2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/210153A 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 $(M,Γ)$ is compact if and only if $M_*$ is an ideal in $M^*$, and a (reduced) $C^*$-algebraic quantum group $(A,Γ)$ is discrete if and only if $A$ is an ideal in $A^{**}$.15 pages; LaTeX2e; minor editsOperator AlgebrasFunctional AnalysisPrimary 46L89; Secondary 22C05, 22D35, 43A99, 46H10, 46L51, 46L65, 47L50, 81R15, 81R50text