Journal of Operator Theory

Volume 60, Issue 2, Fall 2008 pp. 415-428.

Characterizations of compact and discrete quantum groups through second duals

Authors: Volker Runde
Author institution: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Summary: 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

${\mathcal C}_0(G)$ is an ideal in

${\mathcal C}_0(G)^{\ast\ast}$ . 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)

$\cstar$ -algebraic quantum group

$(\A,\Gamma)$ is discrete if and only if

$\A$ is an ideal in

$\A^{\ast\ast}$ .

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