Values of multilinear graded $*$-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 $*$-polynomials on $UT_2$ are homogeneous vector spaces. An analogous result holds for $UT_3$ 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 $UT_n$ $(n\geq 3)$ with the trivial grading is not a vector space, and whose image on $(UT_n)$ $(n\geq 4)$ 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.