On characterizations of amenable $$\hbox {C}^*$$-dynamical systems and new examples

Abstract

We show the equivalence of several amenability type conditions for $$\hbox {C}^*$$ C ∗ -dynamical systems. As an application, based on the Pimsner–Meyer construction, we give a new method to produce amenable actions on simple $$\hbox {C}^*$$ C ∗ -algebras.

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

“…(2) We cannot expect an analog of Suzuki's generalization in [51] for W. Indeed, every action of a countable discrete non-amenable exact group on W is not an amenable (or QAP) action because W is monotracial. (See, for example, [41]. )…”

Every countable group admits amenable actions on stably finite simple C*-algebras

“…This is a consequence of the discovery of amenable C * -dynamical systems on simple C * -algebras [21]. The formulation and characterizations of amenability of (noncommutative) C * -dynamical systems had been unclear for a long time, but this problem was recently settled in [17] (see also [3], [6], [22]). Here we only recall the definition in the discrete group case.…”

“…An action α : Γ A of a discrete group Γ on a C * -algebra A is amenable [2] if and only if there exists a Γ-equivariant conditional expectation ℓ ∞ (Γ) ⊗ A * * → 1 ⊗ A * * when ℓ ∞ (Γ) ⊗ A * * is equipped with the diagonal Γ-action of the left translation action and α * * . For useful characterizations and basic properties of amenability of C * -dynamical systems, we refer the reader to [2], [6], [17]. A recent highlight around this subject is the successful classification results [22], [7], [8] of certain amenable actions on purely infinite simple C * -algebras.…”

“…As an application of new characterizations of amenability, Ozawa and the author have established a powerful method to produce amenable actions on simple C *algebras (see [17], Theorem 6.1). However, in all previously known constructions [21], [17] of amenable actions (of non-amenable groups) on simple C * -algebras, the resulting underlying C * -algebras end up being purely infinite.…”

“…As an application of new characterizations of amenability, Ozawa and the author have established a powerful method to produce amenable actions on simple C *algebras (see [17], Theorem 6.1). However, in all previously known constructions [21], [17] of amenable actions (of non-amenable groups) on simple C * -algebras, the resulting underlying C * -algebras end up being purely infinite. Here we recall that the class of classifiable simple C * -algebras (see e.g., the recent survey [23]) splits into two classes: stably finite C * -algebras and purely infinite C * -algebras.…”