Unitary units and skew elements in group algebras

 Let FG be the group algebra of a group G over a field F and let * denote the canonical involution of FG induced by the map g→g ⁻¹ ,gG. Let Un(FG)={uFG|uu ^* =1} be the group of unitary units of FG. In case char F=0, we classify the torsion groups G for which Un(FG) satisfies a group identity not vanishing on 2-elements. Along the way we actually prove that, in characteristic 0, the unitary group Un(FG) does not contain a free group of rank 2 if FG ⁻ , the Lie algebra of skew elements of FG, is Lie nilpotent. Motivated by this connection we characterize most groups G for which FG ⁻ is Lie nilpotent and char F≠2. Received: 15 July 2002 / Revised version: 28 December 2002 Published online: 24 April 2003 Research partially supported by MURST (Italy) and FAPESP and CNPq (Brazil). Mathematics Subject Classification (2000): Primary 16U60; Secondary 16W10, 20C07