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 G is a locally compact quantum group, we characterise the completely bounded L infinity(G)'-bimodule maps that send C0(G) in terms of the properties of the corresponding elements of the normal Haagerup tensor product L infinity(G)circle times sigma hL infinity(G). As a consequence, we obtain an intrinsic characterisation of the normal completely bounded L infinity(G)'-bimodule maps that leave L infinity invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.