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# Compact Operators in Regular LCQ Groups

Published:2013-03-20

School of Mathematics and Statistics, Carleton University, Ottawa, ON
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## Abstract

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
 MSC Classifications: 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]