  A locally compact group $G$ is compact if and only if $L^1(G)$ is an ideal in
$L^1(G)^{**}$, and the Fourier algebra $A(G)$ of $G$ is an ideal in $A(G)^{**}$
if and only if $G$ is discrete. On the other hand, $G$ is discrete if and only
if $C_0(G)$ is an ideal in $C_0(G)^{**}$. 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 $M_*$ is an ideal in $M^*$, and a
(reduced) $C^*$-algebraic quantum group $(A,\Gamma)$ is discrete if and only if
$A$ is an ideal in $A^{**}$.
