Functional Models for Representations of the Cuntz Algebra

  • Joseph A. Ball
  • Victor Vinnikov
Part of the Operator Theory: Advances and Applications book series (OT, volume 157)


We present a functional model, the elements of which are formal power series in a pair of d-tuples of non-commuting variables, for a row-unitary d-tuple of operators on a Hilbert space. The model is determined by a weighting matrix (called a “Haplitz” matrix) which has both non-commutative Hankel and Toeplitz structure. Such positive-definite Haplitz matrices then serve to classify representations of the Cuntz algebra O d with specified cyclic subspace up to unitary equivalence. As an illustration, we compute the weighting matrix for the free atomic representations studied by Davidson and Pitts and the related permutative representations studied by Bratteli and Jorgensen.


Hilbert Space Unitary Representation Formal Power Series Functional Model Borel Subset 
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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Joseph A. Ball
    • 1
  • Victor Vinnikov
    • 2
  1. 1.Department of MathematicsVirginia TechBlacksburg
  2. 2.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

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