, Volume 16, Issue 1, pp 67–79 | Cite as

A very proper Heisenberg–Lie Banach *-algebra

  • Niels Jakob Laustsen


For each pair of non-zero real numbers q 1 and q 2, Laustsen and Silvestrov have constructed a unital Banach *-algebra \({\fancyscript{C}_{q_1,q_2}}\) which contains a universal normalized solution to the *-algebraic (q 1, q 2)-deformed Heisenberg–Lie commutation relations. We show that for (q 1, q 2) = (−1, 1), this Banach *-algebra is very proper; that is, if \({M\in\mathbb{N}}\) and \({a_1, \ldots, a_M}\) are elements of \({\fancyscript{C}_{-1,1}}\) such that \({\sum_{m=1}^M a_m^*a_m=0}\), then necessarily \({a_1=a_2=\cdots=a_M=0}\).


Heisenberg–Lie commutation relations Banach *-algebra Very proper 

Mathematics Subject Classification (2010)

Primary 46K10 Secondary 43A20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dales H.G.: Banach algebras and automatic continuity. London Math. Soc. Monographs 24. Clarendon Press, Oxford (2000)Google Scholar
  2. 2.
    Kelley J.L., Vaught R.L.: The positive cone in Banach algebras. Trans. Amer. Math. Soc. 74, 44–55 (1953)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Laustsen N.J., Silvestrov S.D.: Heisenberg–Lie commutation relations in Banach algebras. Math. Proc. R. Irish Acad. 109A, 163–186 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Palmer, T.W.: Banach algebras and the general theory of *-algebras, vol. II: Encyclopedia Math. Appl. 79. Cambridge University Press, Cambridge (2001)Google Scholar
  5. 5.
    Sigurdsson G., Silvestrov S.D.: Bosonic realizations of the colour Heisenberg–Lie algebra. J. Nonlinear Math. Phys. 13, 110–128 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Fylde CollegeLancaster UniversityLancasterUK

Personalised recommendations