Hermitian Forms over Ordered ∗ -Fields

Craven, Thomas C.; Smith, Tara L.

doi:10.1006/jabr.1998.7774

Cited by 7 publications

(9 citation statements)

References 26 publications

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“…Thus, the involution * is a nonstandard involution of the first kind. Thus, (ii) is the Lemma 3.3 of Craven (1999). For the reader's convenience, a direct proof is given below.…”

Section: Structure Theorems For Division Ring With An Involutionmentioning

confidence: 95%

Geometry ofn × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Huang

2008

Communications in Algebra

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Let D be a division ring with an involution − and char D = 2, F = x ∈ D x =x . Let n D be the set of all n × n hermitian matrices over D. Two hermitian matrices H 1 and H 2 are said to be "adjacent" if rank H 1 − H 2 = 1. The fundamental theorem of geometry of hermitian matrices over D is proved: If n ≥ 3 and is a bijective map from n D to itself such that preserves the adjacency, then X = t PaX P + 0 ∀X ∈ n D , where a ∈ F * , P ∈ GL n D , and is an automorphism of D which satisfies x = a x a −1 for all x ∈ D. The application of the fundamental theorem to algebra and geometry is discussed. For example, every Jordan isomorphism or additive rank-1-preserving surjective map on n D is characterized.

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“…Thus, the involution * is a nonstandard involution of the first kind. Thus, (ii) is the Lemma 3.3 of Craven (1999). For the reader's convenience, a direct proof is given below.…”

Section: Structure Theorems For Division Ring With An Involutionmentioning

confidence: 95%

Geometry ofn × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Huang

2008

Communications in Algebra

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“…The notation of a * -ordering on a skew field was introduced by S.S. Holland in [11] and studied further by T.C. Craven in [3][4][5][6][7][8][9]. See [1] for a similar notion.…”

Section: Lemma 1 Let R Be a Domain And V A Valuation On R The Relatmentioning

confidence: 99%

Free skew fields have many ∗-orderings

Cimpric

2004

Journal of Algebra

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The notion of a * -ordering on a skew-field with involution was introduced by S. Holland in [J. Algebra 101 (1986) 16-46] as an analogue to the notion of a total ordering on a skew-field and developed further in a series of papers of T. Craven, I. Idris, M. Marshall and T. Smith. While it is well known that every free skew-field has a total ordering (see [J. Lewin, Trans. Amer. Math. Soc. 192 (1974) 339-346]), it has not been known so far whether every free skew-field with some natural involution has a * -ordering. The aim of this paper is to give an explicit construction of a class of * -orderings on free associative algebras and to prove that * -orderings from this class extend uniquely to the corresponding free skew fields. The problem was posed by T.

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“…As but this is not a group under symmetric difference as is normally the case, suggesting that there is no group in this situation to take the place of G in the space of semiorderings (Y, G). A solution to this is proposed in [13]. An analog to the Witt ring is defined by using the ring WS(D, *) in £{YQ, Z) generated by the image of the Witt group W(D, *).…”

Section: P R O O F : Assume a Fixed (Yg)mentioning

confidence: 99%

Abstract theory of semiorderings

Craven

Smith

2005

Bull. Austral. Math. Soc.

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Marshall's abstract theory of spaces of orderings is a powerful tool in the algebraic theory of quadratic forms. We develop an abstract theory for semiorderings, developing a notion of a space of semiorderings which is a prespace of orderings. It is shown how to construct all finitely generated spaces of semiorderings. The morphisms between such spaces are studied, generalising the extension of valuations for fields into this context. An important invariant for studying Witt rings is the covering number of a preordering. Covering numbers are defined for abstract preorderings and related to other invariants of the Witt ring.

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Hermitian Forms over Ordered ∗ -Fields

Cited by 7 publications

References 26 publications

Geometry ofn × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Geometry ofn × n(n ≥ 3) Hermitian Matrices Over Any Division Ring with an Involution and Its Applications

Free skew fields have many ∗-orderings

Abstract theory of semiorderings

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