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 $G$.
Categories: math.OA math.FA