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

TL;DR

This work analyzes two notions of coaction invariance of ideals in a $C^*$-algebra $A$ under a coaction $\delta$ of a locally compact group $G$, linking them to ideals in the crossed product $A\rtimes_{\delta}G$. It shows that for actions, Nilsen invariance aligns with the standard invariance, while for coactions the strong $\delta$-invariance and Nilsen invariance coincide for amenable $G$ but can diverge when $G$ is non-amenable. The authors provide precise criteria: $\mathscr{I}_{\delta}^s(A)=\mathscr{I}_{\delta}^N(A)$ if and only if $\delta$ is normal and the quotient sequence is exact for all strongly invariant $I$, with normality plus $G$-exactness ensuring equality. In the maximal/normal setting, they establish a lattice isomorphism between strongly invariant ideals and crossed-product ideals, and a bijection with Nilsen-invariant ideals via $I\mapsto \text{Res}_{\delta}\circ\text{Ind}_{\delta}(I)$, enabling concrete correspondences and Morita-type insights. These results clarify when invariance notions align and provide explicit lattice-level correspondences under structural conditions on $\delta$ and $G$.

Abstract

Given a coaction $δ$ 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 $A \rtimes_δ G$. In particular, we characterize when these two notions of invariance are equivalent.

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 2.4
• Lemma 2.5
• Definition 2.6
• Theorem 2.7: ndual
• Theorem 2.8
• Lemma 3.1
• proof