Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices

Abstract

Let $F$ be an infinite field and let $f$ be a nonzero multilinear polynomial with coefficients in $F$. We prove that for every positive integer $d$ there exists a positive integer $s$ such that $f(M_s(F))$, the image of $f$ in $M_s(F)$, contains all trace zero $d\times d$ matrices. In particular, the image of $f$ in the algebra of all finitary matrices contains all trace zero finitary matrices.

11 pages, 0 figures, submited to Linear Algebra and Its Applications

References in corpus (2)

• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• Multilinear Polynomials Are Surjective on Algebras With Surjective Inner Derivations

Cited by in corpus (1)

• The Mesyan Conjecture: a restatement and a correction