On locally compact quantum groups whose algebras are factors

In this paper we are interested in examples of locally compact quantum groups (M,Δ) such that both von Neumann algebras, M and the dual , are factors. There is a lot of known examples such that are respectively of type (I∞,I∞) but there is no example with factors of other types. We construct new examples of type (I∞,II∞), (II∞,II∞) and (IIIλ,IIIλ) for each λ∈[0,1]. We also show that there is no such example with M or a finite factor.