Images of polynomials with involution on $2\times 2$ matrices

Abstract

Let $\mathbb{F}$ be a field and let $M_2(\mathbb{F})$ be the algebra of $2\times 2$ matrices endowed with an involution of the first kind. We study the image of multilinear $*$-polynomials evaluated on $M_2(\mathbb{F})$. For the transpose involution over $\mathbb{R}$, we show that the image is either a proper vector subspace or contains a basis of $M_2(\mathbb{R})$. For the symplectic involution over quadratically closed fields or over $\mathbb{R}$, we prove that the image is always a vector space, namely one of $\{0\}$, $\mathbb{F}$, $sl_2(\mathbb{F})$ or $M_2(\mathbb{F})$. As a byproduct, we complete a theorem of Brešar and Klep describing the linear span of the image of a $*$-polynomial on finite dimensional central simple algebras with involution of the first kind. Their result excluded algebras of dimensions 4 and 16; we settle both cases, extending the description to all dimensions greater than 1 (over $\mathbb{R}$ for the transpose involution, and over quadratically closed fields or $\mathbb{R}$ for the symplectic involution). We also classify all Lie skew-ideals of $M_4(\mathbb{F})$ over fields of characteristic zero.