Published online by Cambridge University Press: 20 November 2018
We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if
${{\mathcal{L}}^{\infty }}\left( \mathbb{G} \right)$ contains non-zero compact operators on
${{\mathcal{L}}^{2}}\left( \mathbb{G} \right)$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups
$\mathbb{G}$ where
${{\mathcal{L}}^{1}}\left( \mathbb{G} \right)$ has the Radon-Nikodym property.