Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

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 $\mathcal {A}$ over an arbitrary field $\mathbb {F}$. Our main observation is that $p_a$, the minimal polynomial of $a\in \mathcal {A}$, may depend not only on $a$, but also on the underlying algebra. More precisely, if $\mathcal {A}$ is a subalgebra of $\mathcal {B}$, and if $q_a$ is the minimal polynomial of $a$ in $\mathcal {B}$, then $p_a$ may differ from $q_a$, in which case we have $q_a(t)=tp_a(t)$. In the second section we restrict attention to the case where $\mathbb {F}$ is either the real or the complex numbers, and define $r(a)$, the radius of an element $a$ in $\mathcal {A}$, to be the largest root in absolute value of the minimal polynomial of $a$. We show that $r$ possesses some of the familiar properties of the classical spectral radius. In particular, we prove that $r$ is a continuous function on $\mathcal {A}$. 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 $\mathcal {S}$, a subset of an algebra $\mathcal {A}$, satisfies certain assumptions, and $f$ is a continuous subnorm on $\mathcal {S}$, then $f$ is stable on $\mathcal {S}$ if and only if $f$ majorizes the radius $r$ defined above.