Compact Operators in Regular LCQ 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.