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Spectral Flow for Nonunital Spectral Triples

Published online by Cambridge University Press:  20 November 2018

A. L. Carey ,
V. Gayral ,
J. Phillips ,
A. Rennie  and
F. A. Sukochev
A. L. Carey
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra ACT, 0200 Australia and School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW, 2500 Australia e-mail: alan.carey@anu.edu.au, renniea@uow.edu.au
V. Gayral
Affiliation:
Laboratoire de Mathématiques, Université Reims Champagne-Ardenne, Moulin de la Housse-BP 1039, 51687 Reims France e-mail: victor.gayral@univ-reims.fr
J. Phillips
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria BC e-mail: johnphil@uvic.ca
A. Rennie
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra ACT, 0200 Australia and School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW, 2500 Australia e-mail: alan.carey@anu.edu.au, renniea@uow.edu.au
F. A. Sukochev
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Kensington NSW, 2052 Australia e-mail: f.sukochev@unsw.edu.au
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Abstract

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We prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a ${{C}^{*}}$-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow.

Keywords

46H30spectral triplespectral flowlocal index theorem
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 4 , 01 August 2015 , pp. 759 - 794
Copyright
Copyright © Canadian Mathematical Society 2015

References

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