Duality of Restriction and Induction for $C^{\ast }$-Coactions

Consider a coaction $\delta$ of a locally compact group $G$ on a \cstar algebra $A$, and a closed normal subgroup $N$ of $G$. We prove, following results of Echterhoff for abelian $G$, that Mansfield’s imprimitivity between $A{\times }_{\delta \mathrm{\mid }}G\mathrm{/}N$ and $A{\times }_{\delta }G{\times }_{\text{deltahat},r}N$ implements equivalences between Mansfield induction of representations from $A\times G\mathrm{/}N$ to $A\times G$ and restriction of representations from $A\times G{\times }_{r}N$ to $A\times G$, and between restriction of representations from $A\times G$ to $A\times G\mathrm{/}N$ and Green induction of representations from $A\times G$ to $A\times G{\times }_{r}N$. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.