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A topological approach to unitary spectral flow via continuous enumeration of eigenvalues. (English) Zbl 1481.58011

This paper is a well-written, well-organized and well-detailed. It brings a valuable contribution to the study of the Atiyah-Patodi-Singer (APS) spectral flow (and Toeplitz operators, too).
The authors give an infinite-dimensional version of the well-known result of {T. Kato}: for each continuous \(1\)-parameter family of \(n\times n\)-matrices, the eigenvalues of the family can be chosen continuously. The key ingredient of their result is an alternative approach to the APS spectral flow, based on Kato’s continuous enumeration of eigenvalues (see Theorem 4.12). It is closely related to Dold-Thom’s famous-theorem (in algebraic topology).
In my opinion, it would be interesting to refine the main result into the situation of coefficients in an arbitrary unital \(C^*\)-algebra.

MSC:

58J30 Spectral flows
55Q52 Homotopy groups of special spaces
47A10 Spectrum, resolvent
47A55 Perturbation theory of linear operators
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