Mansfield’s imprimitivity theorem for full crossed products

For any maximal coaction $(A,G,\delta )$ and any closed normal subgroup $N$ of $G$, there exists an imprimitivity bimodule $Y_{G/N}^G(A)$ between the full crossed product $A\times _\delta G\times _{\widehat \delta |}N$ and $A\times _{\delta |}G/N$, together with $\operatorname {Inf}\widehat {\widehat \delta |}-\delta ^{\text {dec}}$ compatible coaction $\delta _Y$ of $G$. The assignment $(A,\delta )\mapsto (Y_{G/N}^G(A),\delta _Y)$ implements a natural equivalence between the crossed-product functors “${}\times G\times N$” and “${}\times G/N$”, in the category whose objects are maximal coactions of $G$ and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of $G$.