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Solvability of algebras of pseudodifferential operators with piecewise smooth coefficients on smooth manifolds


Author: B. A. Plamenevskiĭ
Translated by: The author
Original publication: Algebra i Analiz, tom 21 (2009), nomer 2.
Journal: St. Petersburg Math. J. 21 (2010), 317-351
MSC (2000): Primary 46L45, 47G30
DOI: https://doi.org/10.1090/S1061-0022-10-01097-6
Published electronically: January 26, 2010
MathSciNet review: 2553048
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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 \dots \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.


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Additional Information

B. A. Plamenevskiĭ
Affiliation: Department of Mathematical Physics, Physics Institute, St. Petersburg State University, Ulyanovskaya 1, St. Petersburg 198504, Russia
Email: boris.plamen@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-10-01097-6
Keywords: $C^*$-algebra, stratification, composition series, pseudodifferential operator
Received by editor(s): August 20, 2008
Published electronically: January 26, 2010
Additional Notes: Supported by grant NSh-816.2008.1
Article copyright: © Copyright 2010 American Mathematical Society

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