Exhausting families of representations and spectra of pseudodifferential operators

Abstract

A powerful tool in the spectral theory and the study of Fred-holm conditions for (pseudo)differential operators is provided by families of representations of a naturally associated algebra of bounded operators. Motivated by this approach, we define the concept of an strictly norming family of representations of a C *-algebra A. Let F be a strictly norming family of representations of A. We have then that an abstract differential operator D affiliated to A is invertible if, and only if, φ(D) is invertible for all φ ∈ F. This property characterizes strictly norming families of representations. We provide necessary and sufficient conditions for a family of representations to be strictly norming. If A is a separable C *-algebra, we show that a family F of representations is strictly norming if, and only if, every irreducible representation of A is weakly contained in a representation φ ∈ F. However, this result is not true, in general, for non-separable C *-algebras. 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 P is invertible if, and only if, its Mellin transform P (τ) is invertible, for all τ ∈ R n. The paper is written to be accessible to non-specialists in C *-algebras.