Multihomogeneous Normed Algebras and Polynomial Identities

الأصل · EN

In this paper we consider PI-algebras A 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 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 is multihomogeneus normed algebra and A is a PI-algebra such that the completion of the quotient space /Id(A) is nil, then A is nilpotent. Both applications are extensions of the study initiated in [4].