  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.
