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 Mˆ , are factors. There is a lot of known examples such that (M,Mˆ ) 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 Mˆ a finite factor. © 2006 Elsevier Inc. All rights reserved.