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Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of alocally compact quantum group. We introduce a strictly positive group-likeelement affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures thefailure of $\mathbb{G}$ to act measure-preservingly on $\mathbb{H}$ byconjugation. The triviality of that element is equivalent to the condition that$\mathbb{G}$ and $\mathbb{G}/\mathbb{H}$ have the same modular element, byanalogy with the classical situation. This condition is automatic if$\mathbb{H}\le \mathbb{G}$ is central, and in general implies the unimodularityof $\mathbb{H}$.  We also describe a bijection between strictly positive group-like elements$\delta$ affiliated with $C_0(\mathbb{G})$ and quantum-group morphisms$\mathbb{G}\to (\mathbb{R},+)$, with the closed image of the morphism easilydescribed in terms of the spectrum of $\delta$. This then implies thatproperty-(T) locally compact quantum groups admit no non-obvious strictlypositive group-like elements.

Quantum relative modular functions