Completely bounded maps and invariant subspaces

  • M. Alaghmandan
  • I. G. TodorovEmail author
  • L. Turowska


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.

Mathematics Subject Classification

Primary: 46L89 Secondary: 22D15 47L25 



We are grateful to Jason Crann for a number of fruitful conversations on the topic of this paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCarleton UniversityOttawaCanada
  2. 2.Mathematical Sciences Research CentreQueen’s University BelfastBelfastUK
  3. 3.School of Mathematical SciencesNankai UniversityTianjinChina
  4. 4.Department of Mathematical SciencesChalmers University of Technology and The University of GothenburgGothenburgSweden

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