The relations of the non-commutative coefficient algebra of the unitary group

  • P. Glockner
  • W. von Waldenfels
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1396)


Let Open image in new window be na infinite-dimensional Hilbert space. Any unitary operator U on Open image in new window can be written as a matrix {Upq}o≦p,q≦d whose entries are bounded operators on Open image in new window . The algebra generated by the operator-valued functions Open image in new window is isomorphic to the complex algebra generated by the unit and the noncommutative indeterminates xpq, x*pq with the relations Open image in new window In order to prove this, the corresponding result for the commutative coefficient algebra of the unitary group u(ℂd) is needed, i.e. for the algebra generated by the complex-valued functions Open image in new window Moreover, the following result is obtained: Let F(y1, ..., yn) be a polynomial in the independent non-commutative indeterminates y1, ..., yn and assume that F(A1 ..., An)=0 for all bounded operators A1, ..., An on Open image in new window . Then F ≡ O.


Polynomial Matrix Permutation Symbol Pure Tensor Quantum Stochastic Differential Equation Commutative Polynomial 
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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • P. Glockner
    • 1
  • W. von Waldenfels
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelberg 1

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