Completely bounded maps and invariant subspaces

We provide a description of certain invariance properties of completely bounded bimodule maps in terms of their symbols. If $\mathbb{G}$ is a locally compact quantum group, we characterise the completely bounded $L^{\mathrm{\infty }}(\mathbb{G}{)}^{\mathrm{\prime }}$-bimodule maps that send $C_0(\widehat{\mathbb{G}})$ into $L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$ in terms of the properties of the corresponding elements of the normal Haagerup tensor product $L^{\infty}($\mathbb{G}$) \otimes_{\sigma{\rm h}} L^{\infty}($\mathbb{G}$)$. As a consequence, we obtain an intrinsic characterisation of the normal completely bounded $L^{\mathrm{\infty }}(\mathbb{G}{)}^{\mathrm{\prime }}$-bimodule maps that leave $L^{\mathrm{\infty }}(\widehat{\mathbb{G}})$ invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.