Valuation theory of higher level *-signatures

Around 1980 two generalizations of the theory of linearly ordered fields appeared in the literature: Beckerʹs theory of orderings of higher level on fields (J. Reine Angew. Math. 307/308 (1979)8) and Hollandʹs theory of *-orderings on skew-fields with involution (J. Algebra 101 (1) (1986) 16–46). The aim of this paper is to unify both theories. In Section 1 we define (higher level) *-signatures on domains with involution which correspond to higher level preorderings in Beckerʹs theory. The subclasses of 2-cyclic and cyclic *-signatures correspond to complete preorderings and orderings respectively. We prove a necessary and sufficient condition for extendability of *-signatures from Ore domains to skew-fields of fractions. In Section 2 we define the set of bounded elements of a *-signature on a skew field with involution. If the skew field contains a central element i such that i2=−1 and i*=−i and the *-signature is 2-cyclic then the set of bounded elements is an invariant valuation ring. An example shows that the assumption on i cannot be omitted. In Section 3 we define extended *-signatures and prove that every 2-cyclic *-signature on a skew field D with iset membership, variantZ(D) is a restriction of some extended *-signature. In Section 4 we define extended *-preorderings as positive cones of extended *-signatures. We show that every *-preordering which is a restriction of an extended *-preordering is equal to the intersection of all *-orderings containing it. The assumption iset membership, variantZ(D) is not required. Section 5 presents auxilliary material for the proof of the weak isotropy principle for higher level *-signatures which is given in Section 6.