Minimal polynomials and radii of elements in finitedimensional powerassociative algebras
Author:
Moshe Goldberg
Journal:
Trans. Amer. Math. Soc. 359 (2007), 40554072
MSC (2000):
Primary 15A60, 16B99, 17A05, 17A15
Published electronically:
August 16, 2006
MathSciNet review:
2302523
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finitedimensional powerassociative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have . In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on . In the third and last section, we deal with stability of subnorms acting on subsets of finitedimensional powerassociative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.
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 G. Birkhoff and S. Mac Lane, A Survey of Modern Algebra, Macmillan Publishing Co., New York, 1977. MR 0005093 (3:99h) (review of original 1941 edition)
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 M. Goldberg and W. A. J. Luxemburg, Discontinuous subnorms, Linear and Multilinear Algebra 49 (2001), 124. MR 1888109 (2003a:15026)
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Additional Information
Moshe Goldberg
Affiliation:
Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
Email:
goldberg@math.technion.ac.il
DOI:
http://dx.doi.org/10.1090/S0002994706042966
PII:
S 00029947(06)042966
Keywords:
Finitedimensional powerassociative algebras,
minimal polynomial,
radius of an element in a finitedimensional powerassociative algebra,
norms,
subnorms,
submoduli,
stable subnorms.
Received by editor(s):
December 18, 2005
Received by editor(s) in revised form:
April 17, 2006
Published electronically:
August 16, 2006
Article copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
