Papers
Topics
Authors
Recent
Search
2000 character limit reached

Crossed products of operator algebras: applications of Takai duality

Published 24 May 2017 in math.OA | (1705.08729v4)

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.

Summary

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Continue Learning

We haven't generated follow-up questions for this paper yet.

Collections

Sign up for free to add this paper to one or more collections.