Spectral Invariance for Certain Algebras of Pseudodifferential Operators

Abstract

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.

Boutet de Monvel’s calculus and groupoids I

“…In several papers ( [22], [24], [17], [21]), generalizations of this approach to a larger class of groupoids were achieved. One particular aspect of this theory is that, as A. Connes showed in [8] for smooth manifolds, it is possible to define the analytic index using a groupoid, the tangent groupoid.…”

“…That was done in [17], where the algebras of pseudodifferential operators on continuous family groupoids, which are groupoids whose fibers are smooth manifolds, were defined.…”

“…The guiding principle in that approach is that the central object in global analysis is a groupoid. To study a particular situation, for a class of singular manifolds for instance, one needs to define a groupoid adapted to the context and then use the general pseudodifferential tools for groupoids, as developed in [22], [24], [21], [17], [18]. This has been applied to the context of manifolds with corners, with the goal of studying Melrose's b-calculus (see [22], [24], [21]).…”

An index theorem for manifolds with boundary

“…All the groupoids considered here will be continuous family groupoids [5,10]. Hence we can consider their convolution and C * -algebras without any problem.…”

Contribution of noncommutative geometry to index theory on singular manifolds

“…The article [14] is completely dedicated to these questions. We clarify constructions of algebras stable under holomorphic functional calculus, or more precisely we build sub-algebras stable under holomorphic functional calculus J ⊂ C * (G), and we show that I := Ψ 0 (G) + J is itself stable under holomorphic functional calculus.…”