  We introduce notions of absolutely continuous functionals and representations
on the non-commutative disk algebra $A_n$. Absolutely continuous functionals
are used to help identify the type L part of the free semigroup algebra
associated to a $*$-extendible representation $\sigma$. A $*$-extendible
representation of $A_n$ is ``regular'' if the absolutely continuous part
coincides with the type L part. All known examples are regular. Absolutely
continuous functionals are intimately related to maps which intertwine a given
$*$-extendible representation with the left regular representation. A simple
application of these ideas extends reflexivity and hyper-reflexivity results.
Moreover the use of absolute continuity is a crucial device for establishing a
density theorem which states that the unit ball of $\sigma(A_n)$ is weak-$*$
dense in the unit ball of the associated free semigroup algebra if and only if
$\sigma$ is regular. We provide some explicit constructions related to the
density theorem for specific representations. A notion of singular functionals
is also defined, and every functional decomposes in a canonical way into the
sum of its absolutely continuous and singular parts.
