Non-commutative Clark measures for the free and Abelian Toeplitz algebras

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 Cd. Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed a...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2017-12, Vol.456 (2), p.1062-1100
Main Authors: Jury, M.T., Martin, R.T.W.
Format: Article
Language:English
Subjects:
Aleksandrov–Clark measures
Drury–Arveson space
Non-commutative/free analytic Toeplitz algebra
Schur class
Transfer function realizations
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Summary: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 Cd. 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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2017.07.023