Characterizations of compact and discrete quantum groups through second duals

TL;DR

This paper extends classical characterizations of compact and discrete groups to the setting of locally compact quantum groups, showing that similar ideal properties of duals characterize these quantum group types.

Contribution

It generalizes classical group characterizations to quantum groups, establishing conditions on duals that characterize compactness and discreteness.

Findings

Quantum group $M$ is compact iff $M_*$ is an ideal in $M^*$

Quantum group $A$ is discrete iff $A$ is an ideal in $A^{**}$

Classical group characterizations are special cases of these quantum results

Abstract

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 $A^{**}$.