Quantum relative modular functions

Let H ⊴ G be a closed normal subgroup of locally compact quantum group. We introduce strictly positive group-like element affiliated with L ∞ ( ) that, roughly, measures the failure to act measure-preservingly on by conjugation. The triviality that is equivalent condition and / have same modular element, analogy classical situation. This automatic if ≤ central, in general implies unimodularity . also describe bijection between elements δ C 0 quantum-group morphisms → R , + image morphism easily described terms spectrum then property-(T) groups admit no non-obvious elements.