Lie isomorphisms in *-prime GPI rings with involution (Q1582148)

Let \(R\) and \(S\) be *-prime GPI rings with involution, with respective skew elements \(K\) and \(L\), and with respective extended centroids \(C\) and \(D\). The paper deals with Herstein's long-standing problem of describing the form of a Lie isomorphism \(\alpha\colon[K,K]/[K,K]\cap C\to[L,L]\cap D\). The main result states that, roughly speaking, \(\alpha\) is determined by a related associative isomorphism if both involutions are of the second kind, and that \(\alpha\) cannot exist if the involutions are of different kinds (modulo some low-dimensional counterexamples). We remark that using the theory of functional identities, just recently all of Herstein's Lie map conjectures were completely solved in a series of papers by \textit{K. I. Beidar}, \textit{M. Brešar}, \textit{M. A. Chebotar} and \textit{W. S. Martindale} [On Herstein's Lie map conjectures, I-III, Trans. Am. Math. Soc. 353, No. 10, 4235-4260 (2001; Zbl 1019.16019), J. Algebra 238, No. 1, 239-264 (2001; Zbl 1019.16020), and J. Algebra 249, No. 1, 59-94 (2002; Zbl 1019.16021)].