Signatures for $J$-hermitians and $J$-unitaries on Krein spaces with Real structures

Abstract

For $J$-hermitian operators on a Krein space $(\mathcal{K},J)$ satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of $J$-hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary ${\mathbb{Z}}_2$-invariants are introduced to label their connected components. Related invariants are also analyzed for $J$-unitary operators.

This paper contains and considerably extends the analysis of version 1 of arXiv:1306.1816. The new version 2 of arXiv:1306.1816 only contains the applications

Cited by in corpus (1)

• Space and time dimensions of algebras with applications to Lorentzian noncommutative geometry and quantum electrodynamics