Quantum relative modular functions

TL;DR

This paper introduces a quantum relative modular function that measures how a quantum group acts measure-preservingly on a normal subgroup, revealing new structural insights and implications for property-(T) quantum groups.

Contribution

It defines a strictly positive group-like element to quantify measure-preservation failure and establishes a correspondence with quantum group morphisms, advancing the understanding of quantum group modular theory.

Findings

The triviality of the group-like element aligns with the equality of modular elements.

Central subgroups automatically satisfy the triviality condition.

Property-(T) quantum groups do not admit non-trivial positive group-like elements.

Abstract

Let $\mathbb{H}\trianglelefteq\mathbb{G}$ be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with $L^{\infty}(\mathbb{G})$ that, roughly, measures the failure of $\mathbb{G}$ to act measure-preservingly on $\mathbb{H}$ by conjugation. The triviality of that element is equivalent to the condition that $\mathbb{G}$ and $\mathbb{G}/\mathbb{H}$ have the same modular element, by analogy with the classical situation. This condition is automatic if $\mathbb{H}\le \mathbb{G}$ is central, and in general implies the unimodularity of $\mathbb{H}$. 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 easily described in terms of the spectrum of $\delta$. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.