arXiv subjects

Georges Skandalis

Publications and source records attributed to Georges Skandalis.

17 recordsLinked to original sources

arXiv2026

On the transportation cost norm on finite metric graphs

For a finite metric graph $X=(V,E,\ell)$, where $V$ is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance $d$ on $V$. Namely, for $f$ a function with total sum 0 on $V$, write $f=\sum_{a,b\in V}P(a,b)(δ_a-δ_b)$ where the transportation plan $P$ satisfies $P(a,b)\geq 0$ for $(a,b)\in V\times V$. The cost of $P$ is $W(P):=\sum_{a,b\in V}P(a,b)d(a,b)$ and the transportation norm of $f$ is $\|f\|_{TC}=\min_P W(P)$ where $P$ runs over all transportation plans for $f$. In this semi-survey paper, we give short proofs for the following statements: 1)There always exists an optimal transportation plan supported in $V_+\times V_-$ where $V_+=\{x\in V: f(x)>0\}$ and $V_-=\{x\in V: f(x)<0\}$. If $X$ is a metric tree, we may moreover assume that this plan involves at most $|Supp(f)|-1$ transports. 2) There always exists an optimal transportation plan supported in the set of edges of $X$. 3) Better, there always exists an optimal transportation plan supported in some spanning tree of $X$. We use this to reprove known formulae for the transportation norm when $X$ is either a tree or a cycle.

math.MG↗

arXiv2020

The Baum--Connes conjecture localised at the unit element of a discrete group

We construct a Baum--Connes assembly map localised at the unit element of a discrete group $Γ$. This morphism, called $μ_τ$, is defined in $KK$-theory with coefficients in $\mathbb{R}$ by means of the action of the projection $[τ]\in KK_{\mathbb{R}}^Γ(\mathbb{C},\mathbb{C})$ canonically associated to the group trace of $Γ$. We show that the corresponding $τ$-Baum--Connes conjecture is weaker then the classical one but still implies the strong Novikov conjecture. The right hand side of $μ_τ$ is functorial with respect to the group $Γ$.

math.OA↗

arXiv2019

Lie groupoids, pseudodifferential calculus and index theory

Alain Connes introduced the use of Lie groupoids in noncommutative geometry in his pioneering work on the index theory of foliations. In the present paper, we recall the basic notion involved: groupoids, their C*-algebras, their pseudodifferential calculus... We review several recent and older advances on the involvement of Lie groupoids in noncommutative geometry. We then propose some open questions and possible developments of the subject.

math.OA↗

arXiv2017

Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus

We present natural and general ways of building Lie groupoids, by using the classical procedures of blowups and of deformations to the normal cone. Our constructions are seen to recover many known ones involved in index theory. The deformation and blowup groupoids obtained give rise to several extensions of $C^*$-algebras and to full index problems. We compute the corresponding K-theory maps. Finally, the blowup of a manifold sitting in a transverse way in the space of objects of a Lie groupoid leads to a calculus, quite similar to the Boutet de Monvel calculus for manifolds with boundary.

math.OA↗

arXiv2016

A Baum-Connes conjecture for singular foliations

We consider singular foliations whose holonomy groupoid may be nicely decomposed using Lie groupoids (of unequal dimension). We show that the Baum-Connes conjecture can be formulated in this setting. This conjecture is shown to hold under assumptions of amenability. We examine several examples that can be described in this way and make explicit computations of their K-theory.

math.KT↗

arXiv2015

Bivariant $K$-theory with $R/Z$-coefficients and rho classes of unitary representations

We construct equivariant $KK$-theory with coefficients in $\mathbb{R}$ and $\mathbb{R}/\mathbb{Z}$ as suitable inductive limits over ${\rm II}_1$-factors. We show that the Kasparov product, together with its usual functorial properties, extends to $KK$-theory with real coefficients. Let $Γ$ be a group. We define a $Γ$-algebra $A$ to be $K$-theoretically free and proper (KFP) if the group trace ${\bf tr}$ of $Γ$ acts as the unit element in $KK^Γ_{\mathbb{R}}(A,A)$. We show that free and proper $Γ$-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if $Γ$ is torsion free and satisfies the $KK^Γ$-form of the Baum-Connes conjecture, then every $Γ$-algebra satisfies (KFP). If $α:Γ\to U_n$ is a unitary representation and $A$ satisfies property (KFP), we construct in a canonical way a rho class $ρ_α^A\in KK_{\mathbb{R}/\mathbb{Z}}^{1,Γ}(A,A)$. This construction generalizes the Atiyah-Patodi-Singer $K$-theory class with $\mathbb{R}/\mathbb{Z}$ coefficients associated to $α$.

math.OA↗

arXiv2015

Stability of Lie groupoid C*-algebras

In this paper we generalize a theorem of M. Hilsum and G. Skandalis stating that the $C^*$- algebra of any foliation of non zero dimension is stable. Precisely, we show that the C*-algebra of a Lie groupoid is stable whenever the groupoid has no orbit of dimension zero. We also prove an analogous theorem for singular foliations for which the holonomy groupoid as defined by I. Androulidakis and G. Skandalis is not Lie in general.

math.OA↗

arXiv2014

Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions

The adiabatic groupoid $\mathcal{G}_{ad}$ of a smooth groupoid $\mathcal{G}$ is a deformation relating $\mathcal{G}$ with its algebroid. In a previous work, we constructed a natural action of $\mathbb{R}$ on the C*-algebra of zero order pseudodifferential operators on $\mathcal{G}$ and identified the crossed product with a natural ideal $J(\mathcal{G})$ of $C^*(\mathcal{G}_{ad})$. In the present paper we show that $C^*(\mathcal{G}_{ad})$ itself is a pseudodifferential extension of this crossed product in a sense introduced by Saad Baaj. Let us point out that we prove our results in a slightly more general situation: the smooth groupoid $\mathcal{G}$ is assumed to act on a C*-algebra $A$. We construct in this generalized setting the extension of order $0$ pseudodifferential operators $Ψ(A,\mathcal{G})$ of the associated crossed product $A\rtimes \mathcal{G}$. We show that $\mathbb{R}$ acts naturally on $Ψ(A,\mathcal{G})$ and identify the crossed product of $A$ by the action of the adiabatic groupoid $\mathcal{G}_{ad}$ with an extension of the crossed product $Ψ(A,\mathcal{G})\rtimes \mathbb{R}$. Note that our construction of $Ψ(A,\mathcal{G})$ unifies the ones of Connes (case $A=\mathbb{C} $) and of Baaj ($\mathcal{G}$ is a Lie group).

math.OA↗

arXiv2013

Adiabatic groupoid, crossed product by $\R_+^*$ and Pseudodifferential calculus

We consider the crossed product $G_{ga}$ by $\R_+^*$ of the adiabatic groupoid associated with any Lie groupoid $G$. We construct an explicit Morita equivalence between the exact sequence of order 0 pseudodifferential operators on $G$ and (a restriction of) the natural exact sequence associated with $G_{ga}$. As an important intermediate step, we express a pseudodifferential operator on $G$ as an integral associated to a smoothing operator on the adiabatic groupoid $G_{ad}$ of $G$.

math.FA↗

arXiv2013

Flat bundles, von Neumann algebras and $K$-theory with $\R/\Z$-coefficients

Let $M$ be a closed manifold and $α: π_1(M)\to U_n$ a representation. We give a purely $K$-theoretic description of the associated element $[α]$ in the $K$-theory of $M$ with $\R/\Z$-coefficients. To that end, it is convenient to describe the $\R/\Z$-$K$-theory as a relative $K$-theory with respect to the inclusion of $\C$ in a finite von Neumann algebra $B$. We use the following fact: there is, associated with $α$, a finite von Neumann algebra $B$ together with a flat bundle $\cE\to M$ with fibers $B$, such that $E_\a\otimes \cE$ is canonically isomorphic with $\C^n\otimes \cE$, where $E_α$ denotes the flat bundle with fiber $\C^n$ associated with $α$. We also discuss the spectral flow and rho type description of the pairing of the class $[α]$ with the $K$-homology class of an elliptic selfadjoint (pseudo)-differential operator $D$ of order 1.

math.OA↗

arXiv2010

The analytic index of elliptic pseudodifferential operators on a singular foliation

In previous papers (arxiv:math/0612370 and arxiv:0909.1342) we defined the C*-algebra and the longitudinal pseudodifferential calculus of any singular foliation (M,F). Here we construct the analytic index of an elliptic operator as a KK-theory element, and prove that the same element can be obtained from an "adiabatic foliation" TF on $M \times \R$, which we introduce here.

math.OA↗

arXiv2009

Pseudodifferential calculus on a singular foliation

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation F on a compact manifold M, and we show that it can be naturally understood as a positive, unbounded, self-adjoint operator on L2(M).

math.DG↗

arXiv2007

The holonomy groupoid of a singular foliation

We construct the holonomy groupoid of any singular foliation. In the regular case this groupoid coincides with the usual holonomy groupoid of Winkelnkemper (1983); the same holds in the singular cases of Bigonnet and Pradines (1985) and Debord (2001), which from our point of view can be thought of as being "almost regular". In the general case, the holonomy groupoid can be quite an ill behaved geometric object. On the other hand it often has a nice longitudinal smooth structure. Nonetheless, we use this groupoid to generalize to the singular case Connes' construction of the C*-algebra of the foliation. We also outline the construction of a longitudinal pseudo-differential calculus; the analytic index of a longitudinally elliptic operator takes place in the K-theory of our C*-algebra. In our construction, the key notion is that of a bi-submersion which plays the role of a local Lie groupoid defining the foliation. Our groupoid is the quotient of germs of these bi-submersions with respect to an appropriate equivalence relation.

math.DG↗

arXiv2004

Groups acting properly on "bolic" spaces and the Novikov conjecture

We introduce a class of metric spaces which we call "bolic". They include hyperbolic spaces, simply conneccted complete manifolds of nonpositive curvature, euclidean buildings, etc. We prove the Novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a "bolic", weakly geodesic metric space of bounded geometry.

math.AG↗

arXiv2003

Measurable Kac cohomology for bicrossed products

We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a topological version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.

math.OA↗

arXiv2002

Non-semi-regular quantum groups coming from number theory

In this paper, we study C*-algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi-regular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C*-algebras. On the way, we provide several remarks on C*-algebraic properties of quantum groups and their actions.

math.OA↗

arXiv1998

Unitaires multiplicatifs en dimension finie et leurs sous-objets

A pre-subgroup of a multiplicative unitary $V$ on a finite dimensionnal Hilbert space $H$ is a vector line $L$ in $H$ such that $V(L\otimes L)=L\otimes L$. We show that there are finitely many pre-subgroups, give a Lagrange theorem and generalize the construction of a `bi-crossed product'. Moreover, we establish bijections between pre-subgroups and coideal subalgebras of the Hopf algebra associated with $V$, and therefore with the intermediate subfactors of the associated (depth two) inclusions. Finally, we show that the pre-subgroups classify the subobjects of $(H,V)$.

math.OA↗