Compact Abelian Automorphism Groups of Injective Semi-Finite Factors

Abstract

After the breathtaking breakthrough in the analysis of the structure of injective factors and their automorphism groups by A. Connes during 1975/76, [3,4,5], fine structure analysis of group actions on injective semi-finite factors came into the theory of operator algebras. V. Jones completed a classification of actions of finite groups on an injective II1-factor in his thesis, [13]. A. Ocneanu further supplied an important technical tool, called the stability lemma at infinity, [18], and proved the triviality of the non-commutative second cohomology of amenable group actions on an injective II1-factor. Dualizing Ocneanu’s result, V. Jones classified prime actions of compact abelian groups on an injective II1-factor [14], where an action of an abelian group is said to be prime if it has the entire dual group as its Connes Γ-spectrum. These results can be viewed as a purely non-commutative theory. If we drop the assumption of primeness, then the usual ergodic theory comes into the picture. In these notes, we will present a real blend of commutative and non-commutative ergodic theories based on a recent joint work with V. Jones [15].