2022/01/20 by Michael T. Jury, Robert T. W. Martin, Jury, Michael T. +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2201.08045
openalex publication_date 2022/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 non-commutative 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.