‎generalizing the notion of character amenability for banach‎ ‎algebras‎, ‎we study the concept of $varphi$-connes amenability of‎ ‎a dual banach algebra $mathcal{a}$ with predual $mathcal{a}_*$‎, ‎where $varphi$ is a homomorphism from $mathcal{a}$ onto $bbb c$‎ ‎that lies in $mathcal{a}_*$‎. ‎several characterizations of‎ ‎$varphi$-connes amenability are given‎. ‎we also prove that the‎ ‎follo...

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...

We prove that any free cocycle action of a countable amenable group $\Gamma$ on II$_1$ factor $N$ can be perturbed by inner automorphisms to genuine action. This {\em vanishing cohomology} property, we call $\mathcal V\mathcal C$, is also closed products with amalgamation over finite groups. But beyond this no other examples C$-groups are known. In turn, considering special actions $\Gamma \cur...

We develop a certain piece of functional analysis on general graphs and use it to create what Connes calls a’spectral triple’, i.e. a Hilbert space structure, a representation of a certain (function) algebra and a socalled ’Dirac operator’, encoding part of the geometric/algebraic properties of the graph. We derive in particular an explicit expression for the ’Connesdistance function’ and show ...

We consider the algebra A of bounded operators on L2(Rn) generated by quantizations isometric affine canonical transformations. The includes as subalgebras noncommutative tori all dimensions and toric orbifolds. define spectral triple (A,H,D) with H=L2(Rn,?(Rn)) Euler operator D, a first order differential index 1. show that this has simple dimension spectrum: For every B in ?(A,H,D) Shubin typ...