Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras
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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.
26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysis
26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysis