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Characterizations of compact and discrete quantum groups through second duals. (English) Zbl 1164.22001
Summary: 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 \(\mathcal C_0(G)\) is an ideal in \(\mathcal C_0(G)^{**}\). 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 \((M, \varGamma)\) is compact if and only if \(M_*\) is an ideal in \(M^*\), and a (reduced) \(C^*\)-algebraic quantum group \((A, \varGamma)\) is discrete if and only if \(A\) is an ideal in \(A^{**}\).

MSC:
22C05 Compact groups
22D35 Duality theorems for locally compact groups
43A99 Abstract harmonic analysis
46H10 Ideals and subalgebras
46L51 Noncommutative measure and integration
46L65 Quantizations, deformations for selfadjoint operator algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
47L50 Dual spaces of operator algebras
81R15 Operator algebra methods applied to problems in quantum theory
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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Full Text: arXiv