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On *-representations of λ-deformations of canonical commutation relations

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Ukrainian Mathematical Journal Aims and scope

We study irreducible integrable *-representations of the algebra \( {{\mathfrak{A}}_{{\lambda, 2}}} \) generated by the following relations:

$$ {{\mathfrak{A}}_{{\lambda, 2}}}=\mathbb{C}\left\langle {{a_j},a_j^{*}\left| {a_j^{*}{a_j}=1+{a_j}a_j^{*},\;a_1^{*}{a_2}=\lambda {a_2}a_1^{*},\;{a_2}{a_1}=\lambda {a_1}{a_2},\;j=1,2} \right.} \right\rangle . $$

For this *-algebra, we prove an analog of the von Neumann theorem on the uniqueness of irreducible integrable representation.

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References

  1. O. Bratelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics-2. Equilibrium States. Models in Quantum Statistical Mechanics, Springer, Berlin (2002).

  2. M. Bożejko and R. Speicher, “Completely positive maps on Coxeter groups, deformed commutation relations, and operator spaces,” Mat. Ann., 300, 97–120 (1994).

    Article  MATH  Google Scholar 

  3. Yu. M. Berezansky, Z. G. Sheftel, and G. F. Us, Functional Analysis, Birkhäuser, Basel, etc. (1996).

  4. D. Proskurin, “Stability of a special class of q ij -CCR and extensions of higher-dimensional noncommutative tori,” Lett. Math. Phys., 52, No. 2, 165–175 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  5. P. E. T. Jørgensen, D. P. Proskurin, and Yu. S. Samoilenko, “The kernel of Fock representation of Wick algebras with braided operator of coefficients,” Pacif. J. Math., 198, 109–122 (2001).

    Article  Google Scholar 

  6. P. E. T. Jørgensen, L. M. Schmitt, and R. F. Werner, “Positive representations of general commutation relations allowing Wick ordering,” J. Funct. Anal., 134, 33–99 (1995).

    Article  MathSciNet  Google Scholar 

  7. V. Ostrovskyi and Yu. Samoilenko, “Introduction to the theory of representations of finitely presented algebras,” Rev. Math. Math. Phys., 11 (2000).

  8. V. Ostrovskyi, D. Proskurin, and L. Turowska, “Unbounded representations of q -deformation of Cuntz algebra,” Lett. Math. Phys., 85, No. 2-3, 147–162 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Proskurin, “Homogeneous ideals in Wick *-algebras,” Proc. Amer. Math. Soc., 126, No. 11, 3371–3376 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  10. W. Pusz and S. L. Woronowicz, “Twisted second quantization,” Rep. Math. Phys., 27, 251–263 (1989).

    MathSciNet  Google Scholar 

Download references

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 4, pp. 538–545, April, 2013.

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Proskurin, D.P., Yakymiv, R.Y. On *-representations of λ-deformations of canonical commutation relations. Ukr Math J 65, 593–601 (2013). https://doi.org/10.1007/s11253-013-0797-3

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