Noncommutative rational Clark measures

We characterize the noncommutative Aleksandrov-Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over is C^d defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable 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 noncommutative and contractive rational multipliers. Noncommutative 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 noncommutative rational functions.