Abstract
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.
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Acknowledgements
We are grateful to Jason Crann for a number of fruitful conversations on the topic of this paper.
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Alaghmandan, M., Todorov, I.G. & Turowska, L. Completely bounded maps and invariant subspaces. Math. Z. 294, 471–489 (2020). https://doi.org/10.1007/s00209-019-02255-3
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DOI: https://doi.org/10.1007/s00209-019-02255-3