Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

Kenneth R. Davidson; Jiankui Li; David R. Pitts

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Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

Kenneth R. Davidson·Jiankui Li·David R. Pitts

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Abstract

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.

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BibTeX RIS

Kenneth R. Davidson, Jiankui Li, David R. Pitts. 2004-06-02. Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras. https://arxiv.org/abs/math/0406036

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