The local index formula in semifinite Von Neumann algebras I: Spectral flow

• Published in: Advances in mathematics (New York. 1965) Vol. 202; no. 2; pp. 451 - 516
• Main Authors: Carey, Alan L., Phillips, John, Rennie, Adam, Sukochev, Fyodor A.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-06-2006
• Subjects: Chern character; Cyclic cohomology; Fredholm module; Spectral flow; Von Neumann algebra; Von Neumann algebra; Fredholm module; primary: 19K56; Chern character; Cyclic cohomology; Spectral flow; secondary: 58B30
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2005.03.011

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a
*
-subalgebra
A
of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer–Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a
(
b
,
B
)
-cocycle in the cyclic cohomology of
A
.