# Algebras of pseudodifferential operators on complete manifolds

by

Bernd Ammann, Robert Lauter, Victor Nistor

**Algebras of pseudodifferential operators on complete manifolds**
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Electronic Research Announcements of the AMS **9** (2003), *80-87*.

Melrose has studied examples of non-compact manifolds $M_0$ whose large scale geometry is described by a Lie algebra of vector fields $\VV \subset\Gamma(M;TM)$ on a {\em 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 \cite{aln1}. In this paper, we define and study an algebra $\Psi_{1,0,\VV}^\infty(M_0)$ of pseudodifferential operators canonically associated to a manifold $M_0$ with a Lie structure at infinity $\VV \subset \Gamma(M;TM)$. 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 $\Psi_{1,0,\VV}^\infty(M_0)$ is a ``microlocalization'' of the algebra $\DiffV{*}(M)$ of differential operators with smooth coefficients on $M$ generated by $\VV$ and $\CI(M)$. This proves a conjecture from \cite{meicm}.
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Bernd Ammann, Robert Lauter, Victor Nistor,

The Paper was written on 19.03.2003

Last update 10.7.2003