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Free skew fields have many *-orderings. (English) Zbl 1078.16013
This paper is concerned with skew fields with an involution *, possessing *-compatible orderings, briefly *-orderings. For any integral domain \(R\) with an involution *, a *-ordering is an ordering for which the set \(P\) of positive elements satisfies: (i) if \(a,b\in P\), then \(ab+ba\in P\), (ii) \(rPr^*\subseteq P\) for any \(r\in R\), (iii) \(P\cup-P\) is the set \(S(R)\) of all symmetric elements [cf. S. S. Holland jun., J. Algebra 101, 16-46 (1986; Zbl 0624.06024)]. – A valuation \(v\) on \(R\) with Abelian (additive) value group \(\Gamma\) is called quasi-commutative if \(v(xy-yx)>v(x)+v(y)\) for all \(x,y\in R\) [cf. P. M. Cohn, Skew Fields (Cambridge Univ. Press) (1995; Zbl 0840.16001)]. For any valuation \(v\) on \(R\) an equivalence on \(R\) can be defined by putting \(x\sim y\) if and only if \(v(x)<v(x-y)\). Thus \(v\) is quasi-commutative precisely if all multiplicative commutators are equivalent to \(1\). If \(R\) has an involution * such that \(v(x^*)=v(x)\) for all \(x\in R\), \(v\) is called a *-valuation. Now let \(K\) be a skew field generated by a subring \(R\) and let \(v\) be a \(\mathbb{Z}\)-valued quasi-commutative valuation on \(K\). Then for every \(x\in R^*\) there exist \(a,b\in R^*\) such that \(x\sim ab^{-1}\). This result is used to prove the following
Theorem. Let \(K\) be a skew field with involution * and a quasi-commutative *-valuation with values in \(\mathbb{Z}\cup\{\infty\}\). If \(B\) is a *-subring generating \(K\), then every *-ordering on \(B\) compatible with \(v|R\) extends uniquely to a *-ordering on \(K\) compatible with \(v\).
The result is applied to any complex Lie algebra \(L\), its associative envelope \(U(L)\) and skew field of fractions \(D(L)\) [cf. A. I. Lichtman, J. Algebra 177, No. 3, 870-898 (1995; Zbl 0837.16019), also Cohn, loc. cit., Section 2.6]. This leads to a construction of a class of *-orderings on free associative algebras, each of which extends uniquely to the corresponding skew field, thus answering a question raised by T. C. Craven and T. L. Smith [in J. Algebra 238, No. 1, 314-327 (2001; Zbl 0994.16029)].

MSC:
16K40 Infinite-dimensional and general division rings
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16W80 Topological and ordered rings and modules
12E15 Skew fields, division rings
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