Characterizations of compact and discrete quantum groups through second duals

2005/06/30 by Volker Runde · 1 citation
Mathematics · #math.OA #math.FA #msc:46L89 #msc:22C05 #msc:22D35 #msc:43A99 #msc:46H10 #msc:46L51 #msc:46L65 #msc:47L50 #msc:81R15 #msc:81R50

paper · pdf

published as J. Operator Theory 60 (2008), 415-428 · 15 pages; LaTeX2e; minor edits

arxiv created 2006/06/30 · arxiv updated 2009/12/01

Abstract

A locally compact group G is compact if and only if L¹(G) is an ideal in L¹(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 C₀(G) is an ideal in C₀(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,Γ) is compact if and only if M_* is an ideal in M^*, and a (reduced) C^*-algebraic quantum group (A,Γ) is discrete if and only if A is an ideal in A^**.

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• Amenability of locally compact quantum groups via the long-time behavior of convolution semigroups