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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: 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.

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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.