Amenable Extensions in II1 Factors

Abstract

Amenability is a fundamental in operator algebras. The classification of von Neumann algebras by Alain Connes is a milestone in the theory. The study of amenable subalgebras in II1 factors has led to many important developments such as the computation of the fundamental groups, strong solidity of free group factors, etc. In this thesis we consider a question about amenable extension in II1 factors, namely, given a diffuse amenable subalgebra in a II1 factor, in how many ways it can be extended to some maximal amenable subalgebra? We give two classes of examples where unique amenable extension results are obtained. The key notion we use is a strengthening of Popa’s asymptotic orthogonality property.