We extend the concept of amenability of a Banach algebra A to the case that there is an extra A -module structure on A, and show that when S is an inverse semigroup with subsemigroup E of idempotents, then A = l(S) as a Banach module over A= l(E) is module amenable iff S is amenable. When S is a discrete group, l(E) = C and this is just the celebrated Johnson’s theorem.

We formulate two new criteria of amenability of discrete equivalence relations: in terms of asymptotically invariant families of leafwise probability measures and in terms of isoperimetric properties of leafwise graph structures. These criteria lead to a geometric proof of the Connes{Feldman{Weiss theorem on coincidence of amenability and hyperrniteness for equivalence relations. Nous consid er...

Let G be a locally compact abelian group, and let p 2 1; 1). We show that the Segal algebra S p (G) is always weakly amenable, but that it is amenable only if G is discrete.

We introduce notions of finite presentation and co-exactness which serve as qualitative quantitative analogues finite-dimensionality for operator modules over completely contractive Banach algebras. With these we begin the development a local theory by introducing lifting property, nuclearity, semi-discreteness. For large class prove that property is equivalent to flatness, generalizing space r...

Amenability is a geometric property of convex cones that stronger than facial exposedness and assists in the study error bounds for conic feasibility problems. In this paper we establish numerous properties amenable investigate relationships between amenability other cones, such as niceness projectional exposure. We show compact slice closed cone equivalent to cone, prove several results on pre...