Non-commutative Clark measures for the free and abelian Toeplitz algebras
Abstract
We construct a non-commutative Aleksandrov-Clark measure for any element in the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebra of vector-valued full Fock space over . Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed algebra generated by the component operators of the free shift, the row isometry of left creation operators. This defines a bijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, the norm-closed algebra generated by the free shift. Identifying Drury-Arveson space with symmetric Fock space, we determine the relationship between the non-commutative AC measures for elements of the operator-valued commutative Schur class (the closed unit ball of the WOT-closed Toeplitz algebra generated by the Arveson shift) and the AC measures of their free liftings to the free Schur class.
Keywords
Cite
@article{arxiv.1703.02034,
title = {Non-commutative Clark measures for the free and abelian Toeplitz algebras},
author = {Michael T. Jury and Robert T. W. Martin},
journal= {arXiv preprint arXiv:1703.02034},
year = {2017}
}