Mapping ideals of quantum group multipliers

TL;DR

This paper explores the duality between quantum group convolution maps and multipliers, identifying various mapping ideals and their quantum analogs using advanced operator space and harmonic analysis techniques.

Contribution

It introduces a comprehensive framework linking convolution maps with completely bounded multipliers for a broad class of quantum groups, including new results for group von Neumann algebras.

Findings

Identification of mapping ideals with quantum multipliers

Coincidence of completely nuclear and -multiplier spaces with quantum Bohr compactification

Equivalence of completely compact convolution maps with continuous functions on the quantum Bohr compactification

Abstract

We study the dual relationship between quantum group convolution maps $L^1(\mathbb{G})\rightarrow L^{\infty}(\mathbb{G})$ and completely bounded multipliers of $\widehat{\mathbb{G}}$. For a large class of locally compact quantum groups $\mathbb{G}$ we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with $M_{cb}(L^1(\widehat{\mathbb{G}}))$, yielding a quantum Gilbert representation for completely bounded multipliers. We also identify the mapping ideals of completely integral and completely nuclear convolution maps, the latter case coinciding with $\ell^1(\widehat{b\mathbb{G}})$, where $b\mathbb{G}$ is the quantum Bohr compactification of $\mathbb{G}$. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with $C(b\mathbb{G})$. Our techniques comprise a mixture of operator space theory and abstract harmonic analysis, including Fubini tensor products, the non-commutative Grothendieck inequality, quantum Eberlein compactifications, and a suitable notion of quasi-SIN quantum group, which we introduce and exhibit examples from the bicrossed product construction. Our main results are new even in the setting of group von Neumann algebras $VN(G)$ for quasi-SIN locally compact groups $G$.