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

Given a coaction $\delta$ of a locally compact group $G$ on a ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebra $A{⋊}_{\delta }G$. In particular, we characterize when these two notions of invariance are equivalent.