Spectral flow in Fredholm modules, eta invariants and the JLO cocycle

By Alan L. Carey, J. Phillips. 2003

64 citations recorded by the source corpus.

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer-Fredholm operators in a type $I_{\infty}$ or $II_\infty$ von Neumann algebra ${\mathcal N}$. The framework is that of {\it odd unbounded} $\theta$-{\it summable} {\it Breuer-Fredholm modules} for a unital Banach *-algebra, $\mathcal A$. In the type $II_{\infty}$ 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 type I and type 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 $\mathcal A$.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/292e39e5229918bd07bb497f6d1e91ed7bfb05d9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Fedor Sukochev
2. Alan L. Carey
3. P. Dodds
4. Nurulla Azamov
5. A. Rennie
6. J. Phillips
7. E. Semenov

In Applied Mathematics

1. Fedor Sukochev
2. Alan L. Carey
3. P. Dodds
4. Nurulla Azamov

In Mathematical Physics

1. Alan L. Carey
2. Fedor Sukochev
3. P. Dodds
4. Nurulla Azamov
5. A. Rennie
6. D. Potapov
7. J. Phillips
8. E. Semenov

Citations in the corpus

• Operator Integrals, Spectral Shift, and Spectral Flow (2007, methodology): “The new approach of these papers allows the study of spectral flow between certain unbounded Breuer-Fredholm operators affiliated with a general semifinite von Neumann algebra [ 9 ].”