TY - JOUR

T1 - Non-commutative rational Clark measures

AU - Jury, Michael T.

AU - Martin, Robert T.W.

AU - Shamovich, Eli

N1 - Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - 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 C3 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.

AB - 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 C3 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.

KW - Cuntz algebra

KW - finitely-correlated representations

KW - noncommutative Clark measures

KW - noncommutative rational functions

UR - http://www.scopus.com/inward/record.url?scp=85135612311&partnerID=8YFLogxK

U2 - 10.4153/S0008414X22000384

DO - 10.4153/S0008414X22000384

M3 - Article

AN - SCOPUS:85135612311

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

SN - 0008-414X

ER -