Characterizations of compact and discrete quantum groups through second duals

Abstract

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 $C_0(G)$ is an ideal in $C_0(G{)}^{\ast \ast }$. 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) $C^{\ast }$-algebraic quantum group $(A,\Gamma )$ is discrete if and only if $A$ is an ideal in $A^{\ast \ast }$.

15 pages; LaTeX2e; minor edits

References in corpus (1)

• Locally Compact Quantum Groups. A von Neumann Algebra Approach

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