On locally compact quantum groups whose algebras are factors

Abstract

In this paper we are interested in examples of locally compact quantum groups $(M,\Delta )$ such that both von Neumann algebras, $M$ and the dual $\hat{M}$, are factors. There is a lot of known examples such that $(M,\hat{M})$ are respectively of type $(\mathrm{I}\_\infty ,\mathrm{I}\_\infty )$ but there is no examples with factors of other types. We construct new examples of type $(\mathrm{I}\_\infty ,\mathrm{II}\_\infty )$, $(\mathrm{II}\_\infty ,\mathrm{II}\_\infty )$ and $(\mathrm{III}\_\lambda ,\mathrm{III}\_\lambda )$ for each $\lambda \in [0,1]$. Also we show that there is no such example with $M$ or $\hat{M}$ a finite factor.

20 pages