A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

Let $G$ be a locally compact quantum group and $(M,\alpha )$ a $G$-$W^{⁎}$-algebra. The object of study of this paper is the $W^{⁎}$-category ${\mathrm{Rep}}^G\left(M\right)$ of normal, unital $G$-representations of M on Hilbert spaces endowed with a unitary $G$-representation. This category has a right action of the category $\mathrm{Rep}\left(G\right)={\mathrm{Rep}}^G\left(C\right)$ for which it becomes a right $\mathrm{Rep}\left(G\right)$-module $W^{⁎}$-category. Given another $G$-$W^{⁎}$-algebra $(N,\beta )$, we denote the category of normal ⁎-functors ${\mathrm{Rep}}^G\left(N\right)\to {\mathrm{Rep}}^G\left(M\right)$ compatible with the $\mathrm{Rep}\left(G\right)$-module structure by ${\mathrm{Fun}}_{\mathrm{Rep}\left(G\right)}\left({\mathrm{Rep}}^G\right(N),{\mathrm{Rep}}^G(M\left)\right)$ and we denote the category of $G$-M-N-correspondences, studied in [5], by ${\mathrm{Corr}}^G(M,N)$. We prove that there are canonical functors $P:{\mathrm{Corr}}^G(M,N)\to {\mathrm{Fun}}_{\mathrm{Rep}\left(G\right)}\left({\mathrm{Rep}}^G\right(N),{\mathrm{Rep}}^G(M\left)\right)$ and $Q:{\mathrm{Fun}}_{\mathrm{Rep}\left(G\right)}\left({\mathrm{Rep}}^G\right(N),{\mathrm{Rep}}^G(M\left)\right)\to {\mathrm{Corr}}^G(M,N)$ such that $Q\circ P\cong \mathrm{id}$. We use these functors to show that the $G$-dynamical von Neumann algebras $(M,\alpha )$ and $(N,\beta )$ are equivariantly Morita equivalent if and only if ${\mathrm{Rep}}^G\left(N\right)$ and ${\mathrm{Rep}}^G\left(M\right)$ are equivalent as $\mathrm{Rep}\left(G\right)$-module-$W^{⁎}$-categories. Specializing to the case where $G$ is a compact quantum group, we prove that moreover $P\circ Q\cong \mathrm{id}$, so that the categories ${\mathrm{Corr}}^G(M,N)$ and ${\mathrm{Fun}}_{\mathrm{Rep}\left(G\right)}\left({\mathrm{Rep}}^G\right(N),{\mathrm{Rep}}^G(M\left)\right)$ are equivalent. This is an equivariant version of the Eilenberg-Watts theorem for actions of compact quantum groups on von Neumann algebras.