Analysis on singular spaces: Lie manifolds and operator algebras

Operator Theory, Operator Algebras, and Matrix Theory

Qiao

2018

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We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only if, it is elliptic and all its limit operators, in a sense to be made precise, are invertible. Central to our theoretical results is the concept of a Fredholm groupoid, which is the class of groupoids for which this characterization of the Fredholm condition is valid. We use the notions of exhaustive and strictly spectral families of representations to obtain a general characterization of Fredholm groupoids. In particular, we introduce the class of the so-called groupoids with Exel's property as the groupoids for which the regular representations are exhaustive. We show that the class of "stratified submersion groupoids" has Exel's property, where stratified submersion groupoids are defined by glueing fibered pull-backs of bundles of Lie groups. We prove that a stratified submersion groupoid is Fredholm whenever its isotropy groups are amenable. Many groupoids, and hence many pseudodifferential operators appearing in practice, fit into this framework. This fact is explored to yield Fredholm conditions not only in the above mentioned classes, but also on manifolds that are obtained by desingularization or by blow-up of singular sets.

Section: 2mentioning

confidence: 99%

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

Operator Theory, Operator Algebras, and Matrix Theory

Qiao

2018

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“…It would be interesting to study the corresponding properties for a general Lie group G acting on C u b (G) [34]. Morphisms analogous to the τ χ can be defined also in a groupoid framework [32,38], but they do not have a similar, simple interpretation as strong limits. It would be interesting to understand the connections between the above theorem and the representation theory of groupoids [6,7,15,30,45].…”

Section: Crossed Products and Localizations At Infinitymentioning

confidence: 99%

On the essential spectrum of $N$-body Hamiltonians with asymptotically homogeneous interactions

Georgescu

2017

J. Operator Theory

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ABSTRACT. We determine the essential spectrum of Hamiltonians with N -body type interactions that have radial limits at infinity. This extends the HVZ-theorem, which treats perturbations of the Laplacian by potentials that tend to zero at infinity. Our proof involves C * -algebra techniques that allows one to treat large classes of operators with local singularities and general behavior at infinity. In our case, the configuration space of the system is a finite dimensional, real vector space X, and we consider the algebra E(X) of functions on X generated by functions of the form v • π Y , where Y runs over the set of all linear subspaces of X, π Y is the projection of X onto the quotient X/Y , and v : X/Y → C is a continuous function that has uniform radial limits at infinity. The group X acts by translations on E(X), and hence the crossed product E (X) := E(X) ⋊ X is well defined; the Hamiltonians that are of interest to us are the self-adjoint operators affiliated to it. We determine the characters of E(X). This then allows us to describe the quotient of E (X) with respect to the ideal of compact operators, which in turn gives a formula for the essential spectrum of any self-adjoint operator affiliated to E (X).

“…To summarize the construction of the desingularization, let us denote by φ the natural isomorphism of the following two groupoids: E(S, π, H) U1 U1 (reduction to U 1 ≃ S × (0, 1)) and G U1 U1 = (G 2 ) U1 U1 . Then (25) [ Similar structures arise in other situations; see, for instance, [14,28,19,35,34,43,44,56,53]. See also the discussion at the end of Example 2.4.…”

Section: 1mentioning

confidence: 72%

“…See also the discussion at the end of Example 2.4. Proposition 3.14 is important in Index theory and Spectral theory because it gives rise to exact sequences of algebras [13,44].…”

Section: 1mentioning

confidence: 99%

Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Communications in Analysis and Geometry

2019

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We introduce and study a "desingularization" of a Lie groupoid G along an "A(G)-tame" submanifold L of the space of units M . An A(G)-tame submanifold L ⊂ M is one that has, by definition, a tubular neighborhood on which A(G) becomes a thick pull-back Lie algebroid. The construction of the desingularization [[G : L]] of G along L is based on a canonical fibered pull-back groupoid structure result for G in a neighborhood of the tame A(G)submanifold L ⊂ M . This local structure result is obtained by integrating a certain groupoid morphism, using results of Moerdijk and Mrcun (Amer. J. Math. 2002). Locally, the desingularization [[G : L]] is defined using a construction of Debord and Skandalis (Advances in Math., 2014). The space of units of the desingularization [[G : L]] is [M : L], the blow up of M along L. The space of units and the desingularization groupoid [[G : L]] are constructed using a gluing construction of Gualtieri and Li (IMRN 2014). We provide an explicit description of the structure of the desingularized groupoid and we identify its Lie algebroid, which is important in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example relating our constructions to the so called "edge pseudodifferential calculus." The paper is written such that it also provides an introduction to Lie groupoids designed for applications to analysis on singular spaces.