Abstract
Let be na infinite-dimensional Hilbert space. Any unitary operator U on can be written as a matrix {Upq}o≦p,q≦d whose entries are bounded operators on . The algebra generated by the operator-valued functions is isomorphic to the complex algebra generated by the unit and the noncommutative indeterminates xpq, x*pq with the relations 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 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 . Then F ≡ O.
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References
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Glockner, P., von Waldenfels, W. (1989). The relations of the non-commutative coefficient algebra of the unitary group. In: Accardi, L., von Waldenfels, W. (eds) Quantum Probability and Applications IV. Lecture Notes in Mathematics, vol 1396. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083553
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DOI: https://doi.org/10.1007/BFb0083553
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