Journal of Operator Theory
Volume 78, Issue 2, Fall 2017 pp. 247-279.
Exhaustive families of representations and spectra of pseudodifferential operatorsAuthors: 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 F of a C∗-algebra A is \textit{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.
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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