Orderings and valuations on *-fields. (English) Zbl 0702.16007
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.
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.
Reviewer: P.M.Cohn
MSC:
16K40 | Infinite-dimensional and general division rings |
16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
06F25 | Ordered rings, algebras, modules |
16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
12J10 | Valued fields |
12J15 | Ordered fields |