Exhausting families of representations and spectra of pseudodifferential operators

Families of representations of suitable Banach algebras provide a powerful tool in the study of the spectral theory of (pseudo)differential operators and of their Fredholmness. We introduce the new concept of an exhausting family of representations of a C*-algebra A. An {\em exhausting family} of representations of a C*-algebra A is a set F of representations of A with the property that every irreducible representation of A is weakly contained in some \phi \in F. An exhausting family F of representations of A has the property that `"a \in A is invertible if, and if, \phi(a) is invertible for any \phi \in F." Consequently, the spectrum of a is given by \Spec(a) = \cup_{\phi \in F} \Spec(\phi(a)). In other words, every exhausting family of representations is invertibility sufficient, a concept introduced by Roch in 2003. We prove several properties of exhausting families and we provide necessary and sufficient conditions for a family of representations to be exhausting. Using results of Ionescu and Williams (2009), we show that the regular representations of amenable, second countable, locally compact groupoids with a Haar system form an exhausting family of representations. If $A$ is a separable C*-algebra, we show that a family F of representations of $A$ is exhausting if, and only if, it is invertibility sufficient. However, this result is not true, in general, for non-separable C*-algebras. With an eye towards applications, we extend our results to the case of unbounded operators. A typical application of our results is to parametric families of differential operators arising in the analysis on manifolds with corners, in which case we recover the fact that a parametric operator F is invertible if, and only if, its Mellin transform is invertible. In view of possible applications, we have tried to make this paper accessible to non-specialists in C*-algebras.