The present paper is a short survey on the mathematical basics of Classical Field Theory including the Serre-Swan’ theorem, Clifford algebra bundles and spinor bundles over smooth Riemannian manifolds, Spin-structures, Dirac operators, exterior algebra bundles and Connes’ differential algebras in the commutative case, among other elements. We avoid the introduction of principal bundles and put ...

in this paper among many other things we prove that the topological left amenability and left amenability of a weighted hypergroup (k, ?) are equivalent. for a normal subgroup h of k, we define a weight function ?? on kih and obtain connection between left amenability of (k, ?) and (k|h, ??). let h be a compact subhypergroup of k. we define the weight function on k||h and obtain connection betw...

In this work we examine generalized Connes-Lott models, with C ⊕ C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, wh...

We investigate amenability and weak amenability of the l1-algebra of polynomial hypergroups. We derive conditions for (weak) amenability adapted to polynomial hypergroups and show that these conditions are often not satisfied. However, for the hypergroup induced by Chebyshev polynomials of the first kind we prove amenability.

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...

The work extends the A. Connes’ noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan’s anholonomic frame approach to geometry models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonomic Riemann, Finsler and Lagrange spaces are analyzed. We also present a research...

We study L-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [4]. We give a definition of Lcohomology and show how the study of the first L-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of...

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 this paper by using some conditions, we show that the weak amenability of (2n)-th dual of a Banach algebra A for some $ngeq 1$ implies the weak amenability of A.

In 2005, A. Abdollahi and A. Rejali, studied the relations betweenparadoxical decomposition and configuration for semigroups. In thepresent paper, we introduce one other concept of amenability typeon semigroups and groups which includes amenability of semigroupsand inner-amenability of groups. We have extend the previous knownresults to semigroups and groups satisfying this concept.