Abstract
In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader’s convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.
Mathematics Subject Classification (2010). Primary 58J20; Secondary 58J28, 58J32, 19K56, 46L80, 58J22.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
A. Antonevich, M. Belousov, and A. Lebedev, Functional differential equations II C ∗ -applications, Parts 1, 2, Number 94, 95 in Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, 1998.
A. Antonevich and A. Lebedev, Functional-Differential Equations I C ∗ -Theory, Number 70 in Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman, Harlow, 1994.
A.B. Antonevich. Elliptic pseudodifferential operators with a finite group of shifts, Math. USSR-Izv., 7 (1973), 661–674.
A.B. Antonevich, Linear Functional Equations, Operator Approach, Universitetskoje, Minsk, 1988.
A.B. Antonevich, Strongly nonlocal boundary value problems for elliptic equations, Izv. Akad. Nauk SSSR Ser. Mat., 53(1) (1989), 3–24.
A.B. Antonevich and V.V. Brenner, On the symbol of a pseudodifferential operator with locally independent shifts, Dokl. Akad. Nauk BSSR, 24(10) (1980), 884–887.
A.B. Antonevich and A.V. Lebedev, Functional equations and functional operator equations A C ∗- algebraic approach, in Proceedings of the St. Petersburg Mathematical Society, Vol. VI, Amer. Math. Soc. Transl. Ser. 2 199, 2000, 25–116.
M.F. Atiyah. Elliptic Operators and Compact Groups, Lecture Notes in Mathematics 401, Springer-Verlag, Berlin, 1974.
M.F. Atiyah. K-Theory, Second Edition, The Advanced Book Program, Addison– Wesley, Inc., 1989.
M.F. Atiyah and G.B. Segal, The index of elliptic operators II, Ann. Math. 87 (1968), 531–545.
M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433.
M.F. Atiyah and I.M. Singer, The index of elliptic operators III, Ann. Math. 87 (1968), 546–604.
Ch. Babbage. An assay towards the calculus of functions, part II, Philos. Trans. of the Royal Society 106 (1816), 179–256.
P. Baum and A. Connes, Chern character for discrete groups, in A fˆete of topology, Academic Press, Boston, MA, 1988, 163–232.
A.V. Bitsadze and A.A. Samarskii, On some simple generalizations of linear elliptic boundary problems, Sov. Math., Dokl. 10 (1969), 398–400.
B. Blackadar, K-Theory for Operator Algebras, Second Edition, Mathematical Sciences Research Institute Publications 5, Cambridge University Press, 1998.
T. Carleman, Sur la théorie des équations intégrales et ses applications, Verh. Internat. Math.-Kongr. Zurich. 1, 1932, 138–151.
P.E. Conner and E.E. Floyd, Differentiable Periodic Maps, Academic Press, New York, 1964.
A. Connes, C ∗ -algèbres et géométrie différentielle, C. R. Acad. Sci. Paris Sér. A-B 290(13) (1980), A599–A604.
A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62 (1985), 257–360.
A. Connes, Cyclic cohomology and the transverse fundamental class of a foliation, in Geometric Methods in Operator Algebras, Pitman Res. Notes in Math. 123, Longman, Harlow, 1986.
A. Connes, Noncommutative Geometry, Academic Press Inc., San Diego, CA, 1994.
A. Connes and M. Dubois-Violette, Noncommutative finite-dimensional manifolds I, spherical manifolds and related examples, Comm. Math. Phys. 230(3) (2002), 539– 579.
A. Connes and G. Landi, Noncommutative manifolds, the instanton algebra and isospectral deformations, Comm. Math. Phys. 221(1) (2001), 141–159.
A. Connes and H. Moscovici, Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., E38, Friedr. Vieweg, Wiesbaden, 2008, 57–71.
I.M. Gelfand, On elliptic equations, Russian Math. Surveys 15(3) (1960), 113–127.
A. Gorokhovsky, Characters of cycles, equivariant characteristic classes and Fredholm modules, Comm. Math. Phys. 208(1) (1999), 1–23.
M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73.
A. Jaffe, A. Lesniewski, and K. Osterwalder, Quantum K-theory I. The Chern character, Comm. Math. Phys. 118(1) (1988), 1–14.
T. Kawasaki, The index of elliptic operators over V -manifolds, Nagoya Math. J. 84 (1981), 135–157.
Yu.A. Kordyukov, Transversally elliptic operators on G-manifolds of bounded geometry, Russian J. Math. Phys. 2(2) (1994), 175–198.
Yu.A. Kordyukov, Transversally elliptic operators on G-manifolds of bounded geometry II, Russian J. Math. Phys. 3(1) (1995), 41–64.
Yu.A. Kordyukov, Index theory and non-commutative geometry on foliated manifolds, Russ. Math. Surv. 64(2) (2009), 273–391.
G. Landi and W. van Suijlekom, Principal fibrations from noncommutative spheres, Comm. Math. Phys. 260(1) (2005), 203–225.
G. Luke, Pseudodifferential operators on Hilbert bundles, J. Diff. Equations 12 (1972), 566–589.
A.S. Mishchenko and A.T. Fomenko, The index of elliptic operators over C ∗ -algebras, Izv. Akad. Nauk SSSR Ser. Mat. 43(4) (1979), 831–859, 967.
H. Moscovici, Local index formula and twisted spectral triples, in Quanta of Maths, Clay Math. Proc. 11, Amer. Math. Soc., Providence, RI, 2010, 465–500.
V.E. Nazaikinskii, A.Yu. Savin, and B.Yu. Sternin, Elliptic Theory and Noncommutative Geometry, Operator Theory: Advances and Applications 183, Birkhäuser Verlag, Basel, 2008.
E. Park, Index theory of Toeplitz operators associated to transformation group C ∗ - algebras, Pacific J. Math. 223(1) (2006), 159–165.
A.L.T. Paterson, Amenability, Mathematical Surveys and Monographs 29, American Mathematical Society, Providence, RI, 1988.
G.K. Pedersen, C ∗ -Algebras and Their Automorphism Groups, London Mathematical Society Monographs 14, Academic Press, London–New York, 1979.
D. Perrot. A Riemann-Roch theorem for one-dimensional complex groupoids. Comm. Math. Phys., 218(2):373–391, 2001.
D. Perrot, Localization over complex-analytic groupoids and conformal renormalization, J. Noncommut. Geom. 3(2) (2009), 289–325.
D. Perrot, On the Radul cocycle, Oberwolfach reports (2011), 53–55. DOI: 10.4171/OWR/2011/45.
L.E. Rossovskii, Boundary value problems for elliptic functional-differential equations with dilatation and contraction of the arguments, Trans. Moscow Math. Soc. (2001), 185–212.
A. Savin, E. Schrohe, and B. Sternin, On the index formula for an isometric diffeomorphism, arXiv:1112.5515, 2011.
A. Savin, E. Schrohe, and B. Sternin, Uniformization and an index theorem for elliptic operators associated with diffeomorphisms of a manifold, arXiv:1111.1525, 2011.
A. Savin and B. Sternin, Index defects in the theory of nonlocal boundary value problems and the η-invariant, Sbornik: Mathematics 195(9) (2004), 1321–1358.
A. Savin and B. Sternin, Index of elliptic operators for a diffeomorphism. Journal of Noncommutative Geometry, V. 7, 2013. Preliminary Version: arxiv:1106.4195, 2011.
A.Yu. Savin, On the index of nonlocal elliptic operators for compact Lie groups, Cent. E ur. J. Math. 9(4) (2011), 833–850.
A.Yu. Savin, On the index of nonlocal operators associated with a nonisometric diffeomorphism, Mathematical Notes 90(5) (2011), 701–714.
A.Yu. Savin, On the symbol of nonlocal operators in Sobolev spaces, Differential Equations, 47(6) (2011), 897–900.
A.Yu. Savin and B.Yu. Sternin, Index of nonlocal elliptic operators over C ∗ -algebras, Dokl. Math. 79(3) (2009), 369–372.
A.Yu. Savin and B.Yu. Sternin, Noncommutative elliptic theory. Examples, Proceedings of the Steklov Institute of Mathematics 271 (2010), 193–211.
A.Yu. Savin and B.Yu. Sternin, Nonlocal elliptic operators for compact Lie groups, Dokl. Math. 81(2) (2010), 258–261.
A.Yu. Savin. On the index of elliptic operators associated with a diffeomorphism of a manifold, Doklady Mathematics 82(3) (2010), 884–886.
A.Yu. Savin and B.Yu. Sternin. On the index of noncommutative elliptic operators over C ∗-algebras, Sbornik: Mathematics 201(3) (2010), 377–417.
A.Yu. Savin and B.Yu. Sternin, Nonlocal elliptic operators for the group of dilations, Sbornik: Mathematics 202(10) (2011), 1505–1536.
A.Yu. Savin, B.Yu. Sternin, and E. Schrohe. Index problem for elliptic operators associated with a diffeomorphism of a manifold and uniformization, Dokl. Math. 84(3) (2011), 846–849.
L.B. Schweitzer, Spectral invariance of dense subalgebras of operator algebras, Internat. J. Math. 4(2) (1993), 289–317.
I.M. Singer, Recent applications of index theory for elliptic operators, in Partial Differential Equations, Proc. Sympos. Pure Math., Vol. XXIII, Amer. Math. Soc., Providence, R.I., 1973, 11–31.
A.L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel-Boston-Berlin, 1997.
A.L. Skubachevskii, Nonclassical boundary-value problems I, Journal of Mathematical Sciences 155(2) (2008), 199–334.
A.L. Skubachevskii, Nonclassical boundary-value problems II, Journal of Mathematical Sciences 166(4) (2010), 377–561.
J. Slominska, On the equivariant Chern homomorphism, Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys., 24 (1976), 909–913.
B.Yu. Sternin, On a class of nonlocal elliptic operators for compact Lie groups. Uniformization and finiteness theorem, Cent. E ur. J. Math. 9(4) (2011), 814–832.
M. Vergne, Equivariant index formulas for orbifolds, Duke Math. J. 82(3) (1996), 637–652.
D.P. Williams, Crossed Products of C ∗ -Algebras, Mathematical Surveys and Monographs 134, American Mathematical Society, Providence, RI, 2007.
G. Zeller-Meier, Produits croisés d’une C ∗ -algèbre par un groupe d’automorphismes, J. Math. Pures Appl. 47(9) (1968), 101–239.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Savin, A., Sternin, B. (2013). Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds. In: Molahajloo, S., Pilipović, S., Toft, J., Wong, M. (eds) Pseudo-Differential Operators, Generalized Functions and Asymptotics. Operator Theory: Advances and Applications, vol 231. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0585-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0585-8_1
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0584-1
Online ISBN: 978-3-0348-0585-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)