Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-13T04:01:10.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral Flows of Dilations of Fredholm Operators

Published online by Cambridge University Press:  20 November 2018

Giuseppe De Nittis  and
Hermann Schulz-Baldes
Giuseppe De Nittis
Affiliation:
Department Mathematik, Universit¨at Erlangen-Nürnberg, Germany. e-mail: denittis@math.fau.de, e-mail: schuba@mi.uni-erlangen.de
Hermann Schulz-Baldes
Affiliation:
Department Mathematik, Universit¨at Erlangen-Nürnberg, Germany. e-mail: denittis@math.fau.de, e-mail: 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.

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow that is shown to be equal to the index of the operator. This result is interpreted in terms of the K-theory of an associated mapping cone. It is then extended to connect Z2 indices of odd symmetric Fredholm operators to a Z2-valued spectral flow.

Keywords

19K5646L80spectral flowFredholm operatorsZ2 indices
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 1 , 01 March 2015 , pp. 51 - 68
Copyright
Copyright © Canadian Mathematical Society 2015

References

[APS] Atiyah, M. F., Patodi, V. K., and Singer, I. M., Spectral asymmetry and Riemannian geometry. III. Math. Proc. Cambridge Phil. Soc. 79.(1976), no. 1, 71-99 http://dx.doi.org/10.1017/S0305004100052105 Google Scholar
[AS] Atiyah, M. F. and Singer, I. M., Index theory for skew-adjoint Fredholm operators. Inst. Haute E tudes Sci. Publ. Math. 37(1969), 5–26.Google Scholar
[ASV] Avila, J. C., Schulz-Baldes, H., and Villegas-Blas, C., Topological invariants of edge states for periodic two-dimensional models. Math. Phys. Anal. Geom. 16 (2013), no. 2, 137–170. http://dx.doi.org/10.1007/s11040-012-9123-9 Google Scholar
[ASS] Avron, J., Seiler, R., and Simone, B. index of a pair of projections. J. Funct. Anal. 120 http://dx.doi.org/10.1006/jfan.1994.1031 Google Scholar
[BCP] 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 Sci. Publ., Hackensack, NJ, 2006, pp. 297-352.Google Scholar
[CP] Carey, A. and Phillips, J., Unbounded Fredholm modules and spectral flow. Canad. J. Math. 50(1998), no. 4, 673–718. http://dx.doi.org/10.4153/CJM-1998-038-x Google Scholar
[CPR] Carey, A., Phillips, J., and Rennie, A., Noncommutative Atiyah-Patodi-Singer boundary conditions and index pairings in KK-theory.J. Reine Angew. Math. 643(2010), 59–109 Google Scholar
[Con] Connes, A., Noncommutative geometry. Academic Press Inc., San Diego, CA, 1994.Google Scholar
[DS] De Nittis, G. and Schulz-Baldes, H., Spectralows associated to ux tubes. arxiv:1405.2054Google Scholar
[Hal] Halmos, P. R., Normal dilations and extension of operators. Summa Brasil Math. 2(1950) 125–134;Google Scholar
[HR] Higson, N. and Roe, J., Analytic K-homology. Oxford Mathematical Monographs, Oxford Science Publications, Oxford University Press, Oxford, 2000 Google Scholar
[Kat] Kato, T., Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, New York, 1966.Google Scholar
[Phi1] Phillips, J., Self-adjoint Fredholm operators and spectral flow. Canad. Math. Bull. 39 (1996), no. 4, 460–467. http://dx.doi.org/10.4153/CMB-1996-054-4 Google Scholar
[Phi2] Phillips, J., Spectralow in type I and II factors—a new approach. In: Cyclic cohomology and noncommutative geometry (Waterloo, ON, 1995), Fields Inst. Commun., 17, American Mathematical Society, Providence, RI, 1997, pp. 137–153.Google Scholar
[Put] Putnam, I. F., An excision theorem for the K-theory of C.*algebras. J. Operator Theory 38(1997), no. 1, 151–171.Google Scholar
[Sch] Schröder, H., K-theory for real C.-algebras and applications. Pitman Research Notes in Mathematics Series, z.§, Longman Scientiûc & Technical, Harlow; John Wiley & Sons, New York, 1993.Google Scholar
[SB] Schulz-Baldes, H., Z2 indices of odd symmetric Fredholm operators. arxiv:1311.0379v2Google Scholar