# Signatures, sums of hermitian squares and positive cones on algebras with involution

## TL;DR

This paper explores the extension of real algebraic concepts, including quadratic forms and positive cones, to noncommutative algebras with involution using hermitian forms, aiming to unify and advance the theory.

## Contribution

It introduces a framework connecting real algebra, hermitian forms, and positive cones on algebras with involution, extending classical quadratic form theory to noncommutative settings.

## Key findings

- Develops a coherent theory linking positive cones and hermitian forms
- Extends classical real algebra concepts to noncommutative algebras
- Provides new insights into sums of hermitian squares in algebras with involution

## Abstract

We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.