Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze.
Let \(F\) be a field of characteristic \(q\geq 0\), \(A\) an associative \(F\)-algebra of finite dimension \([A:F]\), \(Z\) the centre of \(A\), and for arbitrary subsets \(\Sigma_1,\Sigma_2\subseteq A\), let \([\Sigma_1,\Sigma_2]\) be the \(F\)-subspace of \(A\) generated by the set of commutators \([\tau_1,\tau_2]\), \(\tau_j\in\Sigma_j\), \(j=1,2\). It is well-known that \(A\) is a Lie algebra over \(F\) with respect to the addition in \(A\) and the Lie product \([x,y]:=xy-yx\), for \(x,y\in A\). The ideals of \(A\) with respect to this product are called Lie ideals of \(A\). Suppose further that \(q\neq 2\), \(Z=F\) and \(A\) is a simple associative algebra with involution \(*\). Denote by \(S\) the set of symmetric elements of \(A\), and by \(K\) is the set of skew-symmetric elements of \(A\) (with respect to \(*\)). By a Lie skew-ideal of \(A\), we mean an \(F\)-subspace \(I\subseteq A\), such that \([K,I]\subseteq I\). The paper under review shows that if \(*\) is of the second kind, then \(0\), \(Z\), \([A,A]\) and \(A\) are all Lie skew-ideals of \(A\). When \(*\) is of the first kind, and \([A:F]\neq 4,16\) or \(q\neq 3\), it proves that the set of Lie skew-ideals of \(A\) consists of \(0\), \(Z\), \(K\), \([S,K]\), \(S\), \(Z+K\), \([A,A]\) and \(A\). As an application, the authors deduce that a polynomial in noncommuting variables over \(F\) is a sum of commutators and a polynomial identity of \(d\times d\) matrices if and only if its values in the matrix algebra \(M_n(F)\) have zero trace.
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