Quantum relative modular functions

Let H⊴G be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with L^∞(G) that, roughly, measures the failure of G to act measure-preservingly on H by conjugation. The triviality of that element is equivalent to the condition that G and G/H have the same modular element, by analogy with the classical situation. This condition is automatic if H≤G is central, and in general implies the unimodularity of H. We also describe a bijection between strictly positive group-like elements δ affiliated with C₀(G) and quantum-group morphisms G→(R,+), with the closed image of the morphism easily described in terms of the spectrum of δ. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.