On~representations of an algebra of pseudodifferential operators with multidimensional discontinuities in the symbols

B. A. Plamenevskii ; V. N. Senichkin

Izvestiya. Mathematics , Tome 31 (1988) no. 1, pp. 143-169

Résumé

This article considers $C^*$-algebras generated by pseudodifferential operators on a smooth $m$-dimensional manifold $\mathscr M$ without boundary. The symbols of the operators are allowed to have discontinuities “of the first kind” along submanifolds of codimension $n$, $1\leqslant n\leqslant m-1$. The operators act in the space $L_2(\mathscr M)$. All irreducible representations (to within equivalence), including two series of infinitedimensional representations, are given for such algebras. Necessary and sufficient conditions for the Fredholm property are determined for arbitrary elements of the algebras. The topology on the spectrum of the algebras is described. A composition series is determined whose successive quotients are isomorphic to algebras of the form $C_0(X)\otimes \mathscr{KH}$, where $X$ is a locally compact space, $C_0(X)$ is the set of continuous functions tending to zero at infinity, and $\mathscr{KH}$ is the ideal of compact operators on a Hilbert space $\mathscr H$. Among the composition series having this property the indicated series is the shortest. Bibliography: 9 titles.