Operator algebras associated to semigroup actions

Abstract

Reflexivity offers a way of reconstructing an algebra from a set of invariant subspaces. It is considered as Noncommutative Spectral Synthesis in association with synthesis problems in commutative Harmonic Analysis. Large classes of algebras are reflexive, the prototypical example being von Neumann algebras. The first example in the nonselfadjoint setting was the algebra of the unilateral shift which was shown by Sarason in the 1960s. Some further examples include the influential work of Arveson on CSL algebras, the Hp Hardy algebras examined by Peligrad, tensor products with the Hardy algebras and nest algebras. The concept of reflexivity was extended by Arveson who introduced the notion of hyperreflexivity. This is a measure of the distance to an algebra in terms of the invariant subspaces. It is a stronger property than reflexivity and examples include nest algebras, the free semigroup algebra and the algebra of analytic Toeplitz operators. Here we consider these questions for the class of w*-semicrossed products, in particular, those arising from actions of the free semigroup and the free abelian semigroup. We show that they are hyperreflexive when the action is implemented by uniformly bounded row operators. Combining our results with those of Helmer, we derive that w*-semicrossed products of factors of any type are reflexive. Furthermore, we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In each case it is also proved that the w*-semicrossed products have the bicommutant property if and only if the initial algebra of the dynamics does also. In addition we are interested in classifying the commuting endomorphisms of B(H) as an important example of dynamics implemented by a Cuntz family. Recall that On does not have a nice representation space in the sense that there is no countable collection of Borel functions that distinguish the unitary invariants. Therefore we focus our attention on the free atomic representations, which Davidson and Pitts classified up to unitary equivalence. Specifically we give a necessary and sufficient condition for an automorphism of B(H) to commute with a cyclic endomorphism.