Jacobson radical and semisimple commutative Banach algebra

Let $A$ be a commutative unital complex algebra (Character and maximal ideal space for the algebra convention and Prime ideals and maximal ideals in a commutative ring for ideals). The Jacobson radical of $A$ is

$\mathrm{rad}(A):=\bigcap \{M:M\text{ is a maximal ideal of }A\},$

the intersection of all maximal ideals of $A$; when $A$ has no maximal ideals the intersection is over the empty family and the radical is $A$ by the convention that an empty intersection is the whole ring. The algebra $A$ is called semisimple when

$\mathrm{rad}(A)=\{0\}\mathrm{.}$

The radical is an ideal: it is the intersection of a family of ideals, hence closed under addition and under multiplication by arbitrary elements of $A$. The definition is stated for commutative unital complex algebras only, which is the class for which the radical is used in this page; it is not the general noncommutative Jacobson radical, and no noncommutative radical characterization is invoked anywhere in this library's Gelfand theory.

• Two equivalent readings for Banach algebras under Choice. Assuming the Axiom of Choice (The Axiom of Choice), for a commutative unital Banach algebra the intersection of all maximal ideals is the same as the intersection of the kernels of all characters, because the maximal ideals are exactly the character kernels (Maximal ideals and characters of a commutative Banach algebra); this equality is used in Kernel of the Gelfand transform is the radical, not assumed here.
• Semisimplicity is an algebraic condition. It says that the maximal ideals separate points of $A$ in the weak sense of having trivial intersection; it does not by itself say anything about the norm, and it is compatible with $\mathrm{\Gamma }$ failing to be isometric.