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Hölder absolute values are equivalent to classical ones

Author(s): E. Muñoz Garcia
Journal: Proc. Amer. Math. Soc. 127 (1999), 1967-1971.
MSC (1991): Primary 12J20; Secondary 12J10, 16W80, 13J99
Posted: March 16, 1999
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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:

[Bou]: N. Bourbaki, Commutative algebra Elements of Mathematics, chapter VI, section 6.1, Addison-Wesley, 1972. MR 50:12997
[Os]: A. Ostrowski, Über einige Lösungen der Funktionalgleichung $\varphi (x).\varphi (y)=\varphi (xy)$ Acta Mathematica, 41, 1917, p. 271-284.
[We]: D. Welsh, Codes and cryptography Oxford Science Publications, 1988. MR 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: 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
Posted: March 16, 1999
Communicated by: Wolmer V. Vasconcelos
Copyright of article: Copyright 1999, American Mathematical Society


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