Mapping ideals of quantum group multipliers

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 $ell^1(widehat{bmathbb{G}})$, where $bmathbb{G}$ 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 $C(bmathbb{G})$. 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 $G$.