Inner Amenability, Cost and Central Sequences in Factors

Specialeforsvar ved Sanne Winther 

Titel: Inner Amenability, Cost and Central Sequences in Factors

Abstract: A group $G$ is said to be inner amenable if the conjugation action of $G$ on $G \setminus \{e\}$ is amenable. In this thesis, we investigate the notions of inner amenability and strong inner amenability of a group (whereas the latter refers to the definition of inner amenability introduced by Tucker-Drob). We show that both amenability and strong inner amenability of a group implies inner amenability, even though none of the two implies the other. We give examples of groups which are inner amenable, but not strongly inner amenable, as well as groups which are inner amenable, but not amenable. We show that inner amenability and strong inner amenability are the same if the considered group is ICC, and we investigate the permanence properties of both inner amenable and strongly inner amenable groups. We use Tarski's Theorem to give another condition equivalent to a group being inner amenable, and we present the result by Effros that an ICC group, whose group von Neumann algebra has property $\Gamma$, is inner amenable. We prove that countable ICC groups with property (T) are not inner amenable. In Chapter 6 we summarize the previously obtained results in three theorems showing equivalent, sufficient and necessary conditions for inner amenability. We introduce the notion of cost very briefly and state the result by Tucker-Drob that strongly inner amenable groups have cost equal to 1

Vejleder: Magdalene Musat
Censor:  Agata Przybyszewska, SDU