Mapping ideals of quantum group multipliers

Abstract

We study the dual relationship between quantum group convolution maps L1(G)→L∞(G) and completely bounded multipliers of Gˆ. For a large class of locally compact quantum groups G, we completely isomorphically identify the mapping ideal of row Hilbert space factorizable convolution maps with Mcb(L1(Gˆ)), 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 coinciding with ℓ1(bGˆ), where bG is the quantum Bohr compactification of G. For quantum groups whose dual has bounded degree, we show that the completely compact convolution maps coincide with C(bG). 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 inner co-amenable quantum group, that we introduce and for which we 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.