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Given a locally compact quantum group   $\mathbb{G}$   and two   $\mathbb{G}$  -  $W^{*}$  -algebras   $\alpha : A\curvearrowleft \mathbb{G}$   and   $\beta : B\curvearrowleft \mathbb{G}$  , we study the notion of equivariant   $W^{*}$  -Morita equivalence   $(A, \alpha )\sim _{\mathbb{G}} (B, \beta )$  , which is an equivariant version of Rieffel’s notion of   $W^{*}$  -Morita equivalence. We prove that important dynamical properties of   $\mathbb{G}$  -  $W^{*}$  -algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra   $L^\infty (\mathbb{K}\backslash \mathbb{G})\subseteq L^\infty (\mathbb{G})$   with dual coideal von Neumann algebra   $L^\infty (\check{\mathbb{K}})\subseteq L^\infty (\check{\mathbb{G}})$  , we use a natural   $\check{\mathbb{G}}$  -  $W^{*}$  -Morita equivalence   $L^\infty (\mathbb{K}\backslash \mathbb{G})\rtimes _\Delta \mathbb{G} \sim _{\check{\mathbb{G}}} L^\infty (\check{\mathbb{K}})$   to relate dynamical properties of   $L^\infty (\mathbb{K}\backslash \mathbb{G})$   with dynamical properties of   $L^\infty (\check{\mathbb{K}})$  . We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski, and Vergnioux, namely that for   $\mathbb{H}$   a closed quantum subgroup of the compact quantum group   $\mathbb{G}$  , coamenability of   $\mathbb{H}\backslash \mathbb{G}$   and relative amenability of   $\ell ^\infty (\check{\mathbb{H}})$   in   $\ell ^\infty (\check{\mathbb{G}})$   are equivalent. Moreover, if   $\mathbb{G}$   is compact, we study the relation between   $\mathbb{G}$  -  $W^{*}$  -Morita equivalence of   $(A, \alpha )$   and   $(B, \beta )$   and   $\mathbb{G}$  -  $C^{*}$  -Morita equivalence of the associated   $\mathbb{G}$  -  $C^{*}$  -algebras   $(\mathcal{R}(A), \alpha )$   and   $(\mathcal{R}(B), \beta )$   of regular elements.

Morita Theory for Dynamical von Neumann Algebras