Noncommutative geometry and classification of elliptic operators

V. E. Nazaikinskii ; A. Yu. Savin ; B. Yu. Sternin

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 29 (2008), pp. 131-164.

Résumé

The computation of a stable homotopic classification of elliptic operators is an important problem of elliptic theory. The classical solution of this problem is given by Atiyah and Singer for the case of smooth compact manifolds. It is formulated in terms of $K$-theory for a cotangent fibering of the given manifold. It cannot be extended for the case of nonsmooth manifolds because their cotangent fiberings do not contain all necessary information. Another Atiyah definition might fit in such a case: it is based on the concept of abstract elliptic operators and is given in term of $K$-homologies of the manifold itself (instead of its fiberings). Indeed, this theorem is recently extended for manifolds with conic singularities, ribs, and general so-called stratified manifolds: it suffices just to replace the phrase “smooth manifold” by the phrase “stratified manifold” (of the corresponding class). Thus, stratified manifolds is a strange phenomenon in a way: the algebra of symbols of differential (pseudodifferential) operators is quite noncommutative on such manifolds (the symbol components corresponding to strata of positive codimensions are operator-valued functions), but the solution of the classification problem can be found in purely geometric terms. In general, it is impossible for other classes of nonsmooth manifolds. In particular, the authors recently found that, for manifolds with angles, the classification is given by a $K$-group of a noncommutative $C^*$-algebra and it cannot be reduced to a commutative algebra if normal fiberings of faces of the considered manifold are nontrivial. Note that the proofs are based on noncommutative geometry (more exactly, the K-theory of $C^*$-algebras) even in the case of stratified manifolds though the results are “classical.” In this paper, we provide a review of the abovementioned classification results for elliptic operators on manifolds with singularities and corresponding methods of noncommutative geometry (in particular, the localization principle in $C^*$-algebras).

Bibliographie

[1] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Usp. mat. nauk, 19:3 (1964), 53–161 | MR | Zbl

[2] Vasilevskii N. L., “Printsipy lokalnosti v teorii operatorov”, Lineinye operatory v funktsionalnykh prostranstvakh, Tez. dokl. Severo-Kavkazskoi regionalnoi konferentsii, Groznyi, 1989, 32–33

[3] Kasparov G. G., “Topologicheskie invarianty ellipticheskikh operatorov. I. $K$-gomologii”, Izv. AN CCCR. Ser. mat., 39:4 (1975), 796–838 | MR | Zbl

[4] Kondratev V. A., “Kraevye zadachi dlya ellipticheskikh uravnenii v oblastyakh s konicheskimi i uglovymi tochkami”, Tr. Mosk. mat. ob-va, 16, 1967, 209–292

[5] Maslov V. P., Operatornye metody, Nauka, M., 1973 | MR

[6] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., Ob obschem printsipe lokalizatsii v $C^*$-algebrakh, podgotovleno k pechati

[7] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., “Psevdodifferentsialnye operatory na stratifitsirovannykh mnogoobraziyakh. I”, Diff. uravn., 43:4 (2007), 519–532 | MR | Zbl

[8] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., “O gomotopicheskoi klassifikatsii ellipticheskikh operatorov na stratifitsirovannykh mnogoobraziyakh”, Dokl. RAN, 408:5 (2006), 591–595 | MR

[9] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., Gomotopicheskaya klassifikatsiya ellipticheskikh operatorov na mnogoobraziyakh s uglami, 2007, podgotovleno k pechati

[10] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., “O gomotopicheskoi klassifikatsii ellipticheskikh operatorov na mnogoobraziyakh s uglami”, Dokl. RAN, 413:1 (2007), 16–19 | MR | Zbl

[11] Nazaikinskii V. E., Savin A. Yu., Sternin B. Yu., Shultse B.-V., “Ob indekse differentsialnykh operatorov na mnogoobraziyakh s rebrami”, Mat. sb., 196:9 (2005), 23–58 | MR | Zbl

[12] Plamenevskii B. A., Senichkin V. N., “Razreshimye algebry operatorov”, Algebra i analiz, 6:5 (1994), 1–87 | MR | Zbl

[13] Plamenevskii B. A., Senichkin V. N., “O klasse psevdodifferentsialnykh operatorov v $\mathbb R^m$ i na stratifitsirovannykh mnogoobraziyakh”, Mat. sb., 191:5 (2000), 109–142 | MR | Zbl

[14] Plamenevskii B. A., Senichkin V. N., “Predstavleniya $C^*$-algebr psevdodifferentsialnykh operatorov na kusochno-gladkikh mnogoobraziyakh”, Algebra i analiz, 13:6 (2001), 124–174 | MR | Zbl

[15] Simonenko I. B., “Novyi obschii metod issledovaniya lineinykh operatornykh uravnenii tipa singulyarnykh integralnykh uravnenii. I”, Izv. AN SSSR. Ser. mat., 29:3 (1965), 567–586 | MR | Zbl

[16] Antonevich A., Lebedev A., Functional differential equations. I. $C^*$-Theory, Longman, Harlow, 1994 | MR | Zbl

[17] Antonevich A., Belousov M., Lebedev A., Functional differential equations. II. $C^*$-Applications, Parts 1, 2, Longman, Harlow, 1998 | MR | Zbl

[18] Arveson W., An invitation to $C^*$-algebras, Springer-Verlag, New York–Heidelberg–Berlin, 1976 | MR | Zbl

[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, Bombay Colloq., 1964, Oxford University Press, London, 1964, 175–186 | MR

[21] Atiyah M. F., Singer I. M., “The index of elliptic operators on compact manifolds”, Bull. Amer. Math. Soc., 69 (1963), 422–433 | DOI | MR | Zbl

[22] Baum P., Douglas R. G., “$K$-homology and index theory”, Oper. Alg. Appl., Amer. Math. Soc., Providence, 1982, 117–173 | MR

[23] Brown L., Douglas R., Fillmore P., “Extensions of $C^*$-algebras and $K$-homology”, Ann. Math. (2), 105 (1977), 265–324 | DOI | MR | Zbl

[24] Blackadar B., $K$-Theory for operator algebras, Cambridge University Press, Cambridge, 1998, 300 pp. | MR | Zbl

[25] Dauns J., Hofmann K. H., Representation of rings by sections, Amer. Math. Soc., Providence, 1968 | MR

[26] Dixmier J., Les $C^*$-algebres et leurs representations, Gauthier-Villars, Paris, 1969 | MR | Zbl

[27] Egorov Yu., Schulze B.-W., Pseudo-differential operators, singularities, applications, Birkhäuser, Boston–Basel–Berlin, 1997 | MR | Zbl

[28] Gohberg I., Krupnik N., One-dimensional linear singular integral equations, Vol. I, II, Birkhäuser, Basel, 1992

[29] Haskell P., “Index theory of geometric Fredholm operators on varieties with isolated singularities”, $K$-Theory, 1:5 (1987), 457–466 | DOI | MR | Zbl

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

[31] Kasparov G., “Equivariant $KK$-theory and the Novikov conjecture”, Inv. Math., 91:1 (1988), 147–201 | DOI | MR | Zbl

[32] Lauter R., Moroianu S., “The index of cusp operators on manifolds with corners”, Ann. Global Anal. Geom., 21:1 (2002), 31–49 | DOI | MR | Zbl

[33] Le Gall P.-Y., Monthubert B., “$K$-theory of the indicial algebra of a manifold with corners”, $K$-Theory, 23:2 (2001), 105–113 | DOI | MR | Zbl

[34] Mazzeo R., “Elliptic theory of differential edge operators. I”, Comm. Partial Differential Equations, 16(10) (1991), 1615–1664 | DOI | MR | Zbl

[35] 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 | DOI | MR | Zbl

[36] Melrose R., “Transformation of boundary problems”, Acta Math., 147 (1981), 149–236 | DOI | MR | Zbl

[37] Melrose R., Analysis on manifolds with corners. Lecture Notes, Preprint, MIT, Cambrige, 1988 | MR

[38] Melrose R., “Pseudodifferential operators, corners, and singular limits”, Proceedings of the International Congress of Mathematicians, Kyoto, Springer, Berlin–Heidelberg–New York, 1990, 217–234 | MR

[39] Melrose R., Mendoza G. A., “Elliptic boundary problems on spaces with conical points” (St.-Jean-de-Monts, 1981), Journées Équ. Dériv. Part., no. 4, 1981, 21 pp.

[40] Melrose R., Nistor V., “$K$-theory of $C^*$-algebras of $b$-pseudodifferential operators”, Geom. Funct. Anal., 8:1 (1998), 88–122 | DOI | MR | Zbl

[41] Melrose R., Piazza P., “Analytic $K$-theory on manifolds with corners”, Adv. Math., 92:1 (1992), 1–26 | DOI | MR | Zbl

[42] Melrose R., Rochon F., Index in $K$-theory for families of fibred cusp operators, , 2005 arXiv: math.DG/0507590 | MR

[43] Monthubert B., “Pseudodifferential calculus on manifolds with corners and groupoids”, Proc. Amer. Math. Soc., 127:10 (1999), 2871–2881 | DOI | MR | Zbl

[44] Monthubert B., “Groupoids of manifolds with corners and index theory”, Groupoids in Analysis, Geometry, and Physics, Amer. Math. Soc., Providence, 2001, 147–157 | MR | Zbl

[45] Monthubert B., “Groupoids and pseudodifferential calculus on manifolds with corners”, J. Funct. Anal., 199:1 (2003), 243–286 | DOI | MR | Zbl

[46] Monthubert B., Nistor V., A topological index theorem for manifolds with corners, , 2005 arXiv: math.KT/0507601 | MR

[47] Nazaikinskii V., Rozenblioum G., Savin A., Sternin B., “Guillemin transform and Toeplitz Representations for operators on singular manifolds”, Spectral Geometry of Manifolds with Boundary, Amer. Math. Soc., Providence, 2005, 281–306 | MR | Zbl

[48] Nazaikinskii V., Savin A., Schulze B.-W., Sternin B., Elliptic theory on singular manifolds, CRC-Press, Boca Raton, 2005 | MR

[49] Nazaikinskii V., Savin A., Sternin B., Elliptic theory on manifolds with corners. I. Dual manifolds and pseudodifferential operators, , 2006 arXiv: math.OA/0608353

[50] Nazaikinskii V., Savin A., Sternin B., Elliptic theory on manifolds with corners. II. Homotopy classification of elliptic operators, , 2006 arXiv: math.OA/0608354

[51] Nazaikinskii V., Savin A., Sternin B., On the homotopy classification of elliptic operators on stratified manifolds, , 2006 arXiv: math.KT/0608332 | MR

[52] Paschke W. L., “$K$-theory for commutants in the Calkin algebra”, Pacific J. Math., 95:2 (1981), 427–434 | MR | Zbl

[53] Pedersen G. K., $C^*$-Algebras and Their Automorphism Groups, Academic Press, London–New York, 1979 | MR | Zbl

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

[55] Schulze B.-W., Pseudodifferential operators on manifolds with singularities, North-Holland, Amsterdam, 1991 | MR | Zbl

[56] Schulze B.-W., Sternin B., Shatalov V., “Structure rings of singularities and differential equations”, Differential Equations, Asymptotic Analysis, and Mathematical Physics, Akademie Verlag, Berlin, 1996, 325–347 | MR

[57] Schulze B.-W., Sternin B., Shatalov V., Differential Equations on Singular manifolds. Semiclassical theory and operator algebras, Wiley-VCH Verlag, Berlin–New York, 1998 | MR

[58] Vasilevski N., “Convolution operators on standard CR-manifolds. II. Algebras of convolution operators on the Heisenberg group”, Integr. Equat. Oper. Theory, 19:3 (1994), 327–348 | DOI | MR | Zbl