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 Izv. Akad. Nauk SSSR Ser. Mat., 1987, Volume 51, Issue 4, Pages 833–859 (Mi izv1321)

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

B. A. Plamenevskii, V. N. Senichkin

Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1988, 31:1, 143–169

Bibliographic databases:

UDC: 517.98
MSC: Primary 47D25, 58G15, 46L99; Secondary 47A10, 47A53, 35S05

Citation: B. A. Plamenevskii, V. N. Senichkin, “On representations of an algebra of pseudodifferential operators with multidimensional discontinuities in the symbols”, Izv. Akad. Nauk SSSR Ser. Mat., 51:4 (1987), 833–859; Math. USSR-Izv., 31:1 (1988), 143–169

Citation in format AMSBIB
\Bibitem{PlaSen87} \by B.~A.~Plamenevskii, V.~N.~Senichkin \paper On~representations of an algebra of pseudodifferential operators with multidimensional discontinuities in the symbols \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1987 \vol 51 \issue 4 \pages 833--859 \mathnet{http://mi.mathnet.ru/izv1321} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=914862} \zmath{https://zbmath.org/?q=an:0656.47044} \transl \jour Math. USSR-Izv. \yr 1988 \vol 31 \issue 1 \pages 143--169 \crossref{https://doi.org/10.1070/IM1988v031n01ABEH001051} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. A. Plamenevskii, V. N. Senichkin, “The spectrum of an algebra of pseudodifferential operators with piecewise smooth symbols”, Math. USSR-Izv., 34:1 (1990), 147–179
2. R. Lauter, “On representations of Ψ* andC *-algebras ofb-pseudo-differential operators on manifolds with corners”, Journal of Mathematical Sciences (New York), 98:6 (2000), 684
3. B. A. Plamenevskii, V. N. Senichkin, “On a class of pseudodifferential operators in $\mathbb R^m$ and on stratified manifolds”, Sb. Math., 191:5 (2000), 725–757
4. B. A. Plamenevskii, “Solvability of the algebra of pseudodifferential operators with piecewise smooth coefficients on a smooth manifold”, St. Petersburg Math. J., 21:2 (2010), 317–351
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