INNER AMENABLE GROUPOIDS AND CENTRAL SEQUENCES

We introduce inner amenability for discrete probability-measure-preserving (p.m.p.) groupoids and investigate its basic properties, examples, and the connection with central sequences in the full group of the groupoid or central sequences in the von Neumann algebra associated with the groupoid. Among other things, we show that every free ergodic p.m.p. compact action of an inner amenable group gives rise to an inner amenable orbit equivalence relation. We also obtain an analogous result for compact extensions of equivalence relations that either are stable or have a nontrivial central sequence in their full group.

On the amenability of a class of Banach algebras with application to measure algebra

Let 𝓛 be a Lau algebra and X be a topologically invariant subspace of 𝓛* containing UC(𝓛). We prove that if 𝓛 has a bounded approximate identity, then strict inner amenability of 𝓛 is equivalent to the existence of a strictly inner invariant mean on X. We also show that when 𝓛 is inner amenable the cardinality of the set of topologically left invariant means on 𝓛* is equal to the cardinality of the set of topologically left invariant means on RUC(𝓛). Applying this result, we prove that if 𝓛 is inner amenable and 〈𝓛2〉 = 𝓛, then the essential left amenability of 𝓛 is equivalent to the left amenability of 𝓛. Finally, for a locally compact group G, we consider the measure algebra M(G) to study strict inner amenability of M(G) and its relation with inner amenability of G.

Inner amenability, property Gamma, McDuff factors and stable equivalence relations

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

Relative Inner Amenability and Relative Property Gamma

Let $H$ be a proper subgroup of a discrete group $G$. We introduce a notion of relative inner amenability of $H$ in $G$, we prove some equivalent conditions and provide examples coming mainly from semidirect products, as well as counter-examples. We also discuss the corresponding relative property gamma for pairs of type II$_1$ factors $N\subset M$ and we deduce from this a characterization of discrete, icc groups which do not have property (T).