Boutet de Monvel’s calculus and groupoids I

Aastrup, Johannes; Melo, Severino T.; Monthubert, Bertrand; Schrohe, Elmar

doi:10.4171/jncg/57

Cited by 15 publications

(39 citation statements)

References 33 publications

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“…In any case, we see that in order to formulate an index problem, we need criteria for the relevant operators to be Fredholm, because it is the condition that P be Fredholm that guarantees that ker(P ), the kernel of P , and coker(P ), the cokernel of P , are finite dimensional. This is also related to the structure of the exact sequence (1).…”

mentioning

confidence: 95%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and there are included some new results as well. In particular, Sections 3 and 4 provide a complete short introduction to analysis on noncompact manifolds that is geared towards a class of manifolds--called "Lie manifolds"--that often appears in practice. Our interest in Lie manifolds is due to the fact that they provide the link between analysis on singular spaces and operator algebras. The groupoids integrating Lie manifolds play an important background role in establishing this link because they provide operator algebras whose structure is often well understood. The initial motivation for the work surveyed here--work that spans over close to two decades--was to develop the index theory of stratified singular spaces. Meanwhile, several other applications have emerged as well, including applications to Partial Differential Equations and Numerical Methods. These will be mentioned only briefly, however, due to the lack of space. Instead, we shall concentrate on the applications to Index theory.Comment: 43 pages, based on my four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 201

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confidence: 95%

Analysis on singular spaces: Lie manifolds and operator algebras

Nistor

2016

Journal of Geometry and Physics

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“…One also sees that the C * -algebra extension of the pseudodifferential operators on a groupoid is directly related to the one naturally associated with the DNC construction and the canonical action of R * + on it ( [1,45]). There is a well defined Morita equivalence between these exact sequences, and the corresponding bimodule gives an alternative definition of the pseudodifferential calculus on a groupoid (cf.…”

Section: Introductionmentioning

confidence: 92%

“…We often identify G (0) with its image in G (1) and make the confusion between G and G (1) . A groupoid G = (G (0) , G (1) , s, r, u, ι, m) will be simply denoted G In particular for A, B ⊂ G (0) , we put G A = {γ ∈ G; r(γ) ∈ A} and G A = {γ ∈ G; s(γ) ∈ A}; we also put G B A = G A ∩ G B .…”

Section: Introductionmentioning

confidence: 99%

Lie groupoids, pseudodifferential calculus, and index theory

Debord¹,

Skandalis²

2019

Advances in Noncommutative Geometry

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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. Lie groupoids and their operators algebrasWe refer to [78,80] for the classical definitions and constructions related to groupoids and their Lie algebroids. The construction of the C * -algebra of a groupoid is due to Jean Renault [100], one can look at his course [101] which is mainly devoted to locally compact groupoids. Lie groupoids GeneralitiesA groupoid is a small category in which every morphism is an isomorphism. Thus a groupoid G is a pair (G (0) , G (1) ) of sets together with structural morphisms:Units and arrows. The set G (0) denotes the set of objects (or units) of the groupoid, whereas the set G (1) is the set of morphisms (or arrows). The unit map u : G (0) → G (1) is the injective map which assigns to any object of G its identity morphism.The source and range maps s, r : G (1) → G (0) are (surjective) maps equal to identity in restriction toThe inverse ι : G (1) → G (1) is an involutive map which exchanges the source and range:for α ∈ G, (α −1 ) −1 = α and s(α −1 ) = r(α), where α −1 denotes ι(α)

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“…So the operator 1 2 + K is Fredholm. Hence we can apply the classical Fredholm theory to the operator 1 2 + K to solve the Dirichlet problem. If the boundary is C 1 , then the integral operator K on ∂Ω is still compact [15] on L 2 (∂Ω), but that no longer holds if there are singularities on the boundary [13,14,18,19,24,25,33,34,35].…”

Section: Introductionmentioning

confidence: 98%

“…Groupoids and groupoid C * -algebras have appeared useful in the analysis over singular spaces and, in particular, spaces with conical singularities, see for instance [1,2,9,10,11,20,21,37,39,42].…”

Section: Introductionmentioning

confidence: 99%

Layer potentials C*-algebras of domains with conical points

Carvalho¹,

Qiao

2012

centr.eur.j.math.

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To a domain with conical points Ω, we associate a natural C *algebra that is motivated by the study of boundary value problems on Ω, especially using the method of layer potentials. In two dimensions, we allow Ω to be a domain with ramified cracks. We construct an explicit groupoid associated to ∂Ω and use the theory of pseudodifferential operators on groupoids and its representations to obtain our layer potentials C * -algebra. We study its structure, compute the associated K-groups, and prove Fredholm conditions for the natural pseudodifferential operators affiliated to this C * -algebra.

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Boutet de Monvel’s calculus and groupoids I

Cited by 15 publications

References 33 publications

Analysis on singular spaces: Lie manifolds and operator algebras

Analysis on singular spaces: Lie manifolds and operator algebras

Lie groupoids, pseudodifferential calculus, and index theory

Layer potentials C*-algebras of domains with conical points

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