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

Keywords:

locally compact quantum groups, regularity, compact operators locally compact quantum groups, regularity, compact operators

MSC Classifications:

46L89 show english descriptions Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22] 46L89 - Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]

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