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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
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Abstract

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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.

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