On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols

This article deals with a $C^*$-algebra $\mathscr A'$ generated by pseudodifferential operators whose symbols can have discontinuities “of the first kind” at a finite number of points. The set of points of discontinuity depends on the operator, and after completion of the algebra $\mathscr A/\mathscr K$, where $\mathscr K$ the ideal of compact operators, there appear classes (elements of the quotient algebra) whose symbols have dense sets of singularities. A complete set of irreducible representations is determined for the quotient algebra $\mathscr A/\mathscr K$, and the Jacobson topology on the spectrum is described. The same problems are solved also for the algebra $\mathscr A$. It is established that $\mathscr A$ and $\mathscr A/\mathscr K$ are algebras of type I.
Bibliography: 7 titles.