The Baum-Connes conjecture states that, for a discrete group G, the K-homology groups of the classifying space for proper G-action is isomorphic to the K-groups of the reduced group C-algebra of G [3, 2]. A positive answer to the Baum-Connes conjecture would provide a complete solution to the problem of computing higher indices of elliptic operators on compact manifolds. The rational injectivit...

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the t...

In 1965, Thompson defined the groups F, T , and V [8]. Thompson’s group V is the group of right-continuous bijections v of [0, 1] that map dyadic rational numbers to dyadic rational numbers, that are differentiable except at finitely many dyadic rational numbers, and such that, on each interval on which v is differentiable, v is affine with derivative a power of 2. The group F is the subgroup o...

–a notion of amenability for topological semigroups is introduced. a topological semigroup s iscalled johnson amenable if for every banach s -bimodule e , every bounded crossed homomorphism froms to e* is principal. in this paper it is shown that a discrete semigroup s is johnson amenable if and only if1(s) is an amenable banach algebra. also, we show that if a topological semigroup s is johns...

The classification of type II1 factors (of discrete groups) was initiated by Murray and von Neumann [MvN] who distinguished the hyperfinite type II1 factor R from the group factor LFr of the free group Fr on r ≥ 2 generators. Thirty years later, Connes [Co2] proved uniqueness of the injective type II1 factor. Thus, the group factor LΓ of an ICC amenable group Γ is isomorphic to the hyperfinite ...

let a be a banach algebra. a is called ideally amenable if for every closed ideal i of a, the first cohomology group of a with coefficients in i* is trivial. we investigate the closed ideals i for which h1 (a,i* )={0}, whenever a is weakly amenable or a biflat banach algebra. also we give some hereditary properties of ideal amenability.

A kind of Dirac-Connes operator defined in the framework of Connes’ NCG is introduced on discrete abelian groups; it satisfies a Junk-free condition, and bridges the NCG composed by Dimakis, MüllerHoissen and Sitarz and the NCG of Connes. Then we apply this operator to d-dimensional lattices. [email protected] [email protected]

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X;), preserving the measure. If T 2 Aut (X;) is an automorphism of the equivalence relation deened by G then T can be extended to an automorphism T of the II 1-factor M = L 1 (X;)oG. We prove that if T commutes with the action of G then H(T) = h(T), where H(T) is the Connes-Sttrmer entropy of T , and h(T) is the K...

Let a countable amenable group G acts freely and ergodically on a Lebesgue space (X;), preserving the measure. If T 2 Aut (X;) is an automorphism of the equivalence relation deened by G then T can be extended to an automorphism T of the II 1-factor M = L 1 (X;)oG. We prove that if T commutes with the action of G then H(T) = h(T), where H(T) is the Connes-Sttrmer entropy of T , and h(T) is the K...