*-valuations and ordered *-fields

Holland, Samuel S.

doi:10.1090/S0002-9947-1980-0583853-8

$^{\ast }$-valuations and ordered $^{\ast }$-fields
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Trans. Amer. Math. Soc. 262 (1980), 219-243 Request permission

Erratum: Trans. Amer. Math. Soc. 267 (1981), 333.

Abstract:

We generalize elementary valuation theory to *-fields (division rings with involution), apply the generalized theory to the task of ordering *-fields, and give some applications to Hermitian forms.

References

A. A. Albert, On involutorial associative division algebras, Scripta Math. 26 (1963), 309–316 (1963). MR 179202
Ichiro Amemiya and Huzihiro Araki, A remark on Piron’s paper, Publ. Res. Inst. Math. Sci. Ser. A 2 (1966/1967), 423–427. MR 0213266, DOI 10.2977/prims/1195195769
Reinhold Baer, Linear algebra and projective geometry, Academic Press, Inc., New York, N.Y., 1952. MR 0052795
P. M. Cohn, Free rings and their relations, London Mathematical Society Monographs, No. 2, Academic Press, London-New York, 1971. MR 0371938
Herbert Gross, Quadratic forms in infinite-dimensional vector spaces, Progress in Mathematics, vol. 1, Birkhäuser, Boston, Mass., 1979. MR 537283
Herbert Gross and Hans A. Keller, On the definition of Hilbert space, Manuscripta Math. 23 (1977/78), no. 1, 67–90. MR 482101, DOI 10.1007/BF01168586
Samuel S. Holland Jr., Orderings and square roots in *-fields, J. Algebra 46 (1977), no. 1, 207–219. MR 447200, DOI 10.1016/0021-8693(77)90402-1
Hans Arwed Keller, Ein nicht-klassischer Hilbertscher Raum, Math. Z. 172 (1980), no. 1, 41–49 (German). MR 576294, DOI 10.1007/BF01182777

Allgemeine Bewertungstheorie

167

G. B. Mathews, Theory of numbers, Chelsea Publishing Co., New York, 1961. 2nd ed. MR 0126402
Karl Mathiak, Bewertungen nicht kommutativer Körper, J. Algebra 48 (1977), no. 2, 217–235 (German). MR 485810, DOI 10.1016/0021-8693(77)90304-0
R. P. Morash, Orthomodularity and non-standard constructions, Glasnik Mat. Ser. III 10(30) (1975), no. 2, 231–239 (English, with Serbo-Croatian summary). MR 398929
A. Prestel, Quadratische Semi-Ordnungen und quadratische Formen, Math. Z. 133 (1973), 319–342 (German). MR 337913, DOI 10.1007/BF01177872
A. Prestel, Euklidische Geometrie ohne das Axiom von Pasch, Abh. Math. Sem. Univ. Hamburg 41 (1974), 82–109 (German). MR 343155, DOI 10.1007/BF02993503
F. Radó, Non-injective collineations on some sets in Desarguesian projective planes and extension of non-commutative valuations, Aequationes Math. 4 (1970), 307–321. MR 270265, DOI 10.1007/BF01844161

Simultaneous linear equations over a division algebra

28

W. John Wilbur, On characterizing the standard quantum logics, Trans. Amer. Math. Soc. 233 (1977), 265–282. MR 468710, DOI 10.1090/S0002-9947-1977-0468710-X