What does it mean to say that a set is radical in Banach Algebra

Question

Let $B$ be a commutative unital Banach algebra. Let $$R(B)=\{x\in B: r(x)=0\}$$ be a radical of $B$. Show that $R(B)$ is closed ideal of $B$.

Here I am confused with the term radical. What does it mean?

Please help.

Answer 1

If I had to guess, we should have $$R(B)=\{x\in B:\forall r\in\Omega(B),r(x)=0\},$$ where $\Omega(B)$ is the character space of $B$ (i.e. the set of all nonzero multiplicative functionals on $B$). In which case, in the notation I've seen, $R(B)$ is the radical of $B$, which is defined as the intersection of all maximal ideals of $B$.