Aleksandrov-Clark theory for the Drury-Arveson space

TL;DR

This paper extends Aleksandrov-Clark theory to the multivariable Drury-Arveson space, establishing canonical extensions of Aleksandrov-Clark maps and characterizing their properties within a non-commutative operator algebra framework.

Contribution

It introduces a canonical 'tight' extension of Aleksandrov-Clark maps to the Cuntz-Toeplitz operator system in the multivariable setting, generalizing previous results.

Findings

Existence of a canonical tight extension of Aleksandrov-Clark maps.

Characterization of all extensions of Aleksandrov-Clark maps.

Application of the extension to generalize earlier results.

Abstract

Recent work has demonstrated that Clark's theory of unitary perturbations of the backward shift restricted to a deBranges-Rovnyak subspace of Hardy space on the disk has a natural extension to the several variable setting. In the several variable case, the appropriate generalization of the Schur class of contractive analytic functions is the closed unit ball of the Drury-Arveson multiplier algebra and the Aleksandrov-Clark measures are necessarily promoted to positive linear functionals on a symmetrized subsystem of the Cuntz-Toeplitz operator system A + A*, where A is the non-commutative disk algebra. We continue this program for vector-valued Drury-Arveson space by establishing the existence of a canonical `tight' extension of any Aleksandrov-Clark map to the full Cuntz-Toeplitz operator system. We apply this tight extension to generalize several earlier results and we characterize all extensions of the Aleksandrov-Clark maps.