Completely bounded maps and invariant subspaces

TL;DR

This paper characterizes certain invariant properties of completely bounded bimodule maps on quantum groups using their symbols, extending known results to a more general setting with a unified approach.

Contribution

It provides a unified intrinsic characterization of invariant completely bounded bimodule maps on locally compact quantum groups, extending previous separate results.

Findings

Characterization of bimodule maps via their symbols.

Extension of results to non-commutative quantum groups.

Unification of commutative and co-commutative cases.

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 $L^{\infty}(\mathbb{G}) \otimes_{\sigma{\rm h}} L^{\infty}(\mathbb{G})$. 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.