Ozawa's class 𝒮 for locally compact groups and unique prime factorization of group von Neumann algebras

We study class 𝒮 for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class 𝒮 and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class 𝒮 is a measure equivalence invariant.

FIXED POINT CHARACTERISATION FOR EXACT AND AMENABLE ACTION

Let $G$ be a finitely generated group acting on a compact Hausdorff space ${\mathcal{X}}$. We give a fixed point characterisation for the action being amenable. As a corollary, we obtain a fixed point characterisation for the exactness of $G$.

Pointwise ergodic theorems beyond amenable groups

We prove pointwise and maximal ergodic theorems for probability-measure-preserving (PMP) actions of any countable group, provided it admits an essentially free, weakly mixing amenable action of stable typeIII$_1$. We show that this class contains all irreducible lattices in connected semi-simple Lie groups without compact factors. We also establish similar results when the stable type isIII$_\lambda $,$0 \lt \lambda \lt 1$, under a suitable hypothesis. Our approach is based on the following two principles. First, we show that it is possible to generalize the ergodic theory of PMP actions of amenable groups to include PMP amenable equivalence relations. Secondly, we show that it is possible to reduce the proof of ergodic theorems for PMP actions of a general group to the proof of ergodic theorems in an associated PMP amenable equivalence relation, provided the group admits an amenable action with the properties stated above.

An example of an amenable action from geometry

Let M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.