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

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

Published electronically:
August 16, 2006

MathSciNet review:
2302523

Full-text PDF

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.

**[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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
15A60,
16B99,
17A05,
17A15

Retrieve articles in all journals with MSC (2000): 15A60, 16B99, 17A05, 17A15

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.