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Transactions of the American Mathematical Society

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Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

Author: Moshe Goldberg
Journal: Trans. Amer. Math. Soc. 359 (2007), 4055-4072
MSC (2000): Primary 15A60, 16B99, 17A05, 17A15
Published electronically: August 16, 2006
MathSciNet review: 2302523
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Abstract | References | Similar Articles | 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 finite-dimensional power-associative algebra $ \mathcal{A}$ over an arbitrary field $ \mathbb{F}$. Our main observation is that $ p_a$, the minimal polynomial of $ a\in\mathcal{A}$, may depend not only on $ a$, but also on the underlying algebra. More precisely, if $ \mathcal{A}$ is a subalgebra of $ \mathcal{B}$, and if $ q_a$ is the minimal polynomial of $ a$ in $ \mathcal{B}$, then $ p_a$ may differ from $ q_a$, in which case we have $ q_a(t)=tp_a(t)$.

In the second section we restrict attention to the case where $ \mathbb{F}$ is either the real or the complex numbers, and define $ r(a)$, the radius of an element $ a$ in $ \mathcal{A}$, to be the largest root in absolute value of the minimal polynomial of $ a$. We show that $ r$ possesses some of the familiar properties of the classical spectral radius. In particular, we prove that $ r$ is a continuous function on $ \mathcal{A}$.

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative 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 $ \mathcal{S}$, a subset of an algebra $ \mathcal{A}$, satisfies certain assumptions, and $ f$ is a continuous subnorm on $ \mathcal{S}$, then $ f$ is stable on $ \mathcal{S}$ if and only if $ f$ majorizes the radius $ r$ defined above.

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

  • [B] J. C. Baez, The octonions, Bull. Amer. Math. Soc. (N. S.) 39 (2002), 145-205. MR 1886087 (2003f:17003)
  • [BM] 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)
  • [D] L. E. Dickson, On quaternions and their generalization and the history of the Eight Square Theorem, Ann. of Math. 20 (1918-1919), 155-171. MR 1502549
  • [G] M. Goldberg, Stable norms--from theory to applications and back, Linear Algebra Appl. 404 (2005), 223-250. MR 2149661 (2006d:15045)
  • [GGL] M. Goldberg, R. Guralnick and W. A. J. Luxemburg, Stable subnorms II, Linear and Multilinear Algebra 51 (2003), 209-219. MR 1976865 (2004c:17001)
  • [GL1] M. Goldberg and W. A. J. Luxemburg, Stable subnorms, Linear Algebra Appl. 307 (2000), 89-101. MR 1741918 (2001m:15065)
  • [GL2] M. Goldberg and W. A. J. Luxemburg, Discontinuous subnorms, Linear and Multilinear Algebra 49 (2001), 1-24. MR 1888109 (2003a:15026)
  • [GL3] M. Goldberg and W. A. J. Luxemburg, Stable norms, Pac. J. Math. 215 (2004), 15-27. MR 2060492 (2005e:17001)
  • [HLP] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
  • [R] J. J. Rotman, Advanced Modern Algebra, Pearson Education, Upper Saddle River, New Jersey, 2002. MR 2043445 (2005b:00002)

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Additional Information

Moshe Goldberg
Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel

Keywords: Finite-dimensional power-associative algebras, minimal polynomial, radius of an element in a finite-dimensional power-associative 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.

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