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integral formula

Journal of the Institute of Mathematics of Jussieu (1)

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SPECTRAL FLOW AND WINDING NUMBER IN VON NEUMANN ALGEBRAS
Charlotte Wahl
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 7 / Issue 3 / July 2008

Published online by Cambridge University Press:

27 February 2008, pp. 589-619

Print publication:

July 2008
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We introduce a new topology, weaker than the gap topology, on the space of self-adjoint operators affiliated to a semifinite von Neumann algebra. We define the real-valued spectral flow for a continuous path of self-adjoint Breuer–Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips's analytical definition and derive integral formulae for the spectral flow for certain paths of unbounded operators with common domain, generalizing those of Carey and Phillips. Furthermore, we prove the homotopy invariance of the real-valued index. As an example we consider invariant symmetric elliptic differential operators on Galois coverings.