Journal of Noncommutative Geometry

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Volume 2, Issue 2, 2008, pp. 141–193
DOI: 10.4171/JNCG/18

Published online: 2008-06-30

The Chern character of semifinite spectral triples

Alan L. Carey[1], John Phillips[2], Adam Rennie[3] and Fyodor A. Sukochev

(1) The Australian National University, Canberra, Australia
(2) University of Victoria, Canada
(3) University of Copenhagen

In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra $\mathcal{A}$ of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of $\mathcal{A}$. In this paper we show that this resolvent cocycle ‘almost’ represents the Chern character and assuming analytic continuation properties for zeta functions we show that the associated residue cocycle, which appears in our statement of the local index theorem does represent the Chern character.

Keywords: Von Neumann algebra, Fredholm module, cyclic cohomology, Chern character, spectral flow

Carey Alan, Phillips John, Rennie Adam, Sukochev Fyodor: The Chern character of semifinite spectral triples. J. Noncommut. Geom. 2 (2008), 141-193. doi: 10.4171/JNCG/18