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

Plamenevskiĭ, B.

doi:10.1090/S1061-0022-10-01097-6

Solvability of algebras of pseudodifferential operators with piecewise smooth coefficients on smooth manifolds
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by B. A. Plamenevskiĭ
Translated by: The author

St. Petersburg Math. J. 21 (2010), 317-351

DOI: https://doi.org/10.1090/S1061-0022-10-01097-6

Published electronically: January 26, 2010

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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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