Comparing Two Notions of Coaction Invariance of Ideals in $\mathrm{C}^*$-Algebras

TL;DR

This paper investigates two different notions of coaction invariance of ideals in C*-algebras under group coactions and characterizes when these notions coincide, enhancing understanding of ideal structures in crossed products.

Contribution

It provides a characterization of the conditions under which two notions of coaction invariance of ideals are equivalent in the context of C*-algebra crossed products.

Findings

Identifies conditions for equivalence of coaction invariance notions.

Clarifies the relationship between ideal invariance and crossed product ideals.

Enhances understanding of ideal structure in coaction settings.

Abstract

Given a coaction $\delta$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_{\delta} G$. In particular, we characterize when these two notions of invariance are equivalent.