Algebras of pseudodifferential operators on complete manifolds

In several influential works, Melrose has studied examples of non-compact manifolds $M_0$ whose large scale geometry is described by a Lie algebra of vector fields $\mathcal V \subset \Gamma(M;TM)$ on a compactification of $M_0$ to a manifold with corners $M$. The geometry of these manifolds--called ``manifolds with a Lie structure at infinity''--was studied from an axiomatic point of view in a previous paper of ours. In this paper, we define and study an algebra $\Psi_{1,0,\mathcal V}^\infty(M_0)$ of pseudodifferential operators canonically associated to a manifold $M_0$ with a Lie structure at infinity $\mathcal V \subset \Gamma(M;TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to the algebras that we introduce. In particular, the algebra $\Psi_{1,0,\mathcal V}^\infty(M_0)$ is a ``microlocalization'' of the algebra ${\rm Diff}^{*}_{\mathcal V}(M)$ of differential operators with smooth coefficients on $M$ generated by $\mathcal V$ and $\mathcal{C}^\infty(M)$. This proves a conjecture of Melrose (see his ICM 90 proceedings paper).