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
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.