Commutativity of Banach Algebras Characterized by Primitive Ideals and Spectra

Abstract

This study is an attempt to prove the following main results.

Let A be a Banach algebra and U = A ⊕ C be the unitization of A. By                        

we denote the set of all primitive ideals of U such as P so that the quotient

algebra  is commutative. In this article, we prove that if A is semi-prime

so that dim(  ⩽ 1, then A is commutative. As another result on

commutativity of Banach algebras, the following conclusion will be proved:

Let A be a semi-simple Banach algebra. Then, A is commutative if and only

if S(a) = {φ(a) : φ  ⋲ } ∪ {0} or S(a) = {φ(a) : φ  ⋲ } for every a ⋲ A,

where S(a) and  denote the spectrum of an element a ⋲ A, and the set of

all non-zero multiplicative linear functionals on A, respectively