##### Department of Mathematics,

University of California San Diego

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### Algebra Seminar

## Moshe Goldberg

#### Deapartment of Mathematics \\ Technion \\ Institute of Technology

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

##### Abstract:

We begin by revisiting the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra $\cal A$ over an arbitrary field $\mathbb{F}$. Our main observation is that $p_a$, the minimal polynomial of $a\in\cal A$, may depend not only on $a$, but also on the underlying algebra. Restricting attention to the case where $\mathbb{F}$ is either $\mathbb{R}$ or $\mathbb{C}$, we proceed to define $r(a)$, the {\it radius} of an element $a$ in $\cal A$, to be the largest root in absolute value of the minimal polynomial of $a$. As it is, $r$ possesses some of the familiar properties of the classical spectral radius. In particular, $r$ is a continuous function on $\cal A$. In the third part of the talk we discuss 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 about the radius $r$. Our main new result states that if $\cal S$, a subset of an algebra $\cal A$, satisfies certain assumptions, and $f$ is a continuous subnorm on $\cal S$, then $f$ is stable on $\cal S$ if and only if $f$ majorizes the radius \nolinebreak$r$.

Host: Efim Zelmanov

### August 8, 2006

### 4:00 PM

### AP&M 7321

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