Definition 7.4.14 (Spectrum).label The space $\OM(A)$ is called the spectrum of A. Each member of $\OM(A)$ is called a character of A. The kernel $\scr{F}\inv(0)$ of the Gelfand representation $\scr{F}$ is called the radical of A. If $\scr{F}\inv(0)=\curl{0}$ then $A$ is said to be semisimple. In other words, an abelian semisimple Banach algebra $A$ is isomorphic to a subalgebra of the abelian $C^{*}$-algebra $C_{\infty}(\OM)$ of all continuous functions on a locally compact space $\OM$ vanishing at infinity.