Proposition 8

Let A be a commutative Banach algebra. The following four sets are equal:

(i) The kernel of the Gelfand transform;

(ii) The set of elements $x$ of A such that Sp$'_A(x) =\{0\}$;

(iii) The set of quasi-nilpotent elements of A;

(iv) The radical of A.

Let us denote these sets by $N_1,N_2,N_3,N_4$, respectively. One has $N_1=$ $N_2$ (Prop. 6, a)), and $N_2= N_3($I, p. 28, Cor. 5). By definition, the set $N_4$ is the intersection of the regular maximal ideals of A; it is therefore the intersection of the kernels of the characters of A (th. 2 of I, p. 30), which is equal to $N_1$.