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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Hölder absolute values are equivalent
to classical ones


Author: E. Muñoz Garcia
Journal: Proc. Amer. Math. Soc. 127 (1999), 1967-1971
MSC (1991): Primary 12J20; Secondary 12J10, 16W80, 13J99
Published electronically: March 16, 1999
MathSciNet review: 1487331
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Abstract | References | Similar Articles | Additional Information

Abstract: We study generalized absolute values on a field or a commutative ring with unit element satisfying an approximate triangle inequality and an approximate multiplicative property. We prove that they are always Hölder equivalent to an absolute value. This implies geometric rigidity results for Lipschitz and Hölder deformations of metric rings.


References [Enhancements On Off] (What's this?)

  • [Bou] Nicolas Bourbaki, Elements of mathematics. Commutative algebra, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass., 1972. Translated from the French. MR 0360549 (50 #12997)
  • [Os] A. Ostrowski, Über einige Lösungen der Funktionalgleichung $\varphi (x).\varphi (y)=\varphi (xy)$ Acta Mathematica, 41, 1917, p. 271-284.
  • [We] Dominic Welsh, Codes and cryptography, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1988. MR 959137 (89i:94001)

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Additional Information

E. Muñoz Garcia
Affiliation: Department of Mathematics, University of California, Los Angeles, California 90024
Email: munoz@math.ucla.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04758-9
PII: S 0002-9939(99)04758-9
Keywords: Valuations, Ostrowski's Theorem, H\"{o}lder deformations, metric spaces
Received by editor(s): June 25, 1997
Received by editor(s) in revised form: October 14, 1997
Published electronically: March 16, 1999
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society