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(Ms(F)), the image of f in Ms(F), contains all trace zero d×d matrices. In particular, the image of f in the algebra of all finitary matrices contains all trace zero finitary matrices.