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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
##### Keywords:
quantum groups; factors; crossed product
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##### References:
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