EXHAUSTIVE FAMILIES OF REPRESENTATIONS AND SPECTRA OF PSEUDODIFFERENTIAL OPERATORS

A family of representations F of a C*-algebra A is exhaustive if every irreducible representation of A is weakly contained in some φ ∈ F. Such an F has the property that “a ∈ A is invertible if and only if φ(a) is invertible for any φ ∈ F”. The regular representations of amenable, second countable, locally compact groupoids form an exhaustive family of representations. If A is a separable C*-algebra, a family F of representations of A is exhaustive if and only if it is strictly spectral. We consider also unbounded operators. A typical application is to parametric pseudodifferential operators.