Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4668755 | Bulletin des Sciences Mathématiques | 2015 | 27 Pages |
Abstract
The adiabatic groupoid Gad of a smooth groupoid G is a deformation relating G with its algebroid. In a previous work, we constructed a natural action of R on the C*-algebra of zero order pseudodifferential operators on G and identified the crossed product with a natural ideal J(G) of Câ(Gad). In the present paper we show that Câ(Gad) 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 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,G) of the associated crossed product AâG. We show that R acts naturally on Ψ(A,G) and identify the crossed product of A by the action of the adiabatic groupoid Gad with an extension of the crossed product Ψ(A,G)âR. Note that our construction of Ψ(A,G) unifies the ones of Connes (case A=C) and of Baaj (G is a Lie group).
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Claire Debord, Georges Skandalis,