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

Authors

Alan L. Carey, Australian National University
John Phillips, University of Victoria
Adam Rennie, University of NewcastleFollow
Fyodor A. Sukochev, Flinders University

RIS ID

77520

Publication Details

Carey, A. L., Phillips, J., Rennie, A. & Sukochev, F. A. (2006). The local index formula in semifinite Von Neumann algebras I: spectral flow. Advances in Mathematics, 202 (2), 451-516.

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.