# Crossed products of operator algebras: applications of Takai duality

## TL;DR

This paper uses Takai duality to analyze crossed products of operator algebras, establishing isomorphisms, ideal correspondences, and studying the Radical and semisimplicity properties in various dynamical systems.

## Contribution

It introduces new stable isomorphisms for semicrossed products, characterizes invariant ideals, and advances understanding of radicals and semisimplicity in operator algebra crossed products.

## Key findings

- Stable isomorphism between semicrossed product and tensor product crossed with group
- Complete lattice isomorphism between invariant ideals
- Crossed product of a radical algebra by a compact abelian group remains radical

## 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\rtimes_{\alpha} \Sigma $ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism \[ \mathcal A\rtimes_{\alpha} \Sigma \sim_{s} \big(\mathcal A \otimes \mathcal K(\mathcal G, \Sigma, \mu)\big)\rtimes_{\alpha\otimes {\rm Ad}\: \rho} \mathcal G, \] where $\mathcal K(\mathcal G, \Sigma, \mu)$ denotes the compact operators in the CSL algebra ${\rm Alg}\:\mathcal L(\mathcal G, \Sigma, \mu)$ 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 $\mathcal A\rtimes_{\alpha} \Sigma$ and the $(\alpha\otimes {\rm Ad}\: \rho)$-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 ${\rm Rad}(\mathcal A \rtimes_\alpha \mathcal G)=({\rm Rad}\:\mathcal A) \rtimes_\alpha \mathcal G$ persists. A broad class of such dynamical systems is identified.