  Let $ G $ be a locally compact group. We study the categories of $
L^{\infty}(G) $-comodules and $ L(G) $-comodules in the setting of dual
operator spaces and the associated crossed products. It is proved that every $
L^{\infty}(G) $-comodule is non-degenerate and saturated, whereas every $ L(G)
$-comodule is non-degenerate if and only if every $ L(G) $-comodule is
saturated if and only if $ G $ has the approximation property in the sense of
Haagerup and Kraus [14]. This allows us to extend known results from the
duality theory of crossed products of von Neumann algebras (such as
Takesaki-duality and the Digernes-Takesaki theorem) to the recent theory of
crossed products of dual operator spaces. As applications, we obtain a
characterization of groups with the approximation property in terms of the
related crossed products improving a recent result of Crann and Neufang [9] and
we generalize a theorem of Anoussis, Katavolos and Todorov [2] providing a less
technical proof of it. Furthermore, this approach provides an answer to a
question raised by the authors in [2].
