Spectra for compact quantum group coactions and crossed products

TL;DR

This paper extends the concepts of Connes spectrum to compact quantum group actions on C*-algebras, providing criteria for simplicity and primeness of crossed products, and demonstrating spectral properties like closure under tensor products.

Contribution

It introduces definitions of Connes spectra for compact quantum group actions and establishes new criteria for the structure of crossed products, generalizing classical results to nonabelian quantum groups.

Findings

Connes spectra are characterized for quantum group actions.

Necessary and sufficient conditions for crossed product primeness and simplicity are provided.

Connes spectra are shown to be closed under tensor products.

Abstract

We present definitions of both Connes spectrum and strong Connes spectrum for actions of compact quantum groups on C*-algebras and obtain necessary and sufficient conditions for a crossed product to be a prime or a simple C*-algebra. Our results extend to the case of compact quantum actions the results in [8] which in turn, generalize results by Connes, Olesen and Pedersen and Kishimoto for abelian group actions. We prove in addition that the Connes spectra are closed under tensor products. These results are new for compact nonabelian groups as well.