KK-theory and Spectral Flow in von Neumann Algebras
Operator Algebras
2007-05-23 v1
Abstract
We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow of the path coincides with the value of trace on the associated K-class. Given a semifinite spectral triple (A,H,D) relative to a semifinite von Neumann algebra N, we construct a class [D] in KK^1(A,N') such that, for a unitary u in A, the von Neumann spectral flow between D and u*Du is equal to the Kasparov product of [u] and [D].
Cite
@article{arxiv.math/0701326,
title = {KK-theory and Spectral Flow in von Neumann Algebras},
author = {Jens Kaad and Ryszard Nest and Adam Rennie},
journal= {arXiv preprint arXiv:math/0701326},
year = {2007}
}