Multihomogeneous Normed Algebras and Polynomial Identities

TL;DR

This paper demonstrates that any PI-algebra over real or complex numbers can be represented as a Banach algebra with the same polynomial identities, and explores conditions under which such algebras are nilpotent.

Contribution

It proves the existence of Banach PI-algebras sharing polynomial identities with given PI-algebras and introduces multihomogeneous norms for analyzing nilpotency.

Findings

Existence of Banach algebra with same polynomial identities as any PI-algebra

If a normed PI-algebra's completion is nil, then it is nilpotent

Introduction of multihomogeneous norms for algebra analysis

Abstract

In this paper we consider PI-algebras $A$ over $\R$ or $\C$. It is well known that in general such algebras are not normed algebras. In fact, there is a nilpontent commutative algebra which is not a normed algebra, see [1]. Here we address the question of whether it is possible to find a normed PI-algebra $B$ with the same polynomial identities as $A$, and moreover, whether there is some Banach PI-algebra with this property. Our main theorem provides an affirmative answer for this question and moreover we also show the existence of a Banach Algebra with the same polynomial identities as $A$. As a byproduct we prove that if $A$ is a normed PI-algebra and its completion is nil, then $A$ is nilpotent. By introducing the concept of multihomogeneous norm we obtain as an application of our main results that if $\FX$ is multihomogeneus normed algebra and $A$ is a PI-algebra such that the completion of the quotient space $\FX/Id(A)$ is nil, then $A$ is nilpotent. Both applications are extensions of the study initiated in [4].