# Journal of Operator Theory

Volume 78, Issue 2, Fall 2017  pp. 247-279.

Exhaustive families of representations and spectra of pseudodifferential operators

Authors:  Victor Nistor (1) and Nicolas Prudhon (2)
Author institution:(1) Departement de Mathematiques, Universite de Lorraine, UFR MIM, Ile du Saulcy, 57045 METZ, France and Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, 014700 Bucharest, Romania
(2) Departement de Mathematiques, Universite de Lorraine, UFR MIM, Ile du Saulcy, 57045 METZ, France

Summary: A family of representations $\mathcal F$ of a $C^{\ast}$-algebra $A$ is \textit{exhaustive} if every irreducible representation of $A$ is weakly contained in some $\phi \in \mathcal F$. Such an $\mathcal F$ has the property that "$a \in A$ is invertible if and only if $\phi(a)$ is invertible for any $\phi \in \mathcal F$". The regular representations of amenable, second countable, locally compact groupoids form an exhaustive family of representations. If $A$ is a separable $C^{\ast}$-algebra, a family $\mathcal F$ 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.

DOI: http://dx.doi.org/10.7900/jot.2016jul26.2121
Keywords: operator spectrum, essential spectrum, $C^*$-algebra, representations of $C^*$-algebra, self-adjoint operator, pseudodifferential operator, Cayley transform