Completely bounded maps and invariant subspaces

Mahmood Alaghmandan; Ivan Todorov; Lyudmila Turowska

doi:10.1007/s00209-019-02255-3

Completely bounded maps and invariant subspaces

Mahmood Alaghmandan, Ivan Todorov, Lyudmila Turowska

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Abstract

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 ∞(G) 0 -bimodule maps that send C0(Gˆ ) into L ∞(Gˆ ) in terms of the properties of the corresponding elements of the normal Haagerup tensor product L ∞(G) ⊗σ h L ∞(G). As a consequence, we obtain an intrinsic characterisation of the normal completely bounded L ∞(G) 0 -bimodule maps that leave L ∞(Gˆ ) invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.

Original language	English
Number of pages	19
Journal	Mathematische Zeitschrift
Early online date	12 Feb 2019
DOIs	https://doi.org/10.1007/s00209-019-02255-3
Publication status	Early online date - 12 Feb 2019

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Completely bounded maps and invariant subspaces
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AB - 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 ∞(G) 0 -bimodule maps that send C0(Gˆ ) into L ∞(Gˆ ) in terms of the properties of the corresponding elements of the normal Haagerup tensor product L ∞(G) ⊗σ h L ∞(G). As a consequence, we obtain an intrinsic characterisation of the normal completely bounded L ∞(G) 0 -bimodule maps that leave L ∞(Gˆ ) invariant, extending and unifying results, formulated in the current literature separately for the commutative and the co-commutative cases.

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