We apply the classical technique on cyclic objects of Alain Connes to various objects, in particular to the higher Chow complex of S. Bloch to prove a Connes periodicity long exact sequence involving motivic cohomology groups. The Cyclic higher Chow groups and the Connes higher Chow groups of a variety are defined in the process and various properties of them are deduced from the known properti...

Differential structure of a d-dimensional lattice, which essentially is a noncommutative exterior algebra, is defined using both reductions in 1-order plus 2-order of universal differential calculus in the context of NCG developed by Dimakis et al , and the formalism adopting a Dirac-Connes operator proposed by us recently within Connes’ NCG. Metric structure on this lattice and on differential...

In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...

Let ! be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin Dirac operator of a spin manifold with positive scalar curvature and fundamental group !; (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group ! . The invariants we consider are more preci...

For $p,q\in [1,\infty)$, we study the isomorphism problem for $p$- and $q$-convolution algebras associated to locally compact groups. While it is well known that not every group can be recovered from its von Neumann algebra, show this case $\mathrm{CV}_p(G)$ of $p$-convolvers $\mathrm{PM}_p(G)$ $p$-pseudomeasures, $p\neq 2$. More generally, if isometrically isomorphic $\mathrm{CV}_q(H)$, with $...

The Atiyah conjecture for a discrete group G states that the L-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of G. Here we establish conditions under which the Atiyah conjecture for a torsion-free group G implies the Atiyah conjecture for every fi...

Abstract We give a new criterion for solvability of group equations, providing proofs various generalizations the Kervaire–Laudenbach conjecture Connes-embeddable groups.

The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing s...

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this mor...