We construct algebras of pseudodifferential operators on a continuous family groupoid $\mathcal{G}$ that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on $\mathcal{G}$ as a dense subalgebra and reflect the smooth structure of the groupoid $\mathcal{G}$, when $\mathcal{G}$ 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.

One of our main results is to reduce the construction of spectrally invariant algebras of order 0 pseudodifferential operators to the analogous problem for regularizing operators. We then show that, in the case of the generalized ‘cusp’-calculi$c_n$, $n\ge2$, it is possible to construct algebras of regularizing operators that are closed under holomorphic functional calculus and consist of smooth kernels. For $n=1$, this was shown not to be possible by the first author in an earlier paper.

AMS 2000 Mathematics subject classification: Primary 35S05. Secondary 35J15; 47G30; 58J40; 46L87