Continuous Trace Groupoid $C^*$-Algebras, II.

“…It is stated without proof in [16,Remark 4.10], and it is, unfortunately, used without comment in [9,10]. We include the statement and proof here for the reader's convenience.…”

Section: Proposition 28 Suppose That G Is a Second Countable Locallmentioning

confidence: 99%

Continuous-trace groupoid 𝐶*-algebras. III

Muhly

Renault

Williams

1996

Trans. Amer. Math. Soc.

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“…In this event, in fact, C * (G; A) is strongly Morita equivalent to C 0 (X/H). Another precedent to this investigation is the work of the second two authors in [10]. Here, the hypothesis is that G is a principal groupoid and the bundle A is again the trivial line bundle.…”

Section: Questionmentioning

confidence: 99%

“…The proof of the necessity in Theorem 1 is modeled on the proofs in [10] and [11], which in turn are inspired by ideas in [5]. There are, however, a number of new difficulties that must be overcome.…”

Section: Necessity In Theoremmentioning

confidence: 99%

“…We now assume the action of G on G (0) is not proper and use Lemma 2.6 of [10] to choose a sequence {γ n } ⊆ G such that γ n → ∞ in the sense that {γ n } eventually escapes each compact subset of G, and such that r(γ n ), s(γ n ) → z, for some z ∈ G (0) . We shall fix this sequence for the remainder of the proof.…”

Section: Corollary 4 Assume That G Is Principal and Thatmentioning

confidence: 99%

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Continuous-trace groupoid crossed products

Fulman

Muhly

Williams

2003

Proc. Amer. Math. Soc.

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Fredholm Conditions on Non-compact Manifolds: Theory and Examples

Carvalho

Nistor

Qiao

2018

Operator Theory, Operator Algebras, and Matrix Theory

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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.

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Continuous Trace Groupoid $C^*$-Algebras, II.

Cited by 52 publications

References 2 publications

Continuous-trace groupoid 𝐶*-algebras. III

Continuous-trace groupoid 𝐶*-algebras. III

Continuous-trace groupoid crossed products

Fredholm Conditions on Non-compact Manifolds: Theory and Examples

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