C*-algebraic methods in spectral theory

Contact:
Serge Richard (richard@math.nagoya-u.ac.jp), Rm. 237 in Sci. Bldg. A

• Description : the pdf file

• Class dates :
  April 11, 18, 25
  May 2, 9, 16, 23, 30
  June 13, 20, 27
  July 4, 11, 18, 25
  

• Lecture notes : full document, 103 pages
  C*-algebraic methods in spectral theory
  

• Lecture notes : each chapter separately
  Title
  Contents
  Chapter 1: Linear operators on a Hilbert space
  Chapter 2: C*-algebras
  Chapter 3: Crossed product C*-algebras
  Chapter 4: Schroedinger operators and essential spectrum
  Chapter 5: Twisted crossed product C*-algebras
  Chapter 6: Pseudodifferential calculus
  Chapter 7: Magnetic systems
  Bibliography
  

• Students reports :
  Some exercises from B. Ding
  Some exercises from T. Fujitani
  The GNS representation from K. Miyabe
  The spectral theorem form Y. Sawada
  

• Additional material :
  Introduction , first lecture
  An introduction to C*-algebras, lecture notes written by Prof. Ivan F. Wilde (King's College London)
  A nice old paper on multiplier algebra, written by Prof. Robert C. Busby
  Cross product C*-algebras by locally compact groups, lecture notes written by P. Skoufranis (UCLA)
  C*-algebras, group actions and crossed products, lecture notes written by Piotr M. Soltan (University of Warsaw)
  

• References : (electronic version available upon request)
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  [ABG96] W.O. Amrein, A. Boutet de Monvel, V. Georgescu, C_o-groups, commutator methods and spectral theory of N-body Hamiltonians, Birkhauser, Basel, 1996.
  [AMP02] W.O. Amrein, M. Mantoiu, R. Purice, Propagation properties for Schroedinger operators affiliated with certain C*-algebras, Ann. Henri Poincaré 3 (2002), no. 6, 1215-1232.
  [BS70] R. Busby, H. Smith, Representations of twisted group algebras, Trans. Amer. Math. Soc. 149, 503-537, 1970.
  [DG13] E.B. Davies, V. Georgescu, C*-algebras associated with some second order differential operators, J. Operator Theory 70 (2013), no. 2, 437-450.
  [Fol95] G.B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.
  [GI02] V. Georgescu, A. Iftimovici, Crossed products of C*-algebras and spectral analysis of quantum Hamiltonians, Comm. Math. Phys. 228 (2002), no. 3, 519-560.
  [GI06] V. Georgescu, A. Iftimovici, Localizations at infinity and essential spectrum of quantum Hamiltonians I: General theory, Rev. Math. Phys. 18 (2006), no. 4, 417-483.
  [LMR10] M. Lein, M. Mantoiu, S. Richard, Magnetic pseudodifferential operators with coefficients in C*-algebras, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 755-788.
  [Man02] M. Mantoiu, C*-algebras, dynamical systems at infinity and the essential spectrum of generalized Schroedinger operators, J. Reine Angew. Math. 550 (2002), 211-229.
  [Man013] M. Mantoiu, On the essential spectrum of phase-space anisotropic pseudodifferential operators, Math. Proc. Cambridge Philos. Soc. 154 (2013), no. 1, 29-39.
  [MP04] M. Mantoiu, R. Purice, The magnetic Weyl calculus, J. Math. Phys. 45 (2004), no. 4, 1394-1417.
  [MPR05] M. Mantoiu, R. Purice, S. Richard, Twisted crossed products and magnetic pseudodifferential operators, in Advances in operator algebras and mathematical physics, 137-172, Theta Ser. Adv. Math., 5, Theta, Bucharest, 2005.
  [MPR07] M. Mantoiu, R. Purice, S. Richard, Spectral and propagation results for magnetic Schroedinger operators; a C*-algebraic framework, J. Funct. Anal. 250 (2007), no. 1, 42-67.
  [Mur90] G.J. Murphy, C*-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.
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  [PR89] J.A. Packer, I. Raeburn, Twisted crossed products of C*-algebras, Math. Proc. Camb. Phyl. Soc. 106, 293-311, 1989.
  [PR90] J.A. Packer, I. Raeburn, Twisted crossed products of C*-algebras II, Math. Ann. 287, 595-612, 1990.
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  [Ped89] G. Pedersen, Analysis now, Graduate Texts in Mathematics, 118, Springer-Verlag, New York, 1989.
  [Ric05] S. Richard, Spectral and scattering theory for Schroedinger operators with Cartesian anisotropy, Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, 73-111.
  [Sch73] L. Schwartz, Théorie des distributions, nouvelle edition, Hermann, 1973.
  [Tak02] M. Takesaki, Theory of operator algebras I, Reprint of the first (1979) edition, Encyclopaedia of Mathematical Sciences 124, Operator Algebras and Non-commutative Geometry 5. Springer-Verlag, Berlin, 2002.
  [Tes09] G. Teschl, Mathematical methods in quantum mechanics, with applications to Schroedinger operators, Graduate Studies in Mathematics 99, American Mathematical Society, Providence, RI, 2009.
  [Tom87] J. Tomiyama, Invitation to C*-algebras and topological dynamics, World Scientific Advanced Series in Dynamical Systems 3, World Scientific Publishing Co., Singapore, 1987.
  [W-O93] N.E. Wegge-Olsen, K-theory and C*-algebras, a friendly approach, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993.
  [Wil07] D.P. Williams, Crossed products of C*-algebras, Mathematical Surveys and Monographs, 134. American Mathematical Society, Providence, RI, 2007.