Geodesic
On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols
Izvestiya. Mathematics, Tome 23 (1984) no. 3, pp. 525-544 
B. A. Plamenevskii; V. N. Senichkin. On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols. Izvestiya. Mathematics, Tome 23 (1984) no. 3, pp. 525-544. http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a6/
@article{IM2_1984_23_3_a6,
     author = {B. A. Plamenevskii and V. N. Senichkin},
     title = {On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols},
     journal = {Izvestiya. Mathematics},
     pages = {525--544},
     year = {1984},
     volume = {23},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a6/}
}
TY  - JOUR
AU  - B. A. Plamenevskii
AU  - V. N. Senichkin
TI  - On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols
JO  - Izvestiya. Mathematics
PY  - 1984
SP  - 525
EP  - 544
VL  - 23
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a6/
LA  - en
ID  - IM2_1984_23_3_a6
ER  - 
%0 Journal Article
%A B. A. Plamenevskii
%A V. N. Senichkin
%T On the spectrum of $C^*$-algebras generated by pseudodifferential operators with discontinuous symbols
%J Izvestiya. Mathematics
%D 1984
%P 525-544
%V 23
%N 3
%U http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a6/
%G en
%F IM2_1984_23_3_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Plamenevskii B. A., Senichkin V. N., “O spektre $C^*$-algebr, porozhdennykh psevdodifferentsialnymi operatorami s izolirovannymi osobennostyami simvolov”, Dokl. AN SSSR, 261:6 (1981), 1304–1306 | MR | Zbl

[2] Plamenevskii B. A., “Ob ogranichennosti singulyarnykh integralov v prostranstvakh s vesom”, Matem. sb., 76:4 (1968), 573–592 | MR | Zbl

[3] Diksme Zh., $C^*$-algebry i ikh predstavleniya, Nauka, M., 1974 | MR

[4] Plamenevskii B. A., “Ob algebrakh, porozhdennykh psevdodifferentsialnymi operatorami s izolirovannymi osobennostyami simvolov”, Dokl. AN SSSR, 248:2 (1979), 298–302 | MR | Zbl

[5] Plamenevskii B. A., “Ob algebre psevdodifferentsialnykh operatorov v prostranstvakh s vesovymi normami”, Matem. sb., 106:2 (1978), 296–320 | MR | Zbl

[6] Plamenevskii B. A., “O meromorfnykh psevdodifferentsialnykh operator-funktsiyakh”, Izvestiya vuzov, “Matematika”, 1978, no. 4, 79–90 | MR | Zbl

[7] Gokhberg I. Ts., Krupnik N. Ya., “Ob algebre, porozhdennoi odnomernymi singulyarnymi integralnymi operatorami s kusochno-nepreryvnymi koeffitsientami”, Funkts. analiz i ego pril., 4:3 (1970), 26–36 | MR | Zbl