Moulay-Tahar Benameur

Type II non-commutative geometry. I. Dixmier trace in von Neumann algebras

Abstract We define the notion of Connes–von Neumann spectral triple and consider the associated index problem. We compute the analytic Chern–Connes character of such a generalized spectral triple and prove the corresponding local formula for its Hochschild class. This formula involves the Dixmier trace for II ∞ von Neumann algebras. In the case of foliations, we identify this Dixmier trace with the corresponding measured Wodzicki residue.

Conformal invariants of twisted Dirac operators and positive scalar curvature

Abstract For a closed, spin, odd dimensional Riemannian manifold ( Y , g ) , we define the rho invariant ρ s p i n ( Y , E , H , [ g ] ) for the twisted Dirac operator ⁄ ∂ H E on Y , acting on sections of a flat Hermitian vector bundle E over Y , where H = ∑ i j + 1 H 2 j + 1 is an odd-degree closed differential form on Y and H 2 j + 1 is a real-valued differential form of degree 2 j + 1 . We prove that it only depends on the conformal class [ g ] of the metric g . In the special case when H is a closed 3-form, we use a Lichnerowicz–Weitzenbock formula for the square of the twisted Dirac operator, which in this case has no first order terms, to show that ρ s p i n ( Y , E , H , [ g ] ) = ρ s p i n ( Y , E , [ g ] ) for all | H | small enough, whenever g is conformally equivalent to a Riemannian metric of positive scalar curvature. When H is a top-degree form on an oriented three dimensional manifold, we also compute ρ s p i n ( Y , E , H ) .

A higher Lefschetz formula for flat bundles

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

Index type invariants for twisted signature complexes and homotopy invariance

For a closed, oriented, odd dimensional manifold

Leafwise homotopies and Hilbert-Poincaré complexes I. Regular HP-complexes and leafwise pull-back maps

X

Gap-labelling conjecture with nonzero magnetic field

, we define the rho invariant

Transverse noncommutative geometry of foliations

\rho(X,E,H)

On the Lefschetz problem in non commutative geometry

for the twisted odd signature operator valued in a flat hermitian vector bundle

An analytic approach to spectral flow in von Neumann algebras

E

Index Theory and Non-Commutative Geometry I. Higher Families Index Theory

, where