Homomorfismos complexos em álgebras de Banach não comutativas

Abstract

In the present dissertation we will explore the relationship between the set of non-zero multiplicative linear functionals (the spectrum), and the set of maximal ideals in a unital Banach algebra. For a given commutative unital Banach algebra A, we will construct the basic foundation in order to demonstrate that there is a bijection between the spectrum and the set of all maximal ideals of A. Furthermore, we will present an example of a unital non-commutative Banach algebra, where such bijection does not exist - in fact we will prove that its spectrum is actually empty. On the other hand, we will show that the algebra of all upper triangular matrices UTm(A) (with entries in A) possesses the bijection property even though it is a non-commutative Banach algebra.