2014/12/05 by Claire Debord, Debord, Claire, Georges Skandalis +1 · 1 citation
Mathematics · Medicine · #46L89 #58H05 #58J22 #Advanced Operator Algebra Research #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Operator Algebras (math.OA) #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.1412.1998
openalex publication_date 2014/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The adiabatic groupoid \Gad of a smooth groupoid \G\nis a deformation relating \G with its algebroid. In a previous work,\nwe constructed a natural action of \ℝ on the C*-algebra of zero order\npseudodifferential operators on \G and identified the crossed\nproduct with a natural ideal J(\G) of C^*(\Gad). In\nthe present paper we show that C^*(\Gad) itself is a\npseudodifferential extension of this crossed product in a sense introduced by\nSaad Baaj. Let us point out that we prove our results in a slightly more\ngeneral situation: the smooth groupoid \G is assumed to act on a\nC*-algebra A. We construct in this generalized setting the extension of order\n0 pseudodifferential operators \Ψ(A,\G) of the associated\ncrossed product A rtimes \G. We show that \ℝ acts\nnaturally on \Ψ(A,\G) and identify the crossed product of A by\nthe action of the adiabatic groupoid \Gad with an extension of\nthe crossed product \Ψ(A,\G) rtimes \ℝ. Note that our\nconstruction of \Ψ(A,\G) unifies the ones of Connes (case\nA=\ℂ ) and of Baaj (\G is a Lie group).\n