Amenability and the Extension Property

Amenability and the Liouville Property

We present a new approach to the amenability of groupoids (both in the measure theoretical and the topological setups) based on using Markov operators. We introduce the notion of an invariant Markov operator on a groupoid and show that the Liouville property (absence of non-trivial bounded harmonic functions) for such an operator implies amenability of the groupoid. Moreover, the groupoid actio...

Module-Amenability on Module Extension Banach Algebras

Let $A$ be a Banach algebra and $E$ be a Banach $A$-bimodule then $S = A oplus E$, the $l^1$-direct sum of $A$ and $E$ becomes a module extension Banach algebra when equipped with the algebras product $(a,x).(a^prime,x^prime)= (aa^prime, a.x^prime+ x.a^prime)$. In this paper, we investigate $triangle$-amenability for these Banach algebras and we show that for discrete inverse semigroup $S$ with...

Cyclic amenability of Lau product and module extension Banach algebras

Recently, some results have been obtained on the (approximate) cyclic amenability of Lau product of two Banach algebras. In this paper, by characterizing of cyclic derivations on Lau product and module extension Banach algebras, we present general necessary and sufficient conditions for those to be (approximate) cyclic amenable. This not only provides new results on (approximate) cyclic amenabi...

Relative property A and relative amenability for countable groups

We define a relative property A for a countable group with respect to a finite family of subgroups. Many characterizations for relative property A are given. In particular a relative bounded cohomological characterization shows that if G has property A relative to a family of subgroups H and if each H ∈ H has property A, then G has property A. This result leads to new classes of groups that hav...

Property a as Metric Amenability and Its Applications to Geometry