‎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...

In a recent paper, the authors proved that no spin foliation on compact enlargeable manifold with Hausdorff homotopy graph admits metric of positive scalar curvature its leaves. This result extends groundbreaking results Lichnerowicz, Gromov and Lawson, Connes non-existence metrics curvature. this paper we review in more detail material needed for proof our theorem extend to non-compact manifol...

In this paper we investigate the index theory of transversally elliptic pseudo-differential operators in the framework of noncommutative geometry. We give examples of spectral triples in the sense of Alain Connes and Henri Moscovici in [12] that are transversally elliptic but nonelliptic. We prove that these spectral triples satisfy the conditions which ensure the Connes-Moscovici local index f...

let s be a locally compact foundation semigroup with identity and                          be its semigroup algebra. let x be a weak*-closed left translation invariant subspace of    in this paper, we prove that  x  is invariantly  complemented in   if and  only if  the left ideal  of    has a bounded approximate identity. we also prove that a foundation semigroup with identity s is left amenab...

In this paper we study the concept of ph-biatness ofa Banach algebra A, where ph is a continuous homomorphism on A.We prove that if ph is a continuous epimorphism on A and A hasa bounded approximate identity and A is ph- biat, then A is ph-amenable. In the case where ph is an isomorphism on A we showthat the ph- amenability of A implies its ph-biatness.

Let and be randomly chosen subsets of the first positive integers cardinalities , such that their sumset has size . We show asymptotically almost surely are fully contained in arithmetic progressions with same common difference approximately also prove a counting theorem for pairs sets arbitrary abelian groups. The results hold Our main tool is an asymmetric version method hypergraph containers...