Amenability — From the Banach-Tarski paradoxon to Gromov’s monster groups
– Campus Golm, Haus 9, Raum 2.22 + Zoom
Kristin Courtney (Uni Münster) und Siegfried Echterhoff (Uni Münster)
14:00 Kristin Courtney (Uni Münster): Approximations on Groups
14:45 Tee und Kaffee Pause
15:15 Siegfried Echterhoff (Uni Münster): Amenable group actions on spaces and operator algebras
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Kristin Courtney (Uni Münster): Approximations on Groups
The Banach-Tarski paradox says that a solid ball in three dimensional space can be decomposed into finitely many disjoint pieces which can then be reassembled to form two identical copies of the original ball. At first glance, this paradox seems to contradict the law of conservation of mass, when, in reality, it is reflecting the non-amenability of the free group which is acting on the sphere. Amenability is exactly the property for groups which prevents such paradoxical actions. This property admits numerous different characterizations ranging from analytic to geometric to combinatorial in flavor. I will introduce the so-called group C*-algebras, and in that context, describe amenability as a finite dimensional approximation property, known as nuclearity. Nuclearity is not the only finite dimensional approximation property inspired by and closely related to a group theoretical analogue, and the interplay of such properties leads to deep insights in the study of groups and operators.
Siegfried Echterhoff (Uni Münster): Amenable group actions on spaces and operator algebras
The concepts of amenable group actions are direct generalizations of the concept of an amenable group as discussed in the lecture by Kristin Courtney. Amenable actions on measure spaces were introduced by Zimmer in the 1980’s in order to study rigidity properties of lattices in Lie groups. Various notions of amenability were later introduced for actions on topological spaces and on operator algebras, where the latter can be viewed as non-commutative analogues of measure or topological spaces. These amenable actions have important applications in the interplay between group theory and operator algebras. After a gentle introduction to various notions of amenable actions we plan to give an overview over some exciting recent results.