Ordered ∗-Rings

Craven, Thomas C.; Smith, Tara L.

doi:10.1006/jabr.2000.8644

Cited by 12 publications

(17 citation statements)

References 7 publications

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“…Note that the natural epimorphisms φ n : R n+1 → R n are * -homomorphisms. The same proof as in [10,Lemma 2.2] shows that the involution on R n extends uniquely to an involution of S n = (R n ) U n . It is easy to verify that the natural epimorphisms φ n : S n+1 → S n are * -homomorphisms.…”

Section: Proposition 5 Every Involution On L Has a Canonical Extensimentioning

confidence: 73%

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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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Section: Proposition 5 Every Involution On L Has a Canonical Extensimentioning

confidence: 73%

“…See [1] for a similar notion. For domains, * -orderings were introduced by M. Marshall in [15], see also [10,12,16].…”

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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“…Can the results of this paper be extended to * -orderings? See [24,23,7]. (3) As noted by Ringel, [26], U + is an iterated skew polynomial ring so further analysis of Sper(U + ) is possible.…”

Section: Final Comments and Open Problemsmentioning

confidence: 99%

Real spectra of quantum groups

CIMPRI

2004

Journal of Algebra

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Abstract. The only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring R q [x, y], see [21]. The aim of this paper is to describe the real spectra of quantum affine rings k q [x 1 , . . . , x n ] where k is a formally real affine R-algebra and q ∈ M n (R + ). As a by-product we describe the real spectra of quantized enveloping algebra U q (sl 2 (R)) and quantum special linear group O q (SL 2 (R)). Formal reality and semireality is characterized for the following classes of quantum groups: quantum affine rings, quantized enveloping algebras, quantized function algebras, quantized Weyl algebras.

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“…Craven and Smith [17] have taken the first steps toward extending the work of Craven [14] to * -rings. They analyze how a given * -ordering P , as a subset of the symmetric elements, can extend to a multiplicatively closed ordering on a larger set of elements.…”

Section: Extending the Theory To Ringsmentioning

confidence: 99%