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Algebra i Analiz, 2009, Volume 21, Issue 2, Pages 214–263 (Mi aa1011)  

This article is cited in 1 scientific paper (total in 1 paper)

Solvability of the algebra of pseudodifferential operators with piecewise smooth coefficients on a smooth manifold

B. A. Plamenevskii


Abstract: On a smooth compact manifold $\mathcal M$ without boundary, the $C^*$-algebra $\mathcal A$ generated on $L_2(\mathcal M)$ by the operators of two classes is considered. One class consists of zero order pseudodifferential operators with smooth symbols. The other class comprises the operators of multiplication by functions (“coefficients”) that may have discontinuities along a given collection of submanifolds (with boundary) of various dimensions; the submanifolds may intersect under nonzero angles. The situation is described formally by a stratification of the manifold $\mathcal M$. All the equivalence classes of irreducible representations of $\mathcal A$ are listed with a detailed proof. A solving composition series in $\mathcal A$ is constructed. This is a finite sequence of ideals $\{0\}=I_{-1}\subset I_0\subset…\subset I_N=\mathcal A$ whose subquotients $I_j/I_{j-1}$ are isomorphic to algebras of continuous functions with compact values; such operator-valued functions are defined on locally compact spaces and tend to zero at infinity.

Keywords: $C^*$-algebra, stratification, composition series, pseudodifferential operator.

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English version:
St. Petersburg Mathematical Journal, 2010, 21:2, 317–351

Bibliographic databases:

MSC: 46L45, 47G30
Received: 20.08.2008

Citation: B. A. Plamenevskii, “Solvability of the algebra of pseudodifferential operators with piecewise smooth coefficients on a smooth manifold”, Algebra i Analiz, 21:2 (2009), 214–263; St. Petersburg Math. J., 21:2 (2010), 317–351

Citation in format AMSBIB
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\by B.~A.~Plamenevskii
\paper Solvability of the algebra of pseudodifferential operators with piecewise smooth coefficients on a~smooth manifold
\jour Algebra i Analiz
\yr 2009
\vol 21
\issue 2
\pages 214--263
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2553048}
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\transl
\jour St. Petersburg Math. J.
\yr 2010
\vol 21
\issue 2
\pages 317--351
\crossref{https://doi.org/10.1090/S1061-0022-10-01097-6}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vasilyev V., “on the Asymptotic Expansion of Certain Plane Singular Integral Operators”, Bound. Value Probl., 2017, 16  crossref  mathscinet  isi  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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