Geodesic
$C^*$-algebras of bisingular operators with discontinuous coefficients
Izvestiya. Mathematics , Tome 63 (1999) no. 2, pp. 367-399

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We investigate the Fredholm property in the $C^*$-algebra of bisingular operators with coefficients having discontinuities along smooth curves using a modification of the Gokhberg–Krupnik local principle, and study some canonical algebras of singular and bisingular operators and present an algorithm for calculating their index. 
@article{IM2_1999_63_2_a4,
     author = {L. I. Sazonov},
     title = {$C^*$-algebras of bisingular operators with discontinuous coefficients},
     journal = {Izvestiya. Mathematics },
     pages = {367--399},
     publisher = {mathdoc},
     volume = {63},
     number = {2},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_2_a4/}
}
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L. I. Sazonov. $C^*$-algebras of bisingular operators with discontinuous coefficients. Izvestiya. Mathematics , Tome 63 (1999) no. 2, pp. 367-399. http://geodesic.mathdoc.fr/item/IM2_1999_63_2_a4/