Exactness of the Fock Space Representation of the q-Commutation Relations

  • Matthew Kennedy
  • Alexandru NicaEmail author


We show that for all q in the interval (−1, 1), the Fock representation of the q-commutation relations can be unitarily embedded into the Fock representation of the extended Cuntz algebra. In particular, this implies that the C*-algebra generated by the Fock representation of the q-commutation relations is exact. An immediate consequence is that the q-Gaussian von Neumann algebra is weakly exact for all q in the interval (−1, 1).


Creation Operator Cuntz Algebra Generalize Brownian Motion Free Entropy Dimension Left Creation Operator 
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.


  1. 1.
    Bozejko M., Speicher R.: An example of a generalized Brownian motion. Commun. Math. Phys. 137(3), 519–531 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Bozejko M., Kummerer B., Speicher R.: q-Gaussian processes: Non-commutative and classical aspects. Commun. Math. Phys. 185(1), 129–154 (1997)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Brown N., Ozawa N.: C*-algebras and finite dimensional approximations. Graduate Studies in Mathematics, Vol. 88. Providence, RI: Amer. Math. Soc., 2008Google Scholar
  4. 4.
    Davidson K.: On operators commuting with Toeplitz operators modulo the compact operators. J. Funct. Anal. 24(3), 291–302 (1977)zbMATHCrossRefGoogle Scholar
  5. 5.
    Dykema K., Nica A.: On the Fock representation of the q-commutation relations. J. fur Reine und Ang. Math. 440, 201–212 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Jorgensen P.E.T., Schmitt L.M., Werner R.F.: q-canonical commutation relations and stability of the Cuntz algebra. Pacific J. Math. 165(1), 131–151 (1994)MathSciNetGoogle Scholar
  7. 7.
    Nou A.: Non-injectivity of the q-deformed von Neumann algebras. Math. Ann. 330(1), 17–38 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Pedersen, G.K.: A commutator inequality. In: Operator Algebras, Mathematical Physics and Low-Dimensional Topology (Istanbul 1991), Research Notes in Mathematics 5, Wellesley, MA: AK Peters, 1993, pp. 233–235Google Scholar
  9. 9.
    Ricard E.: Factoriality of q-gaussian von Neumann algebras. Commun. Math. Phys. 257(3), 659–665 (2005)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. 10.
    Shlyakhtenko D.: Some estimates for non-microstates free dimension, with applications to q-semicircular families. Int. Math. Res. Not. 51, 2757–2772 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shlyakhtenko D.: Lower estimates on microstates free entropy dimension. Anal. PDE 2(2), 119–146 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Sniady P.: Factoriality of Bozejko-Speicher von Neumann algebras. Commun. Math. Phys. 246(3), 561–567 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Voiculescu D., Dykema K., Nica A.: Free random variables. CRM Monograph Series, Vol. 1. Providence, RI: Amer. Math. Soc. 1992Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Pure Mathematics DepartmentUniversity of WaterlooWaterlooCanada

Personalised recommendations