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 \(L^{\infty }(\mathbb {G}) \otimes _{\sigma \mathop {\mathrm{h}}} L^{\infty }(\mathbb {G})\). 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.