Álgebras de Banach zLpd

Abstract

The main purpose of this work is the study of the zero Lie product determined algebras, a concept which is divided into two parallel perspectives: a strictly algebraic perspective and an analytical one. In the algebraic context, our main focus will be to show an example of triangular matrix algebra that is not zLpd. On the way towards this end, we will present the following fact: for n ≥ 2, the matrix algebra Mn(B) is a zLpd algebra, as long as B is a zLpd algebra with unity. For the analytical theory, we will focus on the study of Banach algebras and we will see how the concept of a zLpd Banach algebra relates with classical results from Functional Analysis such as the Hahn-Banach Theorem and the Closed Graph Theorem. The other purpose is to demonstrate that the class of zLpd Banach algebras is quite large, that is, we will present the following result: every C ∗ -algebra is a zLpd Banach algebra. An application of the concept discussed in this work, in the analytical context, will also be presented, involving the concept of commuting linear maps.