Crossed products of operator algebras: applications of Takai duality

Abstract

Let $(\mathcal{G},\Sigma )$ be an ordered abelian group with Haar measure $\mu$, let $(\mathcal{A},\mathcal{G},\alpha )$ be a dynamical system and let $\mathcal{A}{⋊}_{\alpha }\Sigma$ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism \[ \mathcal A\rtimes_α Σ\sim_{s} \big(\mathcal A \otimes \mathcal K(\mathcal G, Σ, μ)\big)\rtimes_{α\otimes {\rm Ad}\: ρ} \mathcal G, \] where $\mathcal{K}(\mathcal{G},\Sigma ,\mu )$ denotes the compact operators in the CSL algebra $\mathrm{Alg}\mathcal{L}(\mathcal{G},\Sigma ,\mu )$ and $\rho$ denotes the right regular representation of $\mathcal{G}$. We also show that there exists a complete lattice isomorphism between the $\hat{\alpha }$-invariant ideals of $\mathcal{A}{⋊}_{\alpha }\Sigma$ and the $(\alpha \otimes \mathrm{Ad}\rho )$-invariant ideals of $\mathcal{A}\otimes \mathcal{K}(\mathcal{G},\Sigma ,\mu )$. Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by $\mathbb{R}$. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems $(\mathcal{A},\mathcal{G},\alpha )$ for which the identity $\mathrm{Rad}(\mathcal{A}{⋊}_{\alpha }\mathcal{G})=(\mathrm{Rad}\mathcal{A}){⋊}_{\alpha }\mathcal{G}$ persists. A broad class of such dynamical systems is identified.

arXiv comment32 pages. More explanation in the proof of Lemma 2.5. Remark 2.6 added to clarify the definition of the semicrossed product and subsequent notation. Lemma 2.9 added to aid in the proof of Proposition 2.10. Several typos corrected