Characterizations of compact and discrete quantum groups through second duals

Details Characterizations of compact and discrete quantum groups through second duals Characterizations of compact and discrete quantum groups through second duals

2005-06-24

arxiv.org/abs/math/0506493v4

J. Operator Theory 60 (2008), 415-428

Mathematics

Operator Algebras

15 pages; LaTeX2e; minor edits

Scientific paper

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 $A^{**}$.

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