2003/04/03 by Bernd Ammann, Ammann, Bernd, Robert Lauter +3
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AP #math.DG
paper · pdf · doi:10.48550/arxiv.math/0304044
openalex publication_date 2003/04/03 · arxiv created 2006/09/26 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Several examples of non-compact manifolds M0 whose geometry at infinity is described by Lie algebras of vector fields V ⊂ Γ(TM) (on a compactification of M0 to a manifold with corners M) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra Ψ1,0,\VV^∞(M0), which is an algebra of pseudodifferential operators canonically associated to a manifold M0 with the Lie structure at infinity V ⊂Γ(TM). We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ1,0,V^∞(M0). We also consider the algebra \DiffV*(M0) of differential operators on M0 generated by V and \CI(M), and show that Ψ1,0,V^∞(M0) is a ``microlocalization'' of \DiffV*(M0). Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra Ψ1,0,V^∞(M0). Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.