Geodesic
On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras
Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 64 (2018) no. 1, pp. 164-179.

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

We consider calculation of a group of stable homotopic classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological $K$-groups of some spaces in the following cases: for boundary-value problems on manifolds with boundaries, for conjugation problems with conditions on a closed submanifold of codimension one, and for nonlocal problems with contractions. 
@article{CMFD_2018_64_1_a9,
     author = {A. Yu. Savin},
     title = {On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {164--179},
     publisher = {mathdoc},
     volume = {64},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a9/}
}
TY  - JOUR
AU  - A. Yu. Savin
TI  - On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras
JO  - Contemporary Mathematics. Fundamental Directions
PY  - 2018
SP  - 164
EP  - 179
VL  - 64
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a9/
LA  - ru
ID  - CMFD_2018_64_1_a9
ER  - 
%0 Journal Article
%A A. Yu. Savin
%T On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras
%J Contemporary Mathematics. Fundamental Directions
%D 2018
%P 164-179
%V 64
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a9/
%G ru
%F CMFD_2018_64_1_a9
A. Yu. Savin. On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras. Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 64 (2018) no. 1, pp. 164-179. http://geodesic.mathdoc.fr/item/CMFD_2018_64_1_a9/

[1] A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, “Smash product of a $C^*$-algegra by endomorphism, algebras of coefficients, and transfer-operators”, Mat. sb., 202:9 (2011), 3–34 (in Russian) | DOI | MR | Zbl

[2] M. I. Vishik, G. I. Eskin, “Convolution elliptic equations in a bounded domain and their applications”, Usp. mat. nauk, 22:1 (1965), 15–76 (in Russian) | MR | Zbl

[3] I. M. Gel'fand, “On elliptic equations”, Usp. mat. nauk, 15:3 (1960), 121–132 (in Russian) | MR | Zbl

[4] I. Ts. Gokhberg, M. G. Kreyn, “Systems of integral equations on semiaxis with kernels depending on difference of arguments”, Usp. mat. nauk, 13:2 (1958), 3–72 (in Russian) | MR | Zbl

[5] V. E. Nazaykinskiy, A. Yu. Savin, B. Yu. Sternin, “On homotopic classification of elliptic operators on stratified manifolds”, Izv. RAN. Ser. Mat., 71:6 (2007), 91–118 (in Russian) | DOI | MR | Zbl

[6] V. E. Nazaykinskiy, A. Yu. Savin, B. Yu. Sternin, “On homotopic classification of elliptic operators on manifolds with corners”, Dokl. RAN, 413:1 (2007), 16–19 (in Russian) | MR | Zbl

[7] V. E. Nazaykinskiy, A. Yu. Savin, B. Yu. Sternin, “Noncommutative geometry and classification of elliptic operators”, Sovrem. mat. Fundam. napravl., 29, 2008, 131–164 (in Russian) | MR

[8] V. E. Nazaykinskiy, A. Yu. Savin, B. Yu. Sternin, “Atiyah–Bott index on stratified manifolds”, Sovrem. mat. Fundam. napravl., 34, 2009, 100–108 (in Russian) | MR

[9] L. E. Rossovskiy, “Boundary-value problems for elliptic functional differential equations with contraction and dilatation of arguments”, Tr. Mosk. mat. ob-va, 62, 2001, 199–228 (in Russian) | MR | Zbl

[10] L. E. Rossovskiy, “Elliptic functional differential equations with contraction and dilatation of arguments of the unknown function”, Sovrem. mat. Fundam. napravl., 54, 2014, 3–138 (in Russian)

[11] L. E. Rossovskiy, A. L. Skubachevskiy, “Solvability and regularity of solutions of some classes of elliptic functional differential equations”, Itogi nauki i tekhn. Sovrem. mat. i ee pril., 66, 1999, 114–192 (in Russian) | MR | Zbl

[12] A. Yu. Savin, B. Yu. Sternin, “Elliptic operators in even subspaces”, Mat. sb., 190:8 (1999), 125–160 (in Russian) | DOI | MR | Zbl

[13] A. Yu. Savin, B. Yu. Sternin, “On the problem of homotopic classification of elliptic boundary-value problems”, Dokl. RAN, 377:2 (2001), 165–169 (in Russian) | MR | Zbl

[14] A. Yu. Savin, B. Yu. Sternin, “Defects of index in the theory of nonlocal boundary-value problems and the $\eta$-invariant”, Mat. sb., 195:9 (2004), 85–126 (in Russian) | DOI | MR | Zbl

[15] A. Yu. Savin, B. Yu. Sternin, “Homotopic classification of elliptic problems associated with actions of discrete groups on manifolds with boundaries”, Ufimsk. mat. zhurn., 8:3 (2016), 126–134 (in Russian) | MR

[16] A. Yu. Savin, B. Yu. Sternin, “Elliptic problems with dilatations and contractions on manifolds with boundaries. C-Theory”, Diff. uravn., 52:10 (2016), 1383–1392 (in Russian) | DOI | Zbl

[17] A. Yu. Savin, B. Yu. Sternin, “Elliptic differential problems with dilatations and contractions on manifolds with boundaries”, Diff. uravn, 53:5 (2017), 665–676 (in Russian) | DOI | MR | Zbl

[18] G. I. Eskin, Boundary-Value Problems for Elliptic Pseudodifferential Equations, Nauka, Moscow, 1973 (in Russian)

[19] Atiyah M. F., “Global theory of elliptic operators”, Proc. of the Int. Symposium on Functional Analysis, University of Tokyo Press, Tokyo, 1969, 21–30 | MR

[20] Atiyah M. F., Bott R., “The index problem for manifolds with boundary”, Differential Analysis: Papers presented at the international colloquium, Oxford University Press, London, 1964, 175–186 | MR

[21] Atiyah M., Patodi V., Singer I., “Spectral asymmetry and Riemannian geometry. III”, Math. Proc. Cambridge Philos. Soc., 79 (1976), 71–99 | DOI | MR | Zbl

[22] Atiyah M. F., Singer I. M., “The index of elliptic operators. I”, Ann. of Math., 87 (1968), 484–530 | DOI | MR | Zbl

[23] Baum P., Douglas R. G., “$K$-homology and index theory”, Operator Algebras and Applications, Am. Math. Soc., 1982, 117–173 | DOI | MR | Zbl

[24] Blackadar B., K-Theory for Operator Algebras, Cambridge University Press, 1998 | MR | Zbl

[25] Higson N., Roe J., Analytic K-homology, Oxford University Press, Oxford, 2000 | MR | Zbl

[26] Hörmander L., The Analysis of Linear Partial Differential Operators, v. III, Springer, Berlin–Heidelberg–New York–Tokyo, 1985 | MR | Zbl

[27] Kwaśniewski B. K., Lebedev A. V., “Crossed products by endomorphisms and reduction of relations in relative Cuntz–Pimsner algebras”, J. Funct. Anal., 264:8 (2013), 1806–1847 | DOI | MR | Zbl

[28] Lescure J.-M., “Elliptic symbols, elliptic operators and Poincaré duality on conical pseudomanifolds”, J. K-Theory, 4:2 (2009), 263–297 | DOI | MR | Zbl

[29] Melo S. T., Nest R., Schrohe E., “$C^*$-structure and $K$-theory of Boutet de Monvel's algebra”, J. Reine Angew. Math., 561 (2003), 145–175 | MR | Zbl

[30] Melo S. T., Schick Th., Schrohe E., “A $K$-theoretic proof of Boutet de Monvel's index theorem for boundary value problems”, J. Reine Angew. Math., 599 (2006), 217–233 | MR | Zbl

[31] Melrose R., Rochon F., “Index in K-theory for families of fibred cusp operators”, K-Theory, 37:1–2 (2006), 25–104 | DOI | MR | Zbl

[32] Monthubert B., Nistor V., “A topological index theorem for manifolds with corners”, Compos. Math., 148:2 (2012), 640–668 | DOI | MR | Zbl

[33] Nazaikinskii V., Savin A., Schulze B.-W., Sternin B., Elliptic Theory on Singular Manifolds, CRC-Press, Boca Raton, 2005 | MR

[34] Pimsner M., Voiculescu D., “Exact sequences for K-groups and Ext-groups of certain cross-product C-algebras”, J. Oper. Theory, 4 (1980), 93–118 | MR | Zbl

[35] Rempel S., Schulze B.-W., “Parametrices and boundary symbolic calculus for elliptic boundary problems without the transmission property”, Math. Nachr., 105 (1982), 45–149 | DOI | MR | Zbl

[36] Savin A., “Elliptic operators on manifolds with singularities and $K$-homology”, K-theory, 34:1 (2005), 71–98 | DOI | MR | Zbl