Absolutely continuous representations and a Kaplansky density theorem for free semigroup algebras

We introduce notions of absolutely continuous functionals and representations on the noncommutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible representation . A ∗-extendible representation of An 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 (An) is weak-∗ dense in the unit ball of the associated free semigroup algebra if and only if is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals isalso defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts. © 2004 Elsevier Inc. All rights reserved