2007/01/11 by Jens Kaad, Kaad, Jens, Ryszard Nest +3 · 15 citations
Mathematics · #Abelian von Neumann algebra #Advanced Operator Algebra Research #Advanced Topics in Algebra #Affiliated operator #Algebra over a field #Algebra representation #Crossed product #FOS: Mathematics #Flow (mathematics) #Geometry #Hilbert space #Ideal (ethics) #Invertible matrix #Jordan algebra #Mathematics #Operator Algebras (math.OA) #Philosophy #Pure mathematics #Spectral Theory in Mathematical Physics #Spectral triple #Trace class #Unitary state #Von Neumann algebra #Von Neumann architecture #Von Neumann's theorem #math.OA
paper · pdf · doi:10.48550/arxiv.math/0701326
published in arXiv (Cornell University) (Cornell University)
arxiv created 2007/01/11 · openalex publication_date 2007/01/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 K0(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 KK1(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].