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 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 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 , 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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Additional Information

**Moshe Goldberg**

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

Email:
goldberg@math.technion.ac.il

DOI:
https://doi.org/10.1090/S0002-9947-06-04296-6

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.