An analytic approach to spectral flow in von Neumann algebras

By M. Benameur, Alan Carey, J. Phillips, A. Rennie, Fedor Sukochev, Krysztof P. Wojciechowski. 2005

69 citations recorded by the source corpus.

The analytic approach to spectral flow is about ten years old. In that time it has evolved to cover an ever wider range of examples. The most critical extension was to replace Fredholm operators in the classical sense by Breuer-Fredholm operators in a semifinite von Neumann algebra. The first part of this paper gives a brief account of this theory extending and refining earlier results. It is then applied in the latter parts of the paper to a series of examples. One of the most powerful tools is an integral formula for spectral flow. This integral formula was known for Dirac operators in a variety of forms ever since the fundamental papers of Atiyah, Patodi and Singer. One of the purposes of this exposition is to make contact with this early work so that one can understand the recent developments in a proper historical context. In addition we show how to derive these spectral flow formulae in the setting of Dirac operators on (non-compact) covering spaces of a compact spin manifold using the adiabatic method. Finally we relate our work to that of Coburn, Douglas, Schaeffer and Singer on Toeplitz operators with almost periodic symbol. We generalise their work to cover the case of matrix valued almost periodic symbols on \R^N using some ideas of Shubin.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/23b8c445f6b07effd47e289f8f28dcbc927b24a0

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In Algebra and Number Theory

1. Fedor Sukochev
2. Alan L. Carey
3. D. Potapov
4. A. Rennie
5. J. Phillips
6. E. Semenov

In Mathematical Physics

1. Fedor Sukochev
2. Alan L. Carey
3. D. Potapov
4. A. Rennie
5. V. Peller
6. J. Phillips
7. E. Semenov

Citations in the corpus

• Dixmier traces and some applications in non-commutative geometry (2006, background): “In this section our object is to describe the classes of positive Connes–Dixmier-measurable operators and positive Dixmier-measurable operators and to show that these two classes actually coincide.”
• Unbounded Fredholm modules and double operator integrals (2008, background): “The result above is of importance for the spectral flow theory, for which we refer to [1, 5, 6].”