Spectral Flow in Fredholm Modules, Eta Invariants and the JLO Cocycle |
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Authors: | Alan Carey and John Phillips |
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Institution: | (1) Mathematical Sciences Institute, Australian National University, Canberra, ACT, Australia;(2) Department of Mathematics and Statistics, University of Victoria, Vic., BC, Canada, V8W 3P4 |
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Abstract: | We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer–Fredholm operators in a type I or II von Neumann algebra N. The framework is that of odd unbounded -summable Breuer–Fredholm modules for a unital Banach *-algebra, A. In the type II case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzler s idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the types I and II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the K-homology and K-theory of A. |
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Keywords: | spectral flow -summable Fredholm module" target="_blank">gif" alt="THgr" align="BASELINE" BORDER="0">-summable Fredholm module eta invariant index |
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