KK -theory and spectral flow in von Neumann algebras

Jens Kaad; Ryszard Nest; Adam Rennie

doi:10.1017/is012003003jkt185

KK -theory and spectral flow in von Neumann algebras

Jens Kaad, Ryszard Nest, Adam Rennie

Institut for Matematiske Fag

13 Citationer (Scopus)

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, t) 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.

Originalsprog	Engelsk
Tidsskrift	Journal of K-Theory
Vol/bind	10
Udgave nummer	2
Sider (fra-til)	241-277
ISSN	1865-2433
DOI	https://doi.org/10.1017/is012003003jkt185
Status	Udgivet - okt. 2012

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10.1017/is012003003jkt185

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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] ∈ 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.",

author = "Jens Kaad and Ryszard Nest and Adam Rennie",

year = "2012",

month = oct,

doi = "10.1017/is012003003jkt185",

language = "English",

volume = "10",

pages = "241--277",

journal = "Journal of K-Theory",

issn = "1865-2433",

publisher = "Cambridge University Press",

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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 -

KK -theory and spectral flow in von Neumann algebras

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