Analytic functionals for the non-commutative disc algebra

TL;DR

This paper extends classical results about analytic functionals from the commutative to the non-commutative setting, exploring their representations, extensions, and obstructions in operator algebras.

Contribution

It generalizes the F. and M. Riesz theorem to multivariate non-commutative algebras and characterizes the existence of weak-* continuous extensions.

Findings

Analytic functionals on the non-commutative disc algebra cannot have singular summands.

Extensions of these functionals are obstructed by the universal structure projection.

Applications to multipliers on Nevanlinna--Pick spaces demonstrate broader relevance.

Abstract

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals must be given as integration against an absolutely continuous measure on the circle. We develop generalizations of this result to the multivariate non-commutative setting, upon reinterpreting the classical result. In one direction, we show that the GNS representation naturally associated to an analytic functional on the Cuntz algebra cannot have any singular summand. Following a different interpretation, we seek weak-$*$ continuous extensions of analytic functionals on the free disc operator system. In contrast with the classical setting, such extensions do not always exist, and we identify the obstruction precisely in terms of the so-called universal structure projection. We also apply our ideas to commutative algebras of multipliers on some complete Nevanlinna--Pick function spaces.