Exercise 7, § 8

Let $A_1$ (resp. $A_2$) be the unital Banach algebra considered in Exercise 5 (resp. 6). Let $A = A_1\times A_2$. Now prove that A is without radical, but that the image of the Gelfand transform of A is neither closed nor dense in $\mathscr{C}(\mathsf{X}(A))$.