Geodesic
Noncommutative elliptic theory. Examples
Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 204-223.

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

We study differential operators with coefficients in noncommutative algebras. As an algebra of coefficients, we consider crossed products corresponding to the action of a discrete group on a smooth manifold. We give index formulas for the Euler, signature, and Dirac operators twisted by projections over the crossed product. The index of Connes operators on the noncommutative torus is computed. 
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A. Yu. Savin; B. Yu. Sternin. Noncommutative elliptic theory. Examples. Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 204-223. http://geodesic.mathdoc.fr/item/TRSPY_2010_271_a14/

[1] Connes A., “$C^*$ algèbres et géométrie différentielle”, C. r. Acad. sci. Paris A, 290:13 (1980), 599–604 | MR | Zbl

[2] Connes A., Noncommutative geometry, Acad. Press, San Diego (CA), 1994 | MR | Zbl

[3] Connes A., Landi G., “Noncommutative manifolds, the instanton algebra and isospectral deformations”, Commun. Math. Phys., 221:1 (2001), 141–159 | DOI | MR | Zbl

[4] Landi G., van Suijlekom W., “Principal fibrations from noncommutative spheres”, Commun. Math. Phys., 260:1 (2005), 203–225 | DOI | MR | Zbl

[5] Connes A., Dubois-Violette M., “Noncommutative finite-dimensional manifolds. I: Spherical manifolds and related examples”, Commun. Math. Phys., 230:3 (2002), 539–579 | DOI | MR | Zbl

[6] Antonevich A.B., Lebedev A.V., “Funktsionalnye i funktsionalno-operatornye uravneniya. $C^*$-algebraicheskii podkhod”, Tr. S.-Peterb. mat. o-va, 6 (1998), 34–140 | MR

[7] Antonevich A.B., “Kraevye zadachi s silnoi nelokalnostyu dlya ellipticheskikh uravnenii”, Izv. AN SSSR. Ser. mat., 53:1 (1989), 3–24 | MR | Zbl

[8] Nazaikinskii V.E., Savin A.Yu., Sternin B.Yu., Elliptic theory and noncommutative geometry: Nonlocal elliptic operators, Oper. Theory: Adv. and Appl., 183, Birkhäuser, Basel, 2008 | MR | Zbl

[9] Nazaikinskii V.E., Savin A.Yu., Sternin B.Yu., “Ob indekse nelokalnykh ellipticheskikh operatorov”, DAN, 420:5 (2008), 592–597 | MR | Zbl

[10] Schweitzer L.B., “Spectral invariance of dense subalgebras of operator algebras”, Intern. J. Math., 4:2 (1993), 289–317 | DOI | MR | Zbl

[11] Kohn J.J., Nirenberg L., “An algebra of pseudo-differential operators”, Commun. Pure and Appl. Math., 18 (1965), 269–305 | DOI | MR | Zbl

[12] Conner P.E., Floyd E.E., Differentiable periodic maps, Acad. Press, New York, 1964

[13] Slominska J., “On the equivariant Chern homomorphism”, Bull. Acad. Pol. Sci. Sér. Sci. Math., Astron. et Phys., 24 (1976), 909–913 | MR | Zbl

[14] Baum P., Connes A., “Chern character for discrete groups”, A fête of topology, Acad. Press, Boston (MA), 1988, 163–232 | MR

[15] Atiyah M.F., Singer I.M., “The index of elliptic operators. III”, Ann. Math. Ser. 2, 87 (1968), 546–604 | DOI | MR | Zbl

[16] Atiyah M.F., Segal G.B., “The index of elliptic operators. II”, Ann. Math. Ser. 2, 87 (1968), 531–545 | DOI | MR | Zbl

[17] Ji R., “Trace invariant and cyclic cohomology of twisted group $C^*$-algebras”, J. Funct. Anal., 130:2 (1995), 283–292 | DOI | MR | Zbl

[18] Baaj S., “Calcul pseudo-différentiel et produits croisés de $C^*$-algèbres. I”, C. r. Acad. sci. Paris. Sér. 1, 307:11 (1988), 581–586 | MR | Zbl

[19] Baaj S., “Calcul pseudo-différentiel et produits croisés de $C^*$-algèbres. II”, C. r. Acad. sci. Paris. Sér. 1, 307:12 (1988), 663–666 | MR | Zbl

[20] Rosenberg J., “Noncommutative variations on Laplace's equation”, Anal. PDE, 1:1 (2008), 95–114 | DOI | MR | Zbl

[21] Rieffel M.A., “$C^*$-algebras associated with irrational rotations”, Pacif. J. Math., 93:2 (1981), 415–429 | DOI | MR | Zbl