Maximal ideals of commutative unital C* algebra using Gelfand-Naimark theorem

Question

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.

Think of kernel of Gelfand transform. If you still have no idea see this answer (theorem 4.2) — Norbert, Jul 01 '13 at 18:13

score 3 · Answer 1 · answered Jul 01 '13 at 13:04

3

Hint: For any space $X$ and any $x\in X$, the set of functions vanishing at $x$ is a maximal ideal in $C(X)$.

answered Jul 01 '13 at 13:04

tomasz

37,896

Maximal ideals of commutative unital C* algebra using Gelfand-Naimark theorem

1 Answers1