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The local index formula in semifinite Von Neumann algebras I: Spectral flow

Authors
Journal
Advances in Mathematics
0001-8708
Publisher
Elsevier
Publication Date
Volume
202
Issue
2
Identifiers
DOI: 10.1016/j.aim.2005.03.011
Keywords
  • Von Neumann Algebra
  • Fredholm Module
  • Cyclic Cohomology
  • Chern Character
  • Spectral Flow
Disciplines
  • Mathematics

Abstract

Abstract We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a * -subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer–Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a ( b , B ) -cocycle in the cyclic cohomology of A .

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