Values of multilinear graded $\ast$-polynomials on upper triangular matrices of small dimension

Abstract

Let $F$ be an algebraically closed field of characteristic different from $2$. We show that the images of multilinear $\ast$-polynomials on $U{T}_2$ are homogeneous vector spaces. An analogous result holds for $U{T}_3$ endowed with non-trivial grading. We further show that these results are optimal, in the following sense: there exist multilinear $+$graded polynomials whose image on $U{T}_n$ $(n\ge 3)$ with the trivial grading is not a vector space, and whose image on $(U{T}_n)$ $(n\ge 4)$ with the ${\mathbb{Z}}_n$-grading is also not a vector space. In particular, an analog of the L'vov-Kaplansky conjecture can not be expected in the setting of algebras with (graded) involutions.