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On locally compact quantum groups whose algebras are factors. (English) Zbl 1121.46056
Summary: We are interested in examples of locally compact quantum groups \((M,\Delta)\) such that both von Neumann algebras, \(M\) and the dual \(\widetilde M\), are factors. There are a lot of known examples such that \((M,\widehat M)\) are respectively of type \((\text{I}_\infty,\text{I}_\infty)\) but there is no example with factors of other types. We construct new examples of type \((\text{I}_\infty,\text{II}_\infty)\), \((\text{II}_\infty,\text{II}_\infty)\) and \((\text{III}_\lambda,\text{III}_\lambda)\) for each \(\lambda\in[0,1]\). We also show that there is no such example with \(M\) or \(\widehat M\) a finite factor.

MSC:
46L65 Quantizations, deformations for selfadjoint operator algebras
46L35 Classifications of \(C^*\)-algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
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