Amenable actions of discrete groups

Amenable Actions and Exactness for Discrete Groups

It is proved that a discrete group G is exact if and only if its left translation action on the Stone-Čech compactification is amenable. Combining this with an unpublished result of Gromov, we have the existence of non exact discrete groups. In [KW], Kirchberg and Wassermann discussed exactness for groups. A discrete group G is said to be exact if its reduced group C-algebra C λ(G) is exact. Th...

Structural Properties of Inner Amenable Discrete Groups

Dual Entropy in Discrete Groups with Amenable Actions

Let G be a discrete group which admits an amenable action on a compact space and γ ∈ Aut(G) be an automorphism. We define a notion of entropy for γ and denote the invariant by ha(γ). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(γ) = hTop(γ̂) where hTop(γ̂) denotes the topological entropy of the induced automorphism γ̂ of the (compact, abelian) dual...

Amenable Actions of Nonamenable Groups

Since 1929 when von Neumann [vN29] introduced the notion of an invariant mean on a group (and more generally on a G-set) there is a permanent interest in the study of the phenomenon known as amenability. Amenable objects like groups, semigroups, algebras, graphs, metric spaces, operator algebras etc. play an important role in different areas of mathematics. A big progress in understanding of th...

Classification of Strongly Free Actions of Discrete Amenable Groups on Strongly Amenable Subfactors of Type Iii0

In the theory of operator algebras, classification of group actions on approximately finite dimensional (AFD) factors has been done since Connes’s work [2]. In subfactor theory, various results on classification of group actions have been obtained. The most powerful results have been obtained by Popa in [16], who classified the strongly outer actions of discrete amenable groups on strongly amen...