Stability index for algebras with involution

TL;DR

This paper introduces the stability index for algebras with involution, building on previous work on signatures of hermitian forms, and explores its properties and implications.

Contribution

It presents a new concept, the stability index, providing an alternative perspective on signatures of hermitian forms over algebras with involution.

Findings

Definition and initial properties of the stability index

Connection between the stability index and total signatures

Enhanced understanding of signatures as continuous functions

Abstract

In earlier work we developed the theory of signatures of hermitian forms over algebras with involution with respect to orderings on the base field of the algebra and obtained in particular that the total signature of a hermitian form is a continuous function from the space of orderings of that field to $\mathbb{Z}$. In this note we give another presentation of signatures and also introduce and study the stability index of algebras with involution.