Values of Noncommutative Polynomials, Lie Skew-Ideals and Tracial Nullstellensätze

A subspace of an algebra with involution is called a {\it Lie skew-ideal} if it is closed under Lie products with \emph{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.

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Published 1 January 2009