Commutativity of Banach Algebras Characterized by Primitive Ideals and Spectra

Amin Hosseini

This study is an attempt to prove the following main results. Let $\mathcal A$ be a Banach algebra and $\mathfrak A=\mathcal A\bigoplus\mathbb C$ be its unitization. By $\prod_c(\mathfrak{A})$, we denote the set of all primitive ideals $\mathcal P$ of $\mathfrak{A}$ such that the quotient algebra $\frac{\mathfrak{U}}{\mathcal{P}}$ is commutative. We prove that if $\mathcal A$ is semi-prime and $\dim(\cap_{\mathcal{P}\in\prod_c(\mathfrak{A})}\mathcal{P})\leq1$, then $\mathcal A$ is commutative. Moreover, we prove the following: Let $\mathcal A$ be a semi-simple Banach algebra. Then, $\mathcal A$ is commutative if and only if $\mathfrak S(a)=\{\varphi(a)\mid\varphi\in\Phi_{\mathcal A}\}\bigcup\{0\}$ or $\mathfrak S(a)=\{\varphi(a)\mid\varphi\in\Phi_{\mathcal A}\}$ for every $a\in\mathcal A$, where $\mathfrak S(a)$ and $\Phi_{\mathcal A}$ denote the spectrum of an element $a\in\mathcal A$, and the set of all non-zero multiplicative linear functionals on $\mathcal A$, respectively.