Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz

TL;DR

This paper classifies Lie skew-ideals in central simple algebras with involution and uses this classification to characterize noncommutative polynomials based on their trace properties in matrix algebras.

Contribution

It provides a complete classification of Lie skew-ideals in central simple algebras with involution and applies this to characterize polynomials via their values in matrix algebras.

Findings

Classification of Lie skew-ideals in central simple algebras with involution

Characterization of polynomials as sums of commutators and identities based on trace

Connection between polynomial values and trace-zero conditions in matrix algebras

Abstract

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of $d\times d$ matrices have zero trace.