Let \(\Psi^{m,0}({\mathcal G})\) be the space of order-\(m\) pseudodifferential operators on a continuous family groupoid \(\mathcal G\). The authors develop a general strategy to embed the algebra \(\Psi^{0,0}({\mathcal G})\) into larger algebras \(A\) that are closed under holomorphic functional calculus and still share some of the interesting properties with the algebra \(\Psi^{0,0}({\mathcal G})\). In fact, it is one of the results of this paper that it usually suffices to embed the algebra \(\Psi^{-\infty,0}({\mathcal G})\) in an algebra \(J\) that is closed under holomorphic functional calculus. Up to some technical conditions, \(A:=\Psi^{0,0}({\mathcal G})+J\) is then an algebra that is closed under holomorphic functional calculus. As an application, one gets a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of noncompact manifolds. For the construction of the algebra \(J\) one suggests three methods: one using semi-ideals, one using commutators and one based on Schwartz spaces on the groupoid.
In the case of the generalized “cusp”-calculi \(c_n\), \(n\geq 2\), one shows that 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.