## Abstract

We characterize the non-commutative Aleksandrov-Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space. Here, the full Fock space over C^{d }is defined as the Hilbert space of square-summable power series in several non-commuting formal variables, and we interpret this space as the noncommutative and multi-variable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov-Clark measure theory for non-commutative and contractive rational multipliers. Non-commutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz-Toeplitz algebra, the unital C*-algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz-Toeplitz and Cuntz algebras and the emerging field of non-commutative rational functions.

Original language | English |
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Pages (from-to) | 1-53 |

Number of pages | 53 |

Journal | Canadian Journal of Mathematics |

Volume | 75 |

Issue number | 5 |

DOIs | |

State | Accepted/In press - 27 Jul 2022 |

## Keywords

- Cuntz algebra
- Noncommutative rational functions
- finitely correlated representations
- noncommutative Clark measures

## ASJC Scopus subject areas

- General Mathematics