Multihomogeneous Normed Algebras and Polynomial Identities
Leandro Cioletti, Jos\'e Ant\^onio Freitas, Dimas Jos\'e, Gon\c{c}alves

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 over or . 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 with the same polynomial identities as , 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 . As a byproduct we prove that if is a normed PI-algebra and its completion is nil, then is nilpotent. By introducing the concept of multihomogeneous norm we obtain as an application of our main results that if is multihomogeneus normed algebra and is a PI-algebra such that the…
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Taxonomy
TopicsMatrix Theory and Algorithms · Polynomial and algebraic computation · Advanced Topics in Algebra
