Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-06-03T02:59:42.316Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Flow Argument Localizing an Odd Index Pairing

Part of: Selfadjoint operator algebras $K$-theory and operator algebras

Published online by Cambridge University Press:  07 January 2019

Terry A. Loring  and
Hermann Schulz-Baldes
Terry A. Loring
Affiliation:
University of New Mexico, Albuquerque, NM 87131, United States Email: loring@math.unm.edu
Hermann Schulz-Baldes
Affiliation:
Department Mathematik, Friedrich-Alexander-Universitaet Erlangen-Nuernberg, 91058 Erlangen, Germany Email: schuba@mi.uni-erlangen.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An odd Fredholm module for a given invertible operator on a Hilbert space is specified by an unbounded so-called Dirac operator with compact resolvent and bounded commutator with the given invertible. Associated with this is an index pairing in terms of a Fredholm operator with Noether index. Here it is shown by a spectral flow argument how this index can be calculated as the signature of a finite dimensional matrix called the spectral localizer.

Keywords

index pairingspectral flowtopological materials

MSC classification

Primary: 19K56: Index theory
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 2 , June 2019 , pp. 373 - 381
Copyright
© Canadian Mathematical Society 2018 

Footnotes

The authors thank the Simons Foundation (CGM 419432), the NSF (DMS 1700102), and the DFG (SCHU 1358/3-4) for financial support.

References

Atiyah, M. F., Patodi, V. K., and Singer, I. M., Spectral asymmetry and Riemannian geometry. III . Math. Proc. Cambridge Philos. Soc. 79(1976), 71–99. https://doi.org/10.1017/S0305004100052105.Google Scholar
Benameur, M.-T., Carey, A. L., Phillips, J., Rennie, A., Sukochev, F. A., and Wojciechowski, K. P., An analytic approach to spectral flow in von Neumann algebras . In: Analysis, geometry and topology of elliptic operators, World Scientific, Hackensack, NJ, 2006, pp. 297–352.Google Scholar
Carey, A. L. and Phillips, J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle . K-Theory 31(2004), 135–194. https://doi.org/10.1023/B:KTHE.0000022922.68170.61.Google Scholar
Connes, A., Noncommutative geometry. Academic Press, San Diego, CA, 1994.Google Scholar
De Nittis, G. and Schulz-Baldes, H., Spectral flows of dilations of Fredholm operators . Canad. Math. Bull. 58(2015), 51–68. https://doi.org/10.4153/CMB-2014-055-3.Google Scholar
Gracia-Bondía, J. M., Várilly, J. C., and Figueroa, H., Elements of noncommutative geometry . Birkhäuser Advanced Texts. Birkhäuser Boston, Boston, MA, 2013.Google Scholar
Grossmann, J. and Schulz-Baldes, H., Index pairings in presence of symmetries with applications to topological insulators . Comm. Math. Phys. 343(2016), 477–513. https://doi.org/10.1007/s00220-015-2530-6.Google Scholar
Loring, T. A., K-theory and pseudospectra for topological insulators . Ann. Physics 356(2015), 383–416. https://doi.org/10.1016/j.aop.2015.02.031.Google Scholar
Loring, T. A. and Schulz-Baldes, H., Finite volume calculations of K-theory invariants . New York J. Math. 22(2017), 1111–1140.Google Scholar
Phillips, J., Self-adjoint Fredholm operators and spectral flow . Canad. Math. Bull. 39(1996), 460–467. https://doi.org/10.4153/CMB-1996-054-4.Google Scholar
Phillips, J., Spectral flow in type I and type II factors—a new approach. In: Cyclic cohomology and noncommutative geometry, Fields Inst. Commun., 17, American Mathematical Society, Proidence, RI, pp. 137–153.Google Scholar
Prodan, E. and Schulz-Baldes, H., Bulk and boundary invariants for complex topological insulators: From K-theory to physics . Springer Int. Pub., Szwitzerland, 2016. https://doi.org/10.1007/978-3-319-29351-6.Google Scholar