Crossed products of dual operator spaces by locally compact groups

For an action $\alpha$ of a locally compact group $G$ on a dual operator space $X$ by w*-continuous completely isometric isomorphisms one can define two generally different notions of crossed products, namely the Fubini crossed product $X\rtimes_{\alpha}^{F}G$ and the spatial crossed product $X\overline{\rtimes}_{\alpha} G$. It is shown that $X\rtimes_{\alpha}^{F}G=X\overline{\rtimes}_{\alpha} G$ if and only if the dual comodule action $\widehat{\alpha}$ of the group von Neumann algebra $L(G)$ on the Fubini crossed product of $X\rtimes_{\alpha}^{F}G$ is non-degenerate. As an application, this yields an alternative proof of the result of Crann and Neufang that the two notions coincide when G satisfies the approximation property (AP) of Haagerup and Kraus. Also, it is proved that the $L(G)$-bimodules $Bim(J^{\perp})$ and $Ran(J)^{\perp}$ defined by Anoussis, Katavolos and Todorov for a left closed ideal J of $L^{1}(G)$ can be identified respectively with a spatial crossed product and a Fubini crossed product of the annihilator of $J$ by $G$. Therefore a necessary and sufficient condition so that $Bim(J^{\perp})=Ran(J)^{\perp}$ is obtained by the main result.