choda, M. Entropy on crossed products and entropy on free products, J. Operator Theory, to appear.Google Scholar
  21. [C7]
    Choda, M. Dynamical entropy for automorphisms of exact C*-algebras, preprint 2000.Google Scholar
  22. [C-H]
    Choda, M. and Hiai, F. Entropy for canonical shifts II. Publ. RIMS Kyoto Univ. 27 (1991), 461–489.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [C-N]
    Choda, M. and Natsume, T. Reduced C*-crossed products by free shifts, Ergod. Th. & Dynam. Sys. 18 (1998), 1075–1096.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Co]
    Connes, A. Entropie de Kolmogoroff Sinai et mechanique statistique quantique, C. R. Acad. Sci. Paris 301 (1985), 1–6.MathSciNetzbMATHGoogle Scholar
  25. [CFW]
    Connes, A., Feldmann, J. and Weiss, B. An amenable equivalence relation is generated by a single transformation, Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.CrossRefzbMATHGoogle Scholar
  26. [CNT]
    Connes, A., Narnhofer, H. and Thirring, W. Dynamical entropy of C*-algebras and von Neumann algebras, Commun. Math. Phys. 112 (1987), 691–719.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [C-S]
    Connes, A. and Størmer, E. Entropy of automorphisms of II1 -von Neumann algebras, Acta Math. 134 (1975), 289–306.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [Cu]
    Cuntz, J. Simple C*-algebras generated by isometries, Commun. Math. Phys. 57 (1977), 173–185.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [De]
    Deaconu, V. Entropy estimates for some C*-endomorphisms, Proc. Amer. Math. Soc. 127(1999), 3653–3658.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [D1]
    Dykema, K. Exactness of reduced amalgamated free products of C*-algebras, preprint 1999.Google Scholar
  31. [D2]
    Dykema, K. Topological entropy of some automorphisms of reduced amalgamated free product C*-algebras, Ergod. Th. & Dynam. Sys., to appear.Google Scholar
  32. [D-S]
    Dykema, K. and Shlyakhtenko, D. Exactness of Cuntz-Pimsner C*-algebras, Proc. Edinburgh Math. Soc, to appear.Google Scholar
  33. [E]
    Enomoto, M., Nagisa M., Watatani, Y. and Yoshida, H. Relative commutant algebras of Powers’ binary shifts on the hyperfinite II1 -factor, Math. Scand. 68 (1991), 115–130.MathSciNetzbMATHGoogle Scholar
  34. [G-Nl]
    Golodets, V. Ya. and Neshveyev, S. Dynamical entropy for Bogoliubov actions of torsion-free abelian groups on the CAR-algebra, Ergod. Th. & Dynam. Sys. 20 (2000), 1111–1125.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [G-N2]
    Golodets, V. Ya. and Neshveyev, S. Entropy of automorphisnes of II1 -factors arising from the dynamical systems theory, preprint 1999.Google Scholar
  36. [G-N3]
    Golodets, V. Ya. and Neshveyev, S. Non-Bernoullian quantum K-systems. Commun. Math. Phys. 195 (1998), 213–232.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [G-Sl]
    Golodets, V. Ya. and Størmer, E. Entropy of C*-dynamical systems defined by bitstreams, Ergod. Th. & Dynam. Sys. 18 (1998), 859–874.CrossRefzbMATHGoogle Scholar
  38. [G-S2]
    Golodets, V. Ya. and Størmer, E. Generators and comparison of entropies of automorphisms of finite von Neumann algebras, J. Funct. Anal. 164 (1999), 110–133.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [H-S]
    Haagerup, U. and Størmer, E. Maximality of entropy in finite von Neumann algebras, Invent. Math. 132 (1998), 433–455.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [H]
    Hiai, F. Entropy for canonical shifts and strong amenability, Int. J. Math. 6 (1995), 381–396.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Hu]
    Hudetz, T. Quantum dynamical entropy revisited, Quantum probability (1997), 241–251, Banach Center Publ. 43.Google Scholar
  42. [J]
    Jones, V.F.R. Index for subfactors, Invent. Math. 72 (1983), 1–25.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [KP]
    Kerr, D. and Pinzari, C. Noncommutative pressure and the variational principle for Cuntz-Krieger-type C*-algebras, preprint 2000.Google Scholar
  44. [K]
    Kirchberg, E. On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35–63.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [L]
    Longo, R. Index of subfactors and statistics of quantum fields, Commun. Math. Phys. 126(1989), 145–155.MathSciNetCrossRefGoogle Scholar
  46. [M]
    Moriya, H. Variational principle and the dynamical entropy of space translations, preprint 1999.Google Scholar
  47. [N-U]
    Nakamura, M. and Umegaki, H. A note on the entropy for operator algebras, Proc. Japan Acad. 37 (1961), 149–154.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [NST]
    Narnhofer, H., Størmer, E. and Thirring, W. C*-dynamical systems for which the tensor product formula for entropy fails, Ergod. Th. & Dynam. Sys. 15 (1995), 96/9/68.CrossRefGoogle Scholar
  49. [N-T1]
    Narnhofer, H. and Thirring, W. Dynamical entropy of quantum systems and their abelian counterpart. On Klauder Path: A field trip, Emch, G.G., Hegerfeldt, G.C., Streit, L. World Scientific; Singapore, 1994.Google Scholar
  50. [N-T2]
    Narnhofer, H. and Thirring, W. C*-dynamical systems that are highly anticom-mutative, Lett. Math. Phys. (1995), 145–154.Google Scholar
  51. [N]
    Neshveyev, S. Entropy of Bogoliubov automorphisnes of CAR and CCR algebras with respect to quasi-free states, Rev. Math. Phys. 13 (2001), 29–50.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [N-Sl]
    Neshveyev, S. and Størmer, E. Entropy in type I algebras, Pacific J. Math., to appear.Google Scholar
  53. [N-S2]
    Neshveyev, S. and Størmer, E. The variational principle for a class of asymptotically abelian C*-algebras, Commun. Math. Phys. 215 (2000), 177–196.CrossRefzbMATHGoogle Scholar
  54. [N-S3]
    Neshveyev, S. and Størmer, E. The McMillan theorem for a class of asymptotically abelian C*-algebras, Ergod. Th. & Dynam. Sys. To appear.Google Scholar
  55. [O-P]
    Ohya, M. and Petz, D. Quantum entropy and its use. Texts and Monographs in Physics, Springer-Verlag, 1993.CrossRefGoogle Scholar
  56. [O]
    Ornstein, D.S. Bernoulli shifts with the same entropy are isomorphic, Adv. Math. 4(1970), 337–352.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [P-S]
    Park, Y.M. and Shin, H.H. Dynamical entropy of space translations of CAR and CCR algebras with respect to quasi-free states, Commun. Math. Phys. 152 (1993), 497–537.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [P-P]
    Pimsner, M. and Popa, S. Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986), 57–106.MathSciNetzbMATHGoogle Scholar
  59. [PWY]
    Pinzari, C., Watatani, Y. and Yonetani, K. KMS states, entropy and the variational principle in full C*-dynamical systems, Commun. Math. Phys, to appear.Google Scholar
  60. [P]
    Powers, R.T. An index theory for semigroups of *-endomorphisms of B(H) and type II1 -factors, Canad. J. Math. 40 (1988), 86–114.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [Po-Pr]
    Powers, R.T. and Price, G. Binary shifts on the hyperfinite II1 -factor, Contemp. Math. 145 (1993), 453–464.MathSciNetCrossRefGoogle Scholar
  62. [Pr]
    Price, G. The entropy of rational Powers shifts, Proc. Amer. Math. Soc. 126 (1998), 1715–1720.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [Q]
    Quasthoff, V. Shift automorphisms of the hyperfinite factor, Math. Nachrichten 131 (1987), 101–106.MathSciNetCrossRefzbMATHGoogle Scholar
  64. [Sa]
    Sauvageot, J. Ergodic properties of the action of a matrix in SL(2, Z) on a non communative torus, preprint.Google Scholar
  65. [S-T]
    Sauvageot, J.-L. and Thouvenot, J.-P. Une nouvelle definition de l’entropie dynamique des systems non commutatifs, Commun. Math. Phys. 145 (1992), 521–542.MathSciNetCrossRefGoogle Scholar
  66. [Sh]
    Shields, P. The theory of Bernoulli shifts, Univ. Chicago Press, 1973.zbMATHGoogle Scholar
  67. [S-S]
    Sinclair, A. and Smith, R.R. The completely bounded approximation property for discrete crossed products, Indiana Univ. Math. J. 46 (1997), 1311–1322.MathSciNetCrossRefzbMATHGoogle Scholar
  68. [S1]
    Størmer, E. Entropy of some automorphisms of the II1 -factor of the free group in infinite number of generators, Invent. Math. 110 (1992), 63–73.MathSciNetCrossRefGoogle Scholar
  69. [S2]
    Størmer, E. Entropy of some inner automorphisms of the hyperfinite II1 -factor, Int. J. Math. 4 (1993), 319–322.CrossRefGoogle Scholar
  70. [S3]
    Størmer, E. States and shifts on infinite free products of C*-algebras, Fields Inst. Commun. 12 (1997), 281–291.Google Scholar
  71. [S4]
    Størmer, E. Entropy of ertdomorphisms and relative entropy in finite von Neumann algebras, J. Funct, Anal. 171 (2000), 34–52.MathSciNetCrossRefGoogle Scholar
  72. [SV]
    Størmer, E. and Voiculescu, D. Entropy of Bogoliubov automorphisms of the canonical anticommutation relations, Commun. Math. Phys. 133 (1990), 521–542.CrossRefGoogle Scholar
  73. [T]
    Thomsen, K. Topological entropy for endomorphisms of local C*-algebras, Commun. Math. Phys. 164 (1994), 181–193.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [Vi]
    Vik, S. Fock representation of the binary shift algebra, Math. Scand., to appear.Google Scholar
  75. [V]
    Voiculescu, D. Dynamical approximation entropies and topological entropy in operator algebras, Commun. Math. Phys. 170 (1995), 249–281.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [VDN]
    Voiculescu, D., Dykema, K. and Nica, A. Free random variables, CRM Monograph Series, Vol. 1, Amer. Math. Soc, Providence, RI, 1992.zbMATHGoogle Scholar
  77. [W]
    Walters, P. An introduction to ergodic theory, Graduate texts in Math. 79, Springer-Verlag, 1982.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

There are no affiliations available

Personalised recommendations