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 $L^{infty}(mathbb{G})$-bimodule maps that send $C_0(hat{mathbb{G}})$ into $L^{infty}(hat{mathbb{G}})$ 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 $L^{infty}(mathbb{G})$-bimodule maps that leave $L^{infty}(hat{mathbb{G}})$ invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.