Automorphism Groups and Invariant Subspace Lattices

Abstract:

Let $(B,\mathbf {R},\alpha )$ be a $C^{*}$- dynamical system and let $A=B^\alpha ([0,\infty ))$ be the analytic subalgebra of $B$. We extend the work of Loebl and the first author that relates the invariant subspace structure of $\pi (A),$ for a $C^{*}$-representation $\pi$ on a Hilbert space $\mathcal {H}_\pi$, to the possibility of implementing $\alpha$ on $\mathcal {H}_\pi .$ We show that if $\pi$ is irreducible and if lat $\pi (A)$ is trivial, then $\pi (A)$ is ultraweakly dense in $\mathcal {L(H}_\pi ).$ We show, too, that if $A$ satisfies what we call the strong Dirichlet condition, then the ultraweak closure of $\pi (A)$ is a nest algebra for each irreducible representation $\pi .$ Our methods give a new proof of a “density” theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid $C^{*}$-algebras.