Amenability

Erwin Schrödinger Institute, Vienna

Amenability

Special semester, February - July 2007

Amenability beyond groups

February 26 - March 17, 2007

Week 1 (February 26 - March 2)

VADIM KAIMANOVICH (Bremen): Amenability of algebraic structures from groups to groupoids

Abstract: This introductory course is aimed at giving general background concerning the notion of amenability. It will begin with the classic definition of amenable groups due to von Neumann and then talk about equivalent definitions (fixed point property, Følner sets, Reiter condition) as well as their generalizations to other algebraic structures (group actions, equivalence relations, pseudogroups and, finally, groupoids).

Schedule of lectures for Week 1

Week 2 (March 5 - March 9)

GABOR ELEK (Budapest): Amenability in Ring Theory

Abstract: Amenability of algebras over a given field can be defined the same fashion as amenability of discrete groups (Mikhael Gromov, Topological Invariants of Dynamical Systems and Spaces of Holomorphic Maps I, Math Phys. Anal. Geom 2 (1999) 323-415). These lectures review the basic definitions and survey some recent results: - Amenable affine algebras; - Amenable skewfields; - Amenable group algebras; - The rank function; - von Neumann's continuous rank algebras and their applications.

ALAIN VALETTE (Neuchatel): Affine isometric actions on Hilbert spaces and amenability

Abstract: We will define affine isometric actions on Hilbert spaces (together with the relevant mild cohomological formalism) and give examples from geometry. We plan to give a proof of the following results: - a group is non-amenable if and only if, whenever an action with linear part the regular representation almost has fixed points, it has fixed points (Guichardet); - every amenable group admits a proper action on a Hilbert space; - if an amenable group in Shalom's class (H_FD) (which contains polycyclic groups) embeds quasi-isometrically into Hilbert space, then it is virtually abelian.

Schedule of lectures for Week 2

Week 3 (March 12 - March 16)

CLAIRE ANANTHARAMAN-DELAROCHE (Orleans): Amenability and exactness for group actions and operator algebras

Abstract: In these lectures we shall introduce the operator algebras associated with groups and group actions. Then we shall study the interactions between amenability properties of groups and group actions and amenability properties of the corresponding operator algebras. We shall focus our attention on boundary amenability and exactness. We shall also discuss the relations between exactness and uniform embeddability in Hilbert spaces and consider some weak form of amenability, intermediate between exactness and usual amenability. With time, we shall survey some specific amenable actions, for instance locally contracting ones (e.g. action of Fuchsian groups on the circle).

MASAKI IZUMI (Kyoto): Non-commutative Poisson boundaries

Abstract: It is well-known that the structure of the Poisson boundary of a group is very much related to amenability (or non-amenability) of the group. In recent works, a non-commutative version of the Poisson boundary, defined for von Neumann algebras, turns out to be very useful in operator algebras. I give a survey of this development and also mention its relationship to amenability of subfactors and quantum groups.

Schedule of lectures for Week 3