Orderings and valuations on *-fields
Let D be a skew field with an involution * and write \(S(D)=\{x\in D|\) \(x^*=x\}\). Further, let v be a valuation on D such that \(v(x^*)=v(x)\) for all \(x\in D\); then the value group \(\Gamma\) is abelian. An element \(s\neq 0\) in D is called smooth if \(s\in S(D)\) and there exists an automorphism \(\alpha_ s\) of \(D_ v\), the residue class field of D such that \(\overline{sx^*s^{-1}}=\alpha_ s[(\alpha_ s^{-1}(\bar x))^*]\) for any \(x\in D\) with \(v(x)=0\), where bars indicate residue classes. If each class \(v^{-1}(\gamma)\), \(\gamma\in \Gamma\) which contains a symmetric element, contains a smooth element, then v is called smooth, and v is strongly smooth if further, \(d+d^*\) is smooth whenever d is a product of smooth elements. The author shows that v is strongly smooth whenever \(v(d-1)>0\) for any commutator d of symmetric elements; in this case the automorphism \(\alpha_ s\) can be taken to be 1. The author studies different types of ordering on D compatible with the valuation v, i.e. such that v(a)\(\geq v(b)\), whenever \(0<a\leq b.\) This ensures that an ordering is induced on the residue class field, which is therefore of characteristic 0. By a Baer ordering of D, the author understands a total ordering of S(D) such that \(1>0\), \(x+y>0\) whenever \(x,y>0\) and for any \(x>0\) and \(d\in D^*\), \(dxd^*>0\). If also \(xy+yx>0\) for any \(x,y>0\), we have a Jordan ordering. The author is concerned with lifting orderings of the residue class field. By a semisection he understands a right inverse of the valuation on \(S(\Gamma)=v(S(D))\). Given a smooth valuation v an \(a^*\)-field D with a smooth semi-section he finds a natural bijection between the Baer orderings on D compatible with v and a certain class of mappings from S(\(\Gamma\))/2\(\Gamma\) to \(Y(D_ v)\times \{\pm 1\}\) where \(Y(D_ v)\) is the set of Baer orderings of \(D_ v\). Similar but more complicated conditions are given for the lifting of Jordan orderings, and the results are illustrated by some examples.
scientific article; zbMATH DE number 4150427