Minimal polynomials and radii of elements in finite-dimensional power-associative algebras
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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
- John C. Baez, The octonions, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 145–205. MR 1886087, DOI 10.1090/S0273-0979-01-00934-X
- Garrett Birkhoff and Saunders MacLane, A Survey of Modern Algebra, The Macmillan Company, New York, 1941. MR 0005093
- L. E. Dickson, On quaternions and their generalization and the history of the eight square theorem, Ann. of Math. (2) 20 (1919), no. 3, 155–171. MR 1502549, DOI 10.2307/1967865
- Moshe Goldberg, Stable norms—from theory to applications and back, Linear Algebra Appl. 404 (2005), 223–250. MR 2149661, DOI 10.1016/j.laa.2005.02.018
- Moshe Goldberg, Robert Guralnick, and W. A. J. Luxemburg, Stable subnorms II, Linear Multilinear Algebra 51 (2003), no. 2, 209–219. MR 1976865, DOI 10.1080/0308108031000078920
- Moshe Goldberg and W. A. J. Luxemburg, Stable subnorms, Linear Algebra Appl. 307 (2000), no. 1-3, 89–101. MR 1741918, DOI 10.1016/S0024-3795(00)00011-2
- Moshe Goldberg and W. A. J. Luxemburg, Discontinuous subnorms, Linear and Multilinear Algebra 49 (2001), no. 1, 1–24. MR 1888109, DOI 10.1080/03081080108818683
- Moshe Goldberg and W. A. J. Luxemburg, Stable subnorms revisited, Pacific J. Math. 215 (2004), no. 1, 15–27. MR 2060492, DOI 10.2140/pjm.2004.215.15
- G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
- Joseph J. Rotman, Advanced modern algebra, Prentice Hall, Inc., Upper Saddle River, NJ, 2002. MR 2043445
Additional Information
- Moshe Goldberg
- Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
- Email: goldberg@math.technion.ac.il
- Received by editor(s): December 18, 2005
- Received by editor(s) in revised form: April 17, 2006
- Published electronically: August 16, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 4055-4072
- MSC (2000): Primary 15A60, 16B99, 17A05, 17A15
- DOI: https://doi.org/10.1090/S0002-9947-06-04296-6
- MathSciNet review: 2302523