Pseudodifferential extensions and adiabatic deformation of smooth groupoid actions
Résumé
The adiabatic groupoid G ad 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 C * (G ad). In the present paper we show that C * (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 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 A G. We show that R acts naturally on Ψ(A, G) and identify the crossed product of A by the action of the adiabatic groupoid G ad with an extension of the crossed product Ψ(A, G) R. Note that our construction of Ψ(A, G) unifies the ones of Connes (case A = C) and of Baaj (G is a Lie group).
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