David R. Pitts — Page 2

arXiv2004

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 $A_n$. 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.

math.OA↗

arXiv2000

Automatic closure of invariant linear manifolds for operator algebras

Kadison's transitivity theorem implies that, for irreducible representations of C*-algebras, every invariant linear manifold is closed. It is known that CSL algebras have this propery if, and only if, the lattice is hyperatomic (every projection is generated by a finite number of atoms). We show several other conditions are equivalent, including the conditon that every invariant linear manifold is singly generated. We show that two families of norm closed operator algebras have this property. First, let L be a CSL and suppose A is a norm closed algebra which is weakly dense in Alg L and is a bimodule over the (not necessarily closed) algebra generated by the atoms of L. If L is hyperatomic and the compression of A to each atom of L is a C*-algebra, then every linear manifold invariant under A is closed. Secondly, if A is the image of a strongly maximal triangular AF algebra under a multiplicity free nest representation, where the nest has order type -N, then every linear manifold invariant under A is closed and is singly generated.

math.OA↗