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 σ. 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 σ(A n ) is weak- * dense in the unit ball of the associated free semigroup algebra if and only if σ 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. 3

In this paper, we show that if L is a completely distributive commutative subspace lattice or a J -subspace lattice, then the space of all bounded derivations of alg L is reflexive. We also study when local automorphisms on some algebras are automorphisms.

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