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

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dc.creatorDavidson, Kenneth R.
dc.creatorLi, Jiankui
dc.creatorPitts, David R.
dc.date2004-06-02
dc.date.accessioned2026-07-07T05:08:48Z
dc.date.available2026-07-07T05:08:48Z
dc.descriptionWe 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.
dc.description26 pages, prepared with LATeX2e, submitted to Journal of Functional Analysis
dc.identifierhttps://arxiv.org/abs/math/0406036
dc.identifierhttp://arxiv.org/abs/math/0406036
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/71412
dc.subjectOperator Algebras
dc.subject47L75, 47L80
dc.titleAbsolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras
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