Free skew fields have many *-orderings

The notion of a *-ordering on a skew-field with involution was introduced by S. Holland in [J. Algebra 101 (1986) 16–46] as an analogue to the notion of a total ordering on a skew-field and developed further in a series of papers of T. Craven, I. Idris, M. Marshall and T. Smith. While it is well known that every free skew-field has a total ordering (see [J. Lewin, Trans. Amer. Math. Soc. 192 (1974) 339–346]), it has not been known so far whether every free skew-field with some natural involution has a *-ordering. The aim of this paper is to give an explicit construction of a class of *-orderings on free associative algebras and to prove that *-orderings from this class extend uniquely to the corresponding free skew fields. The problem was posed by T. Craven and T. Smith in [J. Algebra 238 (2001) 314–327].