Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto?
Context
Page 96 of Kaniuth's "Course on Banach Algebras" gives an example:
Lemma 2.7.10. Let $G$ be a locally compact Abelian group, and suppose that the Gelfand homomorphism $\Gamma : f \to f$ from $L^1(G)$ into $C_0(G)$ is surjective. Then $G$ has to be discrete.
Other suggestions?