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

Published electronically:
January 26, 2010

MathSciNet review:
2553048

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Abstract: On a smooth compact manifold without boundary, the -algebra generated on 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 . All the equivalence classes of irreducible representations of are listed with a detailed proof. A solving composition series in is constructed. This is a finite sequence of ideals whose subquotients 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