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Moulay-Tahar Benameur

Publications and source records attributed to Moulay-Tahar Benameur.

22 records · Page 2Linked to original sources
arXiv2008

Index, eta and rho-invariants on foliated bundles

We study primary and secondary invariants of leafwise Dirac operators on foliated bundles. Given such an operator, we begin by considering the associated regular self-adjoint operator $D_m$ on the maximal Connes-Skandalis Hilbert module and explain how the functional calculus of $D_m$ encodes both the leafwise calculus and the monodromy calculus in the corresponding von Neumann algebras. When the foliation is endowed with a holonomy invariant transverse measure, we explain the compatibility of various traces and determinants. We extend Atiyah's index theorem on Galois coverings to these foliations. We define a foliated rho-invariant and investigate its stability properties for the signature operator. Finally, we establish the foliated homotopy invariance of such a signature rho-invariant under a Baum-Connes assumption, thus extending to the foliated context results proved by Neumann, Mathai, Weinberger and Keswani on Galois coverings.

math.DG↗
arXiv2002

Residues and homology for pseudodifferential operators on foliations

We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras of pseudodifferential operators along the leaves of a foliation.

math.KT↗
arXiv2000

Homology of complete symbols and non-commutative geometry

We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid $\GR$ in terms of the cohomology of $S^*(\GR)$, the cosphere bundle of $A(\GR)$, where $A(\GR)$ is the Lie algebroid of $\GR$. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space $A^*(\GR) \smallsetminus 0 \cong S^*(\GR) \times (0,\infty)$, the dual of $A(\GR)$ with the zero section removed. We use then these results to compute the Hochschild and cyclic homologies of the algebras of complete symbols associated with manifolds with corners, when the corresponding Lie algebroid is rationally isomorphic to the tangent bundle.

math.OA↗