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KK-theory and Spectral Flow in von Neumann Algebras

Jens Kaad·Ryszard Nest·Adam Rennie
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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].

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Jens Kaad, Ryszard Nest, Adam Rennie. 2007-01-11. KK-theory and Spectral Flow in von Neumann Algebras. https://arxiv.org/abs/math/0701326

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