On *-orderable groups

We introduce the class of *-orderable groups as those groups G for which the complex group ring CG with the standard involution is *-orderable. In (Algebras and Representation Theory, to appear) the authors proved that residually torsion-free nilpotent groups are *-orderable and that every *-orderable group is orderable. We prove that being a *-orderable group is a local and residual property and deduce that the class of residually torsion-free nilpotent groups is strictly contained in the class of *-orderable groups. Further, it is proved that *-orderable groups are elementary (and even a quasi-variety) in the language of groups and contain certain normal series. In the last section, we give a countable family of orderable metabelian groups that are not *-orderable.