Values of multilinear graded ⁎-polynomials on upper triangular matrices of small dimension.

Let F be an algebraically closed field of characteristic different from 2. We show that the images of multilinear ⁎-graded 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 ≥ 3) with the trivial grading is not a vector space, and whose image on U T n (n ≥ 4) with the natural 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. [ABSTRACT FROM AUTHOR]