# zbMATH — the first resource for mathematics

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
Full Text: