Amenability in the Theory of Subfactors

We will discuss the motivating background leading to the consideration of the concept of amenability for von Neumann subfactors of nite Jones index and respectively for the combinatorial objects associated with them (standard graphs, standard lattices, paragroups, etc). Then we will review some of the results related to these notions of amenability, outlining the papers 20]{{25] and the lectures on this subject that we gave during 1989-1996. Thus, a number of equivalent characterizations of amenability for subfactors and their graphs are presented, as well as the classiication results that hold true for this class of subfactors. Finally, we will digress on a notion of property T that can be introduced for subfactors and for their combinatorial invariants ((24]) by using one of the tools coming from the study of amenability. For most of the notations and deenitions used in this paper we refer to 20],,21]. In general, the unexplained terms or notations are standard or can be found there. For the proofs, see 20]{{25].