Let $\mathcal{A}$ be a unital, commutative C*-algebra. The radical of $\mathcal{A}$ is defined as

$$\operatorname{Rad}(\mathcal{A}) = \bigcap \{\mathcal{I}\subset \mathcal{A}: \mathcal{I} \text{ is a maximal ideal}\}$$

How can one characterize $\operatorname{Rad}(\mathcal{A})$ using the Gelfand-Naimark theorem only, i.e. without referring to more evolved methods such as representation theory? 

EDIT: (Gelfand-Naimark theorem)

Let $\mathcal{A}$ be a unital commutative C*-algebra. Then the Gelfand transform $\Gamma: \mathcal{A} \rightarrow C(M(\mathcal{A})), \Gamma(A)(m):=m(A)$ is a *-isomorphism.


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