  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 two-sided semi-ideals,
one using commutators, and one based on Schwartz spaces on the groupoid.
