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

Full-text PDF Free Access

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]**John C. Baez,*The octonions*, Bull. Amer. Math. Soc. (N.S.)**39**(2002), no. 2, 145–205. MR**1886087**, 10.1090/S0273-0979-01-00934-X**[BM]**Garrett Birkhoff and Saunders MacLane,*A Survey of Modern Algebra*, Macmillan Company, New York, 1941. MR**0005093****[D]**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**, 10.2307/1967865**[G]**Moshe Goldberg,*Stable norms—from theory to applications and back*, Linear Algebra Appl.**404**(2005), 223–250. MR**2149661**, 10.1016/j.laa.2005.02.018**[GGL]**Moshe Goldberg, Robert Guralnick, and W. A. J. Luxemburg,*Stable subnorms II*, Linear Multilinear Algebra**51**(2003), no. 2, 209–219. MR**1976865**, 10.1080/0308108031000078920**[GL1]**Moshe Goldberg and W. A. J. Luxemburg,*Stable subnorms*, Linear Algebra Appl.**307**(2000), no. 1-3, 89–101. MR**1741918**, 10.1016/S0024-3795(00)00011-2**[GL2]**Moshe Goldberg and W. A. J. Luxemburg,*Discontinuous subnorms*, Linear and Multilinear Algebra**49**(2001), no. 1, 1–24. MR**1888109**, 10.1080/03081080108818683**[GL3]**Moshe Goldberg and W. A. J. Luxemburg,*Stable subnorms revisited*, Pacific J. Math.**215**(2004), no. 1, 15–27. MR**2060492**, 10.2140/pjm.2004.215.15**[HLP]**G. H. Hardy, J. E. Littlewood and G. Pólya,*Inequalities*, Cambridge Univ. Press, Cambridge, 1934.**[R]**Joseph J. Rotman,*Advanced modern algebra*, Prentice Hall, Inc., Upper Saddle River, NJ, 2002. MR**2043445**

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:
http://dx.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.