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For $J$-hermitian operators on a Krein space $(\mathcal{K},J)$ satisfying anadequate Fredholm property, a global Krein signature is shown to be a homotopyinvariant. It is argued that this global signature is a generalization of theNoether index. When the Krein space has a supplementary Real structure, thesets of $J$-hermitian Fredholm operators with Real symmetry can be retracted tocertain 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.

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