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

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 representation σ. 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 σ(An) 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. Free semigroup algebras were introduced in [13] as a method for analyzing the fine structure of n-tuples of isometries with commuting ranges. The C*-algebra generated by such an n-tuple is either the Cuntz algebra On or the Cuntz-Toeplitz algebra En. As such, the free semigroup algebras can be used to reveal the fine spatial structure of representations of these algebras much in the same way as the von Neumann algebra generated by a unitary operator encodes the measure class and multiplicity which cannot be detected in the C*-algebra it generates. This viewpoint yields critical information in the work of Bratteli and Jorgensen [5, 6, 20, 21] who use certain representations of On to construct and analyze wavelet bases. From another point of view, free semigroup algebras can be used to study arbitrary (row contractive) n-tuples of operators. Frahzo [17, 18], Bunce [7] and Popescu [23] show that every (row) contractive n-tuple of operators has a unique minimal dilation to an n-tuple of isometries which is a row contraction, meaning that the ranges are pairwise orthogonal. Thus every row contraction determines a free semigroup algebra. Popescu [26] establishes the n-variable von Neumann inequality which follows immediately from the dilation theorem. Popescu has pursued a program of establishing the analogues of the Sz. Nagy–Foiaş program in the n-variable setting [24, 25, 27]; the latter two papers deal with the free semigroup algebras from this point of view. Free semigroup algebras play the same role for noncommuting operator theory as the weakly closed unital algebra determined by the isometric dilation of a contraction plays for a single operator. 1991 Mathematics Subject Classification. 47L80. Printed on February 1, 2008. First author partially supported by an NSERC grant.