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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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