On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem

Boris Sternin

Open Mathematics, Tome 9 (2011), p. 814-832 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider a class of nonlocal operators associated with an action of a compact Lie group G on a smooth closed manifold. Ellipticity condition and Fredholm property for elliptic operators are obtained. This class of operators is studied using pseudodifferential uniformization, which reduces the problem to a pseudodifferential operator acting in sections of infinite-dimensional bundles.

Détail

Publié le : 2011-01-01

Zbl 1241.58012

EUDML-ID : urn:eudml:doc:269578

Bibliographie

[1] Antonevich A., Lebedev A. Functional-Differential Equations. I. C*-Theory, Pitman Monogr. Surveys Pure Appl. Math., 70, Longman, Harlow, 1994

[2] Antonevich A.B., Lebedev A.V., Functional equations and functional operator equations. A C*-algebraic approach, In: Proceedings of the St. Petersburg Mathematical Society, VI, Amer. Math. Soc. Transl. Ser. 2, 199, American Mathematical Society, Providence, 2000, 25–116 | Zbl 1287.47039

[3] Atiyah M.F., Elliptic Operators and Compact Groups, Lecture Notes in Math., 401, Springer, Berlin-New York, 1974

[4] Atiyah M.F., Singer I.M., The index of elliptic operators. I, Ann. of Math., 1968, 87, 484–530 http://dx.doi.org/10.2307/1970715 | Zbl 0164.24001

[5] Babbage Ch., An essay towards the calculus of functions. II, Philos. Trans. Roy. Soc. London, 1816, 106, 179–256 http://dx.doi.org/10.1098/rstl.1816.0012

[6] Connes A., Cyclic cohomology and the transverse fundamental class of a foliation, In: Geometric Methods in Operator Algebras, Kyoto, 1983, Pitman Res. Notes Math. Ser., 123, Longman, Harlow, 1986, 52–144

[7] Connes A., Noncommutative Geometry, Academic Press, San Diego, 1994

[8] Hu X., Transversally elliptic operators, preprint available at http://arxiv.org/abs/math/0311069 | Zbl 1125.35302

[9] Kordyukov Yu.A., Index theory and non-commutative geometry on foliated manifolds, Russian Math. Surveys, 2009, 64(2), 273–391 http://dx.doi.org/10.1070/RM2009v064n02ABEH004616 | Zbl 1180.58006

[10] Kordyukov Yu.A., Transversally elliptic operators on G-manifolds of bounded geometry, Russ. J. Math. Phys., 1994, 2(2), 175–198 | Zbl 0904.58056

[11] Luke G., Pseudodifferential operators on Hilbert bundles, J. Differential Equations, 1972, 12(3), 566–589 http://dx.doi.org/10.1016/0022-0396(72)90026-5

[12] Nazaikinskii V.E., Savin A.Yu., Sternin B.Yu., Elliptic Theory and Noncommutative Geometry, Oper. Theory Adv. Appl., 183, Birkhäuser, Basel, 2008 | Zbl 1158.58013

[13] Nistor V., Weinstein A., Xu P., Pseudodifferential operators on differential groupoids, Pacific J. Math., 1999, 189(1), 117–152 http://dx.doi.org/10.2140/pjm.1999.189.117 | Zbl 0940.58014

[14] Pedersen G.K., C*-Algebras and their Automorphism Groups, London Math. Soc. Monogr., 14, Academic Press, London-New York, 1979

[15] Savin A.Yu., On the index of nonlocal elliptic operators for compact Lie groups, Cent. Eur. J. Math., 2011, 9(4) | Zbl 1248.58013

[16] Savin A., Sternin B., Boundary value problems on manifolds with fibered boundary, Math. Nachr., 2005, 278(11), 1297–1317 http://dx.doi.org/10.1002/mana.200410308 | Zbl 1078.58014

[17] Singer I.M., Recent applications of index theory for elliptic operators, In: Partial Differential Equations, Proc. Sympos. Pure Math., XXIII, Univ. California, Berkeley, 1971, American Mathematical Society, Providence, 1973, 11–31

[18] Sternin B.Yu., Shatalov V.E., An extension of the algebra of pseudodifferential operators, and some nonlocal elliptic problems, Russian Acad. Sci. Sb. Math., 1995, 81(2), 363–396 http://dx.doi.org/10.1070/SM1995v081n02ABEH003543

[19] Williams D.P., Crossed Products of C*-Algebras, Math. Surveys Monogr., 134, American Mathematical Society, Providence, 2007 | Zbl 1119.46002