# Journal of Operator Theory

Volume 60, Issue 2, Fall 2008 pp. 415-428.

Characterizations of compact and discrete quantum groups through second duals**Authors**: Volker Runde

**Author institution:**Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

**Summary:**A locally compact group $G$ is compact if and only if $L^1(G)$ is an ideal in $L^1(G)^{\ast\ast}$, and the Fourier algebra $A(G)$ of $G$ is an ideal in $A(G)^{\ast\ast}$ if and only if $G$ is discrete. On the other hand, $G$ is discrete if and only if ${\mathcal C}_0(G)$ is an ideal in ${\mathcal C}_0(G)^{\ast\ast}$. We show that these assertions are special cases of results on locally compact quantum groups in the sense of J.\ Kustermans and S.\Vaes. In particular, a von Neumann algebraic quantum group $(\M,\Gamma)$ is compact if and only if $\M_\ast$ is an ideal in $\M^\ast$, and a (reduced) $\cstar$-algebraic quantum group $(\A,\Gamma)$ is discrete if and only if $\A$ is an ideal in $\A^{\ast\ast}$.

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