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KK-theory and spectral flow in von Neumann algebras

Journal Article


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.

Authors


  •   Kaad, Jens (external author)
  •   Nest, Ryszard (external author)
  •   Rennie, Adam C.

Publication Date


  • 2012

Citation


  • Kaad, J., Nest, R. & Rennie, A. C. (2012). KK-theory and spectral flow in von Neumann algebras. Journal of K-Theory, 10 (2), 241-277.

Scopus Eid


  • 2-s2.0-84869191507

Has Global Citation Frequency


Number Of Pages


  • 36

Start Page


  • 241

End Page


  • 277

Volume


  • 10

Issue


  • 2

Place Of Publication


  • United Kingdom

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.

Authors


  •   Kaad, Jens (external author)
  •   Nest, Ryszard (external author)
  •   Rennie, Adam C.

Publication Date


  • 2012

Citation


  • Kaad, J., Nest, R. & Rennie, A. C. (2012). KK-theory and spectral flow in von Neumann algebras. Journal of K-Theory, 10 (2), 241-277.

Scopus Eid


  • 2-s2.0-84869191507

Has Global Citation Frequency


Number Of Pages


  • 36

Start Page


  • 241

End Page


  • 277

Volume


  • 10

Issue


  • 2

Place Of Publication


  • United Kingdom