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KK-theory and spectral flow in von Neumann algebras

Journal Article


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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, 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.

Authors


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

Publication Date


  • 2007

Citation


  • Kaad, J., Nest, R. & Rennie, A. C. (2007). KK-theory and spectral flow in von Neumann algebras. Journal of K-theory, 10 (2), 1-29.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1482&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/477

Number Of Pages


  • 28

Start Page


  • 1

End Page


  • 29

Volume


  • 10

Issue


  • 2

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.

Authors


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

Publication Date


  • 2007

Citation


  • Kaad, J., Nest, R. & Rennie, A. C. (2007). KK-theory and spectral flow in von Neumann algebras. Journal of K-theory, 10 (2), 1-29.

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=1482&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/477

Number Of Pages


  • 28

Start Page


  • 1

End Page


  • 29

Volume


  • 10

Issue


  • 2