Spectrum of \(C^*\)-algebras of pseudodifferential operators with discontinuous symbols in an open manifold. (Russian) Zbl 0599.46079

Plamenevskij, B. A.; Senichkin, V. N.

Probl. Mat. Fiz. 11, 178-210 (1986).

The algebra \({\mathcal A}\) generated by zero order pseudodifferential operators on a smooth open manifold is considered. Symbols of the operators may have discontinuities of the ”first kind”. In the point of such a discontinuity there exists a limit of the symbol depending on the operator, and after completing the algebra the symbols with an everywhere dense set of discontinuities arise. The spectrum of the algebra \({\mathcal A}\) is described, i.e., the set of equivalence classes of irreducible representation with Jacobson’s topology. It is found that \({\mathcal A}\) is a I-type algebra.

Reviewer: O.Dumbrajs

MSC:

46L05	General theory of \(C^*\)-algebras
47Gxx	Integral, integro-differential, and pseudodifferential operators
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

pseudodifferential operators on a smooth open manifold; irreducible representation; Jacobson’s topology