Amenable Pairs of Groups and Ergodic Actions and the Associated von Neumann Algebras

Abstract

Abstract. If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair (X, Y). This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.

Spectral rigidity of group actions on homogeneous spaces

“…There were intensively studied by Eymard in the case of the action of G on a homogeneous space X = G/H in [Eyma72], where such an action (or space) is called amenable. We prefer to call them co-amenable in order to avoid confusion with the well-established notion of an amenable action G X due to Zimmer ([Zimm84]), which in the case X = G/H corresponds to the amenability of H; a unification of both notions by means of an appropriate definition of amenable actions on pairs of measure spaces is given in [Zimm78]. For a further extension of these notions to the context on non-commutative measure spaces (that is, to the context of von Neumann algebras), see [Anan08].…”

On the comparison of norms of convolutors associated with noncommutative dynamics

“…When the conditional expectation is not required to be normal, we are led to the following definition, due to Zimmer [45] for pairs of abelian von Neumann algebras. Let us also recall the definitions of the two following important particular cases.…”

“…When the conditional expectation is not required to be normal, we are led to the following definition, due to Zimmer [45] for pairs of abelian von Neumann algebras.…”

Proper cocycles and weak forms of amenability

“…Definition 1.2. ( [6], [8], [11]) Let G be a locally compact, second countable group that acts in a measure class preserving way on the standard space (X, µ). Then we say that (G, X) is an amenable pair if L ∞ (X, µ) has a G-invariant state.…”