A New Characterization of Amenable Groups

A CHARACTERIZATION OF EXTREMELY AMENABLE SEMIGROUPS

Let S be a discrete semigroup, m (S) the space of all bounded real functions on S with the usualsupremum norm. Let Acm (S) be a uniformly closed left invariant subalgebra of m (S) with 1 c A. We say that A is extremely left amenable if there isamultiplicative left invariant meanon A. Let P = {h ?A: h =|g-1,g | forsome g ?A, s ?S}. It isshown that . A is extremely left amenable if and only ...

Amenable Groups

Throughout we let Γ be a discrete group. For f : Γ → C and each s ∈ Γ we define the left translation action by (s.f)(t) = f(s−1t). Definition 1.1. A group Γ is amenable is there exists a state μ on l∞(Γ) which is invariant under the left translation action: for all s ∈ Γ and f ∈ l∞(Γ), μ(s.f) = μ(f). Example 1.2. Finite groups are amenable: take the state which sends χ{s} to 1 |Γ| for each s ∈ ...

Structural Properties of Inner Amenable Discrete Groups

Some extremely amenable groups

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe’s result, we sh...

Are Unitarizable Groups Amenable?

We give a new formulation of some of our recent results on the following problem: if all uniformly bounded representations on a discrete group G are similar to unitary ones, is the group amenable? In §5, we give a new proof of Haagerup's theorem that, on non-commutative free groups, there are Herz-Schur multipliers that are not coefficients of uniformly bounded representations. We actually prov...