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Integral Formula for Spectral Flow for $p$-Summable Operators

Part of: Infinite-dimensional manifolds $K$-theory and operator algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  07 January 2019

Magdalena Cecilia Georgescu
Magdalena Cecilia Georgescu*
Affiliation:
Department of Mathematics, Ben Gurion University, 8410501 Be’er Sheva, Israel Email: [email protected]
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Abstract

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Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

Keywords

spectral flowp-summable Fredholm module

MSC classification

Primary: 19K56: Index theory
Secondary: 46L87: Noncommutative differential geometry 58B34: Noncommutative geometry (à la Connes)
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 2 , April 2019 , pp. 337 - 379
Copyright
© Canadian Mathematical Society 2018 

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