## Mathematics, Department of

#### Title

ABSOLUTELY CONTINUOUS REPRESENTATIONS AND A KAPLANSKY DENSITY THEOREM FOR FREE SEMIGROUP ALGEBRAS

#### Date of this Version

6-2-2004

#### Citation

2004 Authors

#### Abstract

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra An. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a ∗-extendible represen- tation . 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 func- tionals 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 is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.

## Comments

Published as Kenneth R. Davidson, Jiankui Li, and David R. Pitts, "Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras, Journal of Functional Analysis, 224(1), (2005), 160-191