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KK-Theory and Spectral Flow in von Neumann Algebras

Published online by Cambridge University Press:  04 April 2012

J. Kaad ,
R. Nest  and
A. Rennie
J. Kaad
Affiliation:
Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, jenskaad@hotmail.com
R. Nest
Affiliation:
Institute for Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark, rnest@math.ku.dk
A. Rennie
Affiliation:
Mathematical Sciences Institute, John Dedman Building, Australian National University, Acton 0200, ACT, Australia, adam.rennie@anu.edu.au
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Abstract

We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko(J).

Given a semifinite spectral triple (A, H, D) relative to (N, τ) with A separable, we construct a class [D] ∈ KK1(A, K(N)). For a unitary u ∈ A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.

Keywords

KK-theoryspectral flowvon Neumann algebra
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 2 , October 2012 , pp. 241 - 277
Copyright
Copyright © ISOPP 2012

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