Radii and subnorms on finite-dimensional power-associative algebras

Abstract

In the first part of this paper, we review some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Restricting attention to the case where is either or we briefly discuss r(a), the radius of an element a in which is the largest root in absolute value of the minimal polynomial of a. We then obtain a formula for the radius which is a variant of a well-known result in the context of complex Banach algebras. We prove that if f is a continuous subnorm on then for all

We observe that while this formula holds for certain well-behaved discontinuous subnorms, it fails for others. The paper concludes with a study of the relations between stable subnorms and the above defined radius. For example, we show that if a stable subnorm f is continuous on then f is radially dominant, i.e., f(a)≥ r(a) for all .

Keywords:

• Finite-dimensional power-associative algebras
• Minimal polynomial
• Radius of an element in a finite-dimensional power-associative algebra
• Norms
• Subnorms
• Stable subnorms
• Radial dominance

Keywords:

• 2000 AMS Subject Classifications: 15A60
• 16B99
• 17A05

Notes

¹ As usual, a subnorm f on a finite-dimensional algebra is said to be continuous if it is continuous with respect to the (unique) finite-dimensional norm-topology on .

¹ An algebra is called alternative if the subalgebra generated by any two elements in is associative; hence, an alternative algebra is power-associative.