# 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

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