A locally compact group G is amenable if and only if it has Reiter’s property (Pp) for p = 1 or, equivalently, all p ∈ [1,∞), i.e., there is a net (mα)α of non-negative norm one functions in L(G) such that limα supx∈K ‖Lx−1mα − mα‖p = 0 for each compact subset K ⊂ G (Lx−1mα stands for the left translate of mα by x ). We extend the definitions of properties (P1) and (P2) from locally compact gro...

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...

Abstract Amenability is a notion of facial exposedness for convex cones that stronger than being facially dual complete (or ‘nice’) which is, in turn, merely exposed. Hyperbolicity are family algebraically structured closed contain all spectrahedral (linear sections positive semidefinite cones) as special cases. It known amenable. We establish hyperbolicity As part the argument, we show any fac...