KK -theory and spectral flow in von Neumann algebras

TY - JOUR

T1 - KK -theory and spectral flow in von Neumann algebras

AU - Kaad, Jens

AU - Nest, Ryszard

AU - Rennie, Adam

PY - 2012/10

Y1 - 2012/10

N2 - 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] ∈ KK 1 (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.

AB - 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] ∈ KK 1 (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.

U2 - 10.1017/is012003003jkt185

DO - 10.1017/is012003003jkt185

M3 - Journal article

SN - 1865-2433

VL - 10

SP - 241

EP - 277

JO - Journal of K-Theory

JF - Journal of K-Theory

IS - 2

ER -