Compact Operators in Regular LCQ Groups

TL;DR

This paper characterizes when a regular locally compact quantum group is discrete by examining the presence of non-zero compact operators in its associated von Neumann algebra, providing a classification criterion.

Contribution

It establishes a new characterization of discreteness in regular LCQ groups via compact operators, linking operator theory with quantum group structure.

Findings

Discreteness of regular LCQ groups is equivalent to the existence of non-zero compact operators in $L^ Infty(G)$.

Classifies discrete quantum groups among regular LCQ groups based on the Radon--Nikodym property of $L^1(G)$.

Provides a new operator-theoretic criterion for identifying discrete quantum groups.

Abstract

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $L^\infty(\mathbb{G})$ contains non-zero compact operators on $L^2(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $L^1(\mathbb{G})$ has the Radon--Nikodym property.