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 Izv. Akad. Nauk SSSR Ser. Mat., 1983, Volume 47, Issue 6, Pages 1263–1284 (Mi izv1463)

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

B. A. Plamenevskii, V. N. Senichkin

Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1984, 23:3, 525–544

Bibliographic databases:

UDC: 517.98
MSC: Primary 46H15, 46L99, 58G15; Secondary 35S99, 47G05, 46H10

Citation: B. A. Plamenevskii, V. N. Senichkin, “On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols”, Izv. Akad. Nauk SSSR Ser. Mat., 47:6 (1983), 1263–1284; Math. USSR-Izv., 23:3 (1984), 525–544

Citation in format AMSBIB
\Bibitem{PlaSen83} \by B.~A.~Plamenevskii, V.~N.~Senichkin \paper On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1983 \vol 47 \issue 6 \pages 1263--1284 \mathnet{http://mi.mathnet.ru/izv1463} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=727755} \zmath{https://zbmath.org/?q=an:0604.47031|0551.47023} \transl \jour Math. USSR-Izv. \yr 1984 \vol 23 \issue 3 \pages 525--544 \crossref{https://doi.org/10.1070/IM1984v023n03ABEH001784} 

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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. N. L. Vasilevskii, “Algebras generated by multidimensional singular integral operators and by coefficients admitting discontinuities of homogeneous type”, Math. USSR-Sb., 57:1 (1987), 1–19
2. B. A. Plamenevskii, V. N. Senichkin, “Spectra of $X^*$-algebras of pseudodifferential operators with multidimensional discontinuities in symbols”, Funct. Anal. Appl., 20:4 (1986), 328–329
3. B. A. Plamenevskii, V. N. Senichkin, “On representations of an algebra of pseudodifferential operators with multidimensional discontinuities in the symbols”, Math. USSR-Izv., 31:1 (1988), 143–169
4. 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
5. B. A. Plamenevskii, G. V. Rozenblum, “Pseudodifferential operators with discontinuous symbols: $K$-theory and the index formula”, Funct. Anal. Appl., 26:4 (1992), 266–275
6. 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
7. 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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