On the Question of the Thompson Group

This survey is concerned with the Thompson group F and the von Neumann algebra L(F) associated with F. We first review some basic results on the Thompson groups and group von Neumann algebras. We investigate the structure of the von Neumann algebra W*(F,P) generated by L(F) and a projection P on l2(F). We show that the algebra (not necessarily *) algebraically generated by two generating unitaries of L(F) and the commutant L(F)′ is strong-operator dense in {\\mathcal B}({\\mathcal H}). Furthermore, we will discuss actions of F and F′ on a set of dyadic rational numbers in [0,1] and some outer automorphism of L(F) for the amenability question of F.